I. INTRODUCTION: Observations are taken in the laboratory and from these observations certain conclusions are drawn. Since no observation or series of observations is absolutely accurate, it is often desirable to check the dependability of the conclusions by a study of the errors in the experiment.

Suppose that an experiment on the relation between the pressure and volume of a gas is performed in the laboratory and that the conclusion is the statement of the law that the volume is inversely proportional to the pressure. The experiment does not prove that the law is absolutely accurate but only that within certain limits, determined by the accuracy of the experiment, it has been found to be true. Small departures from the law will always be found and it should be possible to determine whether these departures indicate that the law is not exactly true or whether they are due to unavoidable experimental errors. Even if in this experiment no significant departures were found, observations with more refined apparatus might show conclusively that the law was only an approximation to the truth.

II. SIGNIFICANT FIGURES: In most experiments a detailed study of the probable error is not required. Usually it is sufficient to indicate roughly how accurate the result is. In elementary work all the sure figures and one (but only one) of the estimated figures are recorded so that in merely writing down an observation an estimate of its accuracy is indicated. In Table I are recorded five observations each of the length L, width W, and thickness T, of a block of wood. The first observation of T is 3.57cm. The first two figures are known but the third figure 7 is doubtful. Although the 7 is doubtful it does have significance. We feel reasonably sure that the correct value is between 5 and 9, say, and we have, therefore, three significant figures. The location of the decimal point has nothing to do with the number of significant figures. Whether written as 3.57cm, 35.7mm, or 0.0357m, this item has three significant figures. When a zero serves merely to locate the decimal point it is not a significant figure. However the zero in the third observation of W is the first doubtful figure and is significant. To omit this zero would be wrong, for that would indicate that the preceding 2 was doubtful.

Each value of T has three significant figures and since there is considerable variation in the third figures, three significant figures should be used in expressing the average. In other words, the second 5 in the average is doubtful and the average is known to three significant figures. However, in the observations of W the variations in the third figure are so small that the third figure 1 in the average can hardly be called doubtful and in the average we are justified in keeping four significant figures.

The product LWT is 313.9735020cm³. However, this does not correctly represent the volume. It indicates that all the figures are known except the final zero, and, of course, this is far from true. Since an error of 1%, say. in anyone of the factors will cause a 1% error in the result, the volume cannot be determined to any greater degree of accuracy than the least accurate of the factors. Although it is difficult to make hard and fast rules about significant figures we may say that, in general, in multiplication and division the result should have as many significant figures as the least accurate of the factors. In some cases the answer should have one more significant figure than the least accurate of the factors. For example, in the equation 9.8 x 1.28 = 12.5, if the answer is to be as accurate as the least accurate of the factors, it must have three significant figures although the least accurate factor has only two. An inspection of the equation should make clear why this is true. The rule must be supplemented by the judgment of the experimenter. The volume of the wooden block should, therefore, be recorded as 314cm³. Do not carry worthless figures through a series of computations only to discard them at the end. To save time keep only one or two doubtful figures throughout the computations. This will not affect the accuracy of the result. In addition and subtraction the case is entirely different. Suppose that a certain metal rod is 126.73cm long at 20°C and that experiment shows that when heated to 100°C the increase in length is 0.2138cm. The new length is 126.73cm + 0.2138cm = 126.94cm. Since the numbers to which the 3 and 8 are to be added are unknown (there is no reason to believe they are zeros) the sum is known to 2 decimal places only. From the above illustration it should be clear that when numbers are properly arranged in columns for a computation in addition or subtraction, if any digit in a column of digits is unreliable, the answer in that column is likewise unreliable.

III. CLASSIFICATIONS OF ERRORS: An error that tends to make an observation too high is called a positive error and one that makes it too low a negative error. Errors can be grouped in two general classes, systematic and random. A systematic error is one that always produces an error of the same sign, e. g., one that would tend to make all the observations of some one item too small, say. A random error is one in which positive and negative errors are equally probable. Systematic errors may be subdivided into three groups: instrumental, personal, and external. An instrumental error is an error caused by faulty or inaccurate apparatus. Personal errors are due to some peculiarity or bias of the observer. External errors are caused by external conditions (wind, temperature, humidity, vibration, etc.)

To understand better these various kinds of errors, let us study the errors in a particular experiment. In this experiment two observers, in an attempt to measure the velocity of sound, use a pistol and a 1/20sec stop watch to measure the time required for sound to travel between two points. The first observer fires the pistol and the second observer uses the stop watch to measure the time required for the sound to reach him. This simple experiment may be used to illustrate all the types of errors listed above. It should be possible to determine the sources of error. analyze these errors, and assign each to its proper classification. This is done in the paragraphs immediately following.

(A) Systematic Errors.

(1) Instrumental Error. Since the stop watch is the only instrument used in this experiment, all instrumental errors must come from faults in the watch. A stop watch that did not run at the proper rate would cause an instrumental error. If the watch ran too fast all the observed times would be too high. Readings taken by a stopwatch are also subject to another type of systematic error. It takes an appreciable time merely to start or stop the mechanism of a watch. If the lag at starting is not equal to the lag at stopping, error will result. To reduce instrumental errors the instrument should be checked with an accurate standard and the necessary corrections applied to the observations. If a high degree of accuracy is called for, the instrument may be sent to the National Bureau of Standards for calibration.

(2) Personal Error. There are several ways in which the bias of the observer or his particular method of taking data might produce systematic errors. Two of these will be considered. Owing to the difference in the way he reacts to the flash of the pistol and the report that he hears, an observer might tend always to get time readings which are too large, say. It also might be true that having been warned by the flash of the gun he will be more alert and react more quickly in stopping the watch than in starting it. Personal errors maybe minimized by taking the observations under various conditions and by using several observers working independently.

(3) External Errors. External errors are usually caused by conditions over which the observer has no control. Therefore they cannot be eliminated, but necessary corrections may be applied. In this experiment the error due to wind might be quite serious. To keep this error small the experimenters might follow any of the following procedures: choose a time when there is very little wind, measure the wind velocity and make the necessary corrections, or change positions with each other so that when the results are averaged the errors tend to cancel out.

(B) Random Errors. In this experiment 26 observations of the time required for sound to travel between two points are taken. These observed times are recorded in column II of Table II. Since a systematic error would affect each observation in the same way, the variations in this column are not due to systematic errors. We believe that each of these errors is due to a large number of factors each of which adds its own small contribution to the total error. Since these factors are unknown and variable it is assumed that the resulting error is a matter of chance and therefore positive and negative errors are equally probable. Such errors are called random errors; some authors prefer to call them accidental errors. Owing to the fact that random errors are subject to the laws of chance, their effect on the experiment maybe made quite small by taking a large number of observations. It should be clear why increasing the number of observations has no effect on systematic errors.

IV. PROBABLE ERROR: Since the variations in the observed times t (column II, Table II) are governed by chance, one may apply the laws of statistics to them and arrive at certain definite conclusions about the magnitude of the errors.

No attempt will be made to derive these statistical laws, but the ones that are pertinent to this discussion will be simply stated. Along the horizontal axis of Fig. 1 are plotted observed times and each dot represents one observation. For example, three of the twenty-six observations of time gave 1.60sec. It is clear from this figure that the data tend to cluster about a certain mean value. What value is the one having the highest probability of being correct? To answer this question the methods of statistics are used and although the proof is rather difficult the conclusion is quite simple. It indicates that the best average is obtained by dividing the sum of the t’s by the number of observations n. This is the simple method of averaging with which the reader is already familiar and an average obtained in this way is known as the arithmetic mean a.m. In other words, the arithmetic mean, obtained by dividing the sum of the observed values by the number of observations taken, represents the best value obtainable from a series of observations. The a.m. in this experiment is found to be 1.540sec and is represented by the vertical line in Fig. 1.

The difference between an observation and the arithmetic mean a.m. is called the deviation d and the average deviation a.d. is a measure of the accuracy of the experiment. Obviously, the average deviation is the sum of the deviations divided by the number of observations, a.d. = Σd/n. Deviations of the observations from the a.m. are recorded in column III, in which negative deviations are placed on the left side of the column and positive deviations on the right. Adding separately it is seen that the sum of the positive deviations is approximately equal to the sum of the negative deviations. Ideally they should be exactly equal.

To study the distribution and significance of the deviations, these deviations are plotted in Fig. 2 in much the same way that observations were plotted in Fig. 1. Each dot represents the deviation of one observation. Let us divide the observations into groups by the vertical lines a, b. c, etc., which divide the figure into slices each 1/10 sec wide with zero deviation at the midpoint of the central slice. In the figure the number of observations in each slice is represented by across (X) at the midpoint of the slice and the best smooth curve is drawn through these points. The curve therefore represents the relation between the magnitude of the deviations and the frequency with which they occur. From this graph the following general rules may be inferred:

(1) Positive and negative deviations are equally probable.
(2) Small deviations occur more frequently than large deviations.

Theory indicates that the relation between the probable error p.e. of a single observation, the sum of the deviations Σd (added without regard to sign) and the number n of observations is given by the equation

If this equation is applied to the data in Table II, we find that

This does not mean that no observation will deviate by more than 0.075 sec from the mean. It does mean that the chances are 50 to 50 that the error of a single observation will not exceed 0.075 sec; or. what amounts to the same thing, if a large number of observations are taken, half of these observations will have errors less than this amount. In Fig; 2 vertical lines P and P’ are drawn at d = -0.075 and d = +0.075. It can be seen that halt of the observations lie within these limits. Obviously, the probable error P.E. of the a.m. is less than the probable error p.e. of a single observation. P.E. may be computed by the equation

remembering that a.d. = Σd/n. which in this experiment gives

and one may write as the result of the twenty-six observations of time t that t = 1.540 ± 0.015. Again this does not mean that one can be sure that the correct value is between 1.525 sec and 1.555 sec but that the chances are even that it lies between these limits.

Comparing Eq. (1) with Eq. (3) it is found that the probable error of the mean of n observations is 1/√n times the probable error of a single observation. P .E. = p.e./√n. For example, the result obtained by averaging 9 observations is three times as reliable as a single observation and 81 observations are three times as reliable as 9. Since the accuracy increases as the square root of the number of observations taken, it is evident that an observer is not justified in spending the time required to take a very large number of observations. For most experiments 5 or 10 observations should be sufficient.

V. PROBABLE ERROR-ANOTHER METHOD: The probable error may also be computed from the sum of the squares of the deviations Σd². The squares of the deviations are given in column IV, Table II. This method is more tedious and only slightly more accurate than the method discussed in Section IV. The equations used in this method and the numerical results for this experiment are given below. where the symbols have the same significance they had in Eqs. (1). and (3).

It is seen that this method gives substantially the same probable error as the one used in Section IV.

It was stated in Section IV that the best average of a series of observations is the a.m. It can also be shown that the best average is the value which makes Σd² a minimum. For this reason the branch of mathematics which has been employed in the study of errors is often called The Method of Least Squares.

VI. PERCENTAGE ERROR: In a great many cases one is not so much interested in the numerical error as in the percent of error. For example, in Section IV it was found that the probable error of a single observation is 0.075 sec. It is often desirable to express the error in percent of the thing being measured. The probable percentage error p.p.e. of a single observation is (.075/1.54) 100% or 4.9%; the odds are even that an observation will not deviate more than 4.9% from the mean. Similarly, the probable percentage error P.P.E. of the mean is (.015/1.54) 100% or .97%.

VII. PROPACATION OF ERRORS: So far this discussion has been limited to a study of the errors in a group of observations all measuring the same thing, namely, the time t required for sound to travel between two points. Since in this experiment the observers are interested in the velocity of sound, it will be necessary to measure the distance S between the two points and compute the velocity v from the measured values of t and S. What effect do errors in t and S have upon v? Can general laws be formulated governing the effect of errors in the several items on the computed result? We shall consider only two cases.

Addition and Subtraction. The probable error of the result is the square root of the sum of the squares of the probable errors of the separate items. For example, (12.15 ± 0.03)cm + (8.63 ± 0.04)cm – (6.15 ± 0.05)cm = (14.63 ± .07)cm, since √ (0.03² + 0.04² + 0.05²) = 0.07

Multiplication and Division. When the result is obtained by multiplication and division its probable percentage error is determined by the application of the following two rules:

(1) The P.P.E. of the result is the square root of the sum of the squares of the P.P.E.’s of the factors.
(2) In case a factor is raised to the nth power its P.P.E. should be multiplied by n.

For example, let us assume that the density D of a certain cylinder is computed from the equation D=Mπr2h and that the P.P.E.’s of M, r, and h are 2%, 3%, and 5%, respectively. Then the P.P.E. of D is 8% since ‘√(2² + (2 x 3)² + 5² = 8. From this it is clear that a 3% error in the radius r will more seriously affect the result than a 5% error in the height h. Having determined the P.P.E. of the result, the P.E. may readily be computed.

VIII. PROBABLE ERROR WHEN SYSTEMATIC ERROR IS PRESENT: In the discussion of probable error above only random errors were considered. Systematic errors will always be present and in some cases will be sufficiently large to affect the reliability of the result. The probable value of the systematic error may be determined from a separate experiment or, with an experienced observer, may be estimated. Calling the probable error of a single observation due to random errors r, the probable systematic error r₁, the probable error of the result due to both causes r_o, and the number of observations n, it can be shown that

If r and r₁ cannot be expressed in the same units, probable percentage errors must be used.

Eq. (7) shows that if n is large it is the systematic error that determines the reliability of the result and a very large number of observations would not be justified. In many experiments r₁ is small and only random errors need be considered.

IX. GENERAL: In the study of errors we have applied certain statistical laws to 26 observations. Actually these laws apply accurately only when the number of observations is quite large. It is unusual for 26 observations to agree as well with theory as the ones given in Table II. To those interested in a more rigorous and more complete treatment of errors the following books are recommended:

H. M. Goodwin, Precision of Measurements and Graphical Methods, McGraw-Hill, 1920;

D. Brunt, The Combination of Observations, Cam-bridge University Press, 1931;

G. V. Wendell and W. L. Severinghaus, A Manual of Physical Measurements, Privately Printed, 1918.

J. W. Mellor, Higher Mathematics for Students of Chemistry and Physics, Longmans Green & Co., 1929.

From: Cenco Physics Selected Experiments in Physics (No. 71990-001), Copyright, 2003, Sargent-Welch Scientific Company.

Learning standards covered by this activity:

Major Understanding

• 5.1p The impulse* imparted to an object causes a change in its momentum*.
• 5.1r Momentum is conserved in a closed system. (Note: Testing will be limited to momentum in one dimension.)

The above learning standards were taken from the Core Curriculum Physical Setting/Physics, The University of the State of New York – The State Education Department.

Materials:

• a set of happy sad balls
• wooden mallet
• eye screw hook about ¾” in diameter
• glue gun
• wooden block about 2” x 4” x 12” in size
• ring stand
• lattice clamp
• rod

Procedure:

1. Cut the happy and sad ball in half.
2. Screw the eye screw hook into the top of the handle of the wooden mallet so that the eyehook is parallel to the head of the mallet.
3. Glue half of the happy ball on one end of the head of the mallet.
4. The half the sad ball on the other end of the head of the mallet.
5. Once the glue is dry, set up the apparatus as shown in the diagram below.
6. Pull back the wooden mallet to a certain angle and always use that angle. About 30 to 45 degrees works best, but you may need to play with yours a little bit to see what works best for your set up.
7. Set the wooden block standing up right in front of the wooden mallet, and adjust the distance from the wooden mallet to the block until the happy ball just barely is able to knock over the wooden block.
8. You may want to mark were you positioned your wooden block and your apparatus, to make sure nothing moves.
9. Now you are ready for the demo.
10. Show your students the happy ball hitting the wooden block and knocking the block over, then flip the mallet around so that the sad ball hits the block this time. Show the students that you are releasing the other end of the mallet from the same angle as the happy side.
11. This time the block will not tip over.
12. Have your students come up with some reasons why this happens, you can have them hold the mallet and knock it against the table, they will see that the mallet will bounce from the table on the happy side and will not on the sad side.

Explanation:
Making some simplifications we can use the following calculations to show students the general idea of why the happy ball knocks the block over and the sad ball does not.

Assume that the speed that happy ball hits the block is the same as the speed that the ball bounces off at.

Assume that the speed of the sad ball after it hits the block is zero

J = impulse
p_i = initial momentum of the wooden mallet
p_f = final momentum of the wooden mallet
m = mass of the wooden mallet with the happy sad balls attached
v = the velocity of the mallet just before it hits the wooden block

J = Δp = p_f−p_i

For the happy ball:

J = mv – m(-v)

J = 2mv

* Since momentum is a vector quantity, the initial and final velocities in this case are equal in magnitude but opposite in direction. Therefore the initial velocity has a negative value since it goes in the opposite direction of the final velocity of the mallet.

For the sad ball:

J = mv – m(0)

J = mv

Therefore the impulse of the happy ball is twice that of the impulse of the sad ball. Impulse is force times time, so if we can assume that the time the happy and sad ball are in contact with the wooden block is the same, then we can assume that the average force acting on the block from the happy ball is twice the force applied by the sad ball.

The block is knocked over because there is more force acting on it from the happy ball than from the sad ball.

It is a good discussion for students to analyze how valid these assumptions are, but the simplifications make the math easy for students to understand at a high school level.

Reinforcement Activities:
You can tie many real life examples to this idea of impulse and momentum. For example, if a student was going to get into a car crash and they had a choice of hitting a snow bank or the side of a building, which should they aim for? You can also talk about Karate Chopping a board. If your hand doesn’t make it through the board it hurts more than if you break the board, have the students explain why that is. These questions would be good questions to introduce or follow up the topic.

Have students study conservation of momentum using a ballistic pendulum. Ballistics is used to study the speed of bullets coming out of a gun. This is an excellent example of using Physics in real life. Use a ballistic pendulum to simulate ballistics without firing weapons in your classroom.

Related Products

• Happy and Sad Balls – These polymer balls may look the same but they behave differently, making them the ideal choice for demonstrating the coefficient of restitution.
• Lattice clamp – for lattice building and rod coupling.
• Aluminum plain end support rods – for constructing support lattices.
• Aluminum Rods, 1/2″ diameter, 24″ in length
• Ring Stand base
• CENCO Ballistic Pendulum – using our classic, time-tested physics apparatus, your students can: demonstrate the conservation of momentum, find the initial velocity of the ball, verify the determination of the initial velocity, find the time of flight of the ball, compute kinetic energy, and Investigate projectile motion.
• Basic Student Ballistic Pendulum – a classic physics lab with an impressive military history, the ballistics pendulum was invented in 1742 to measure the speed of bullets.

Learning standards covered by these activities:

Major Understanding
5.1q According to Newton’s Third Law, forces occur in action/reaction pairs. When one object exerts a force on a second, the second exerts a force on the first that is equal in magnitude and opposite in direction.

Alka-Seltzer Rocket Materials:

• black 35 mm film canisters
• Alka-Seltzer tablets
• Water
• A flat surface that can get messy

Procedure:

1. Break an Alka-Seltzer tablet into quarters
2. Put about 1 teaspoon of water into the bottom of the film canister
3. Drop the tablet in and quickly seal the top of the canister with the lid
4. Place the canister upside down on a flat surface and wait about 30 seconds
5. The rocket should launch a meter or two in the air with a loud popping noise

Explanation:
As the Alka-Seltzer mixes with the water it releases carbon-dioxide gas. The gas is confined to a closed space, so therefore builds up a lot of pressure. Eventually the pressure is so great that the top blows off forcing the canister upward. This is a good demonstration for Newton’s 3rd law. For every action, there is an equal and opposite reaction. The gas is pushing up on the top of the canister and the canister is pushing down on the gas.

Balloon Rocket Materials:

• Two chairs
• Tape
• String
• Balloons
• Soda Straws

Balloon Rocket Procedure:

1. Set up two chairs about five meters apart
2. Tie a string to one end of the chair
3. Blow up a balloon and hold the end closed, do not tie the end
4. Tape a straw along it’s vertical axis
5. Thread the string through the straw and tape the other end of the string to the second chair
6. Let go of the end of the balloon and watch the balloon race to the other end of the chair
7. You can add some more fun by having the students design their own balloon using construction paper and tape or change parameters such as the amount of air blown into the balloon or the size of the hole that the air is allowed to escape, to see which student get make the best (quickest) balloon rocket

Explanation:
Just like the Alka-Seltzer rockets, the walls of the balloon are pushing in on the gas and the gas is pushing out on the side of the balloon. All the forces balance until you let go of one end of the balloon, and then the escaping air pushes the balloon forward while the air gets pushed back by the wall of the balloon. It’s Newton’s Third Law again!

Reinforcement Activities:
Have students measure the height of a rocket, time of flight, or analyze if the rocket follows projectile motion. Talk about real world situations and how air resistance affects the flight of an object by having the students design their own rockets. Model rockets and water rockets have excellent applications beyond Newton’s third law and are a lot of fun for student and teacher alike.

Related Products

• Water Rocket and Water Rocket Launcher – Safely and inexpensively introduce students to model rockets. Shatterproof plastic rocket runs on only water and air.
• Alpha III E2X – An excellent beginner level model rocket.
• Launch Pro Altimeter - Accurately record the trajectory of your entire flight and download data directly into your computer of PDA. Designed for use with water powered or solid fuel rockets.
• Soda Bottle Physics Kit – Grab your students’ attention with four easy, safe and entertaining experiments: Hero’s Fountain, Tornado Tube, Water Rocket, and “Squidy” Cartesian Diver. Each supplied with all components and instructions.
• The Launch Pro Ultimate Bottle Rocket System – Unlike any bottle rocket system you’ve ever seen, the Launch Pro system offers a complete range of accessories for converting your used plastic soda bottles into powerful single and two-stage rockets.

Materials

• 5 sheets of tissue paper per student or group
• one glue stick per student
• markers for decorating (optional)
• dry fuel tablets
• 2′ tall stove pipe
• 3 hair dryers (alternate material to replace fuel tablets and stove pipe)
• scissors

Procedure

1. Have students pick out five sheets of tissue paper and a glue stick. It is best if students work in groups of two or three.
2. Have students lay the first sheet of tissue paper on a clean flat surface and apply a solid line of glue stick to one of the long edges of the tissue paper
3. Immediately have students place their next sheet of tissue paper on top of the line of glue so that the two sheets overlap about ¾”.
4. Continue this process until you have one large sheet made up of four sheets of tissue paper glued long side to long side as shown in the diagram below.
5. Then glue the short end of the fifth sheet to the top of first sheet, short side to short side as shown in the diagram below.
6. Glue edge one to edge two with about ¾” overlap so that the tissue paper forms a box shape that is open on both ends.
7. Cover the top edge of the box with the fifth sheet of the tissue paper, gluing heavily around the edge and covering over top of the box to form a seal with no leaks. The bottom of the hot air balloon should remain open.
8. Have student decorate their balloon if they wish, being careful not to rip the balloon. Even the smallest hole may cause the balloon not to fly.
9. Take the balloons outside and have the students hold the bottom edge of the balloon over the heat source. Have student be careful not to touch the heat source, or allow the tissue paper to touch the heat source as the tissue paper might catch on fire. Be ready to extinguish any fires by having water and an extinguisher ready.
   
10. After the inside of the balloon has heated up, have the students count down from three and all let go of the balloon. On a good day the balloons with fly three stories high before losing their air and coming back down.

SAFETY PRECAUTIONS
Make sure that students do not run after the balloon, especially younger children will be looking up and not in front of them and my trip or bash into other students doing the same.

Make sure that students do not try to have the balloons land over their heads, even when the balloon is falling, the air inside the balloon can be very warm

Explanation
As the air inside the balloon is replaced by the hotter air from the fuel tablet or hair dryer, the density of air inside the balloon becomes less dense then the air outside the balloon. Since warm air rises, the air attempts to rise, thus pushing the balloon upward until the air inside the balloon cools down.

This can also be looked at in terms of buoyancy. The buoyant force needs to be greater than the gravitational force of the balloon in order for the balloon to rise. The density of the balloon changes when the air in the balloon heats up. When the weight of air that the balloon displaces is equal in weight to the amount of gravitational force on the balloon and the air inside the balloon, the balloon will begin to float.

Reinforcement Activities
It is often difficult to get the balloons to launch correctly and it takes up class time to have the students build the balloons. There is a simpler, easier, more durable way to perform this experiment, by buying the pre-made kit listed below.

A similar and quite amazing activity is the Giant Solar Bag. This 50 foot bag when filled with air will magically rise off the ground if placed in a direct sunlight and allowed to heat up. This is a very impressive and inexpensive demonstration that illustrates the same concepts above.

Related Products

• Hot Air Balloon Kits (12′, 9′ and 6′ models available)
• Dry Fuel Tablets
• Giant Solar Bag
• 45M Thick Braided Cord for Solar Bag

Learning standards covered by this activity:

Major Understanding

• 4.3f Resonance occurs when energy is transferred to a system at its natural frequency.
• 4.3m When waves of a similar nature meet, the resulting interference may be explained using the principle of superposition. Standing waves are a special case of interference.

Process Skill

• 4.3 iii. identify nodes and antinodes in standing waves
• 4.3 vi Predict the superposition of two waves interfering constructively and destructively (indicating nodes, antinodes, and standing waves)

The above learning standards were taken from the Core Curriculum Physical Setting/Physics, The University of the State of New York, The State Education Department.

Wine Glass Resonance Materials

• wine glass, the thinner the rim of the glass the better
• water
• soap

Procedure

1. Wash your hands with soapy water for best effect.
2. Fill the glass part of the way with water
3. Dip the tip of your finger into the water then slowly and lightly rub the rim of the glass with your moist finger
4. After a few seconds the glass will start to sing
5. You can change the pitch of the note by adding or subtracting water

Singing Rod Materials

• Rosin (from the music room)
• Metal rod from your ring stands (or any other uniform metal rod will do)

Procedure

1. Crush a pea sized amount of rosin on a piece of paper until it is ground into power.
2. Cover the tips of your thumb, index and middle finger of your non-dominate hand with rosin as if you were going to use the rosin to make a finger print.
3. With your other hand grasp the metal rod with your thumb and index finger exactly in the middle of the rod.
4. Stroke one half of the rod with your rosin covered fingers. After four or five strokes a standing wave will build up as the other end of the rod begins to resonate.

Explanation
Every object has a natural frequency. If you drop a set of car keys it makes a recognizable sound, as does dropping a pin. The sound that it makes is the natural frequency of the keys and the pin. You can get the rim of the glass to vibrate at the natural frequency of the glass by rubbing the rim with your finger. Your fingertips cause the glass to vibrate at it’s natural frequency. If this happens over and over again a standing wave builds up and you can get a sound with a lot of volume.

If you cause an object to vibrate at its natural frequency by having another object near vibrating at the same frequency you can set up a standing wave in the first object. This process is called resonance. This is what is happening with the singing rod. Since you are holding the rod at its middle point, the natural frequency of both ends of the rod is the same. As you set up a standing wave with your hand on one end of the rod, there is a standing wave set up due to resonance on the other end of the rod. Be careful, you can get the rod to sing loud enough to be painful to some people’s ears.

Reinforcement Activities
Look on the internet for small video clips of the Tacoma Narrows Bridge Collapse or Galloping Gurdie. Both of these names reference a bridge that collapsed due to resonance. The wind blew across the bridge at the same frequency as the natural frequency of the bridge. The standing wave produced was a couple of meters high and eventually the bridge shook so violently it collapsed. A great video for getting your students thinking and talking about standing waves and resonance.

Myth Busters did an interesting video on resonance trying to shatter a wine glass with their voice. Other Myth Busters videos are available for purchase through Cenco Physics. (See the related products section below.)

Allow students to play with resonance on their own using the differential and sympathetic Tuning fork set. Have students set up a wave on one tuning fork and then match the same frequency with a second tuning fork. When the tuning forks are matched in natural frequency then second fork will ring without being struck.

Related Products

• Standing wave demonstrator – an economical way to create large standing waves.
• Spiral Spring – classroom size.
• String vibrator – Teach students about wavelength, modes of vibration and frequency with this inexpensive, easy-to-use apparatus.
• “Slinky” springs – Superb for illustrating both transverse and longitudinal waves.
• Differential and Sympathetic Tuning Fork Set – Investigate beat phenomena and resonance.
• Mechanical oscillator – Ideal for harmonic motion demonstrations.
• Resonating hoop – As the loop is excited through a range of frequencies, nodes and anitnodes are visible at resonance.
• Resonating reed – Resonance occurs on the six separate reed sections in a variety of modes: fundamental and overtones and at multiple frequencies.
• Wave spring – Demonstrate dramatic longitudinal or transverse waves with this Wave Spring accessory.
• Ripple tank apparatus – You can use this economy ripple tank with complete confidence in its reliability and effectiveness.
• Overhead projection ripple tank set – Your entire class can watch exciting wave experiments right from their seats.
• Dual string vibrator – Compare two different standing waves simultaneously and explain their relationship.
• Mythbusters Videos: Force and Motion | Circular Motion Video – Always intriguing, entertaining, and enlightening, Mythbusters are on a quest to uncover scientific truth.
• Heavy Duty Steel Support Rod – For use with most bases and rod support clamps.

OBJECT: To study the motion of a freely falling body; in particular, to measure g, the acceleration due to gravity.

METHOD: An object is allowed to fall freely, and its positions at the ends of successive equal intervals are recorded on a coated paper strip by means of electric sparks. From these data graphs of distance-time and velocity-time are plotted. The acceleration is determined from the slope of the velocity-time graph.

THEORY: The average speed v of a body is the quotient of the distance s which it traverses and the time t required to travel that distance. In symbols (equation 1):

v = s/t

The instantaneous speed v of an object is defined as the limit of this ratio as the time is made vanishingly small. Symbolically (equation 2):

v = Δs/Δt

where Δs represents a small increment of distance traversed in the corresponding increment of time Δt.

In Fig. 2 curve (a) shows the distance-time relationship for a freely falling body. In any such curve Eq. (2) states that the instantaneous speed is given by the slope of a tangent drawn to the curve at the point for the instant in question. If the speed were constant the slope would be constant and the curve would be a straight line. For a freely falling body this is evidently not true, as the speed, and hence the slope of the curve, is continually increasing.

When the velocity of a body varies, the motion is said to be accelerated. Acceleration is defined as the time rate of change of velocity; in symbols (equation 3):

a = (v_t−v_o)/t

where a represents the average acceleration of a body which changes its velocity from v_o to v_t in the time t. Since acceleration has the dimensions of a velocity divided by a time, the absolute unit in the metric system will be the centimeter per second per second and in the British system the foot per second per second; usually written, cm/sec² and ft/sec².

If a body moves in a straight line, making equal changes of velocity in equal intervals of time, its acceleration must be constant, and it is said to be moving with uniformly accelerated motion. This is the type of motion produced when a constant force acts upon a body which is free to move. The most common example of this is the motion of a freely falling body. This acceleration g is called the “acceleration due to gravity” and has a value of approximately 980cm/sec² or 32.2ft/sec².

The relationships between the three quantities velocity, distance, and time, in uniformly accelerated motion are readily deduced from the above definitions. Eq. (3) yields directly (equation 4):

v_t = v_o = at

which expresses the dependence of vt upon t in terms of the constants vo and a. It is the equation of a straight line, the slope of which is equal to the acceleration.

Since for uniformly accelerated motion the average velocity during an interval t is the arithmetical mean of the terminal velocities, in view of Eq. (1),

s = vt = ((v_t = v_o)/2) t

Substitution of v_t from (4) yields (equation 5):

s = v_{ot + 1/2at}²

When vo = 0, Eq. (5) shows that the distance-time curve is a parabola. The slope of the curve at any point (slope of the tangent) is the velocity at the corresponding instant.

A velocity-time curve for a freely falling body is plotted as curve (b) in Fig. 2. The time interval T is the interval between the sparks. A sample record is shown in Fig. 8. Since the graph is a straight line the velocity changes at a uniform rate.

The slope of this curve ΔvΔt is the acceleration. Since the slope is constant, the acceleration is constant. Hence the average velocity during the time interval is identical with the instantaneous velocity at the middle of that time interval.

In the present experiment the value of g will be determined from the slope of such a velocity-time curve, as plotted from the experimental data.

The principal points in the preceding discussion may be summarized as follows:

(a) The average speed of a body is obtained by dividing the distance which it traverses by the time required to travel that distance.

(b) The instantaneous velocity of an object is the limit approached by the ratio Δs/Δt as Δt approaches zero. This velocity is also equal to the slope of the tangent to the distance-time curve at the point in question.

(c) The acceleration of an object is the time rate of change of its velocity, or a=Δv/Δt. It is also the slope of the tangent to the velocity-time curve at the instant considered.

(d) For a constant acceleration, the velocity-time curve is a straight line and the average velocity of the body is also the actual (or instantaneous) velocity at the midpoint of the time interval used.

APPARATUS: The apparatus consists of two principal units: the fall apparatus and the timing device. As auxiliary apparatus a 6volt storage battery, a 20ohm rheostat, a switch, an impulse counter, a spark coil, a stopwatch or clock, a C-clamp, and a good-quality boxwood or steel metric scale are required.

The fall apparatus provides convenient means for holding the falling body suspended, for releasing it at will, for holding the record strip properly to receive the marks recorded during the fall, and for catching the falling body. The timing device is a unit which produces a series of intense sparks at equal time intervals. Its design permits of easy determination of the length of the interval between sparks.

The fall apparatus is designed so that the fall may be entirely unrestricted, save for air resistance. The marks which define the positions of the body are produced by equally-timed electric sparks which jump from a high-potential vertical wire to the falling body and thence through the record strip to a second vertical wire at ground potential. Fig. 3 shows a horizontal section of this part of the apparatus.

Fig. 4 shows a vertical section of the complete fall apparatus, arranged for operation. The falling body B is a steel cylinder and is shown falling, having been released by the electromagnet M. The latter is energized by current from a storage battery connected through a rheostat and switch. When it is desired to have the body fall, this current is interrupted. The apparatus is firmly secured to a vertical wall or supported from a substantial tripod base and carefully aligned so that the falling body, throughout its path, will remain uniformly distant between the rear wire W₁ and the front wire W₂, and finally fall accurately into the dashpot P.

The latter has about an inch of sand in its bottom, and its sides are heavily lined with felt. A prepared paper coated on one side with paraffin, constitutes the record strip. A roll of this paper is carried in a holder at F. When a record is to be made the end of the strip is pulled through the opening at G, thence upward over the wire W₁ and back through the opening at K. It is held smooth and taut against W₁ by a weighted clip C.

The secondary of the spark coil is connected to E.

The Timing Device or Spark Timer, Fig. 5, consists of an electrically maintained vibrating steel bar provided with electric contacts for making and breaking a circuit at equal intervals, the length of one interval being the full period of the bar.

Two sets of contacts are provided, one of which is used in maintaining the vibration by opening and closing the circuit through an electromagnet; the other set, independent of and insulated on one side from the first, opens and closes the primary of a spark coil for producing the timed sparks.

Electrical connections on the apparatus are as shown in Fig. 6. A 6volt storage battery is connected to the two center binding posts V. The primary of a spark coil is connected to the “spark coil” binding posts S, the secondary being connected to the wires in the fall apparatus. A spark gap, attached to one of the secondary terminals of the spark coil, may be adjusted to insure passage of sparks at peak voltage.

The make-and-break contacts on the spark coil should be carefully closed so that the breaker cannot vibrate. The spark timer should be rigidly clamped near its center to the table and its frame grounded. The Impulse Counter, used to measure the frequency of the vibrations, is connected to the binding posts I.

The Impulse Counter or Interval Timer, Fig. 7, counts the electrical impulses which produce the sparks. Each impulse causes a sweep hand to move one division on the dial, one complete revolution of the pointer representing 60 impulses. A small pointer records the number of whole revolutions of the sweep hand, counting up to 30 revolutions. A push-button key must be depressed to complete the circuit through the counter. By rotating the push-button it may be locked down in the operating position. A stopwatch or clock is used to measure the time of a suitable number of impulses registered on the counter and hence to determine the time interval between sparks.

PROCEDURE:
Experimental: It will be assumed that the fall apparatus has been properly aligned so that the falling body will remain equally distant from the two wires and will accurately strike the center of the pocket. This important and delicate adjustment should be made only under the personal supervision of the instructor.

Energize the electromagnet by closing the switch connecting the storage battery through the 20ohm rheostat to the binding posts on the electromagnet. With all the resistance cut out, suspend the body from the electromagnet.

Then, holding one hand just under the body, increase the resistance until it is released. Now again decrease the resistance slightly until the body will just hang from the electromagnet. When the body hangs motionless, open the switch and the body should fall directly into the pocket.

Use suitable precautions to prevent the body from becoming damaged by striking any object.

With the body not hanging draw the sensitized paper through the opening at the lower end of the casting, then upward and back through the upper opening. The light coated side of the paper must be on the outside. Attach the weighted clip to the end of the paper to hold it taut.

Make the necessary electrical connections to the timing device as indicated in Fig. 6. The vibrating bar and stationary contacts should be adjusted so as to be in alignment. The stationary contacts are adjusted so that when the bar is at rest there will be a gap from 0.25mm to 0.5mm between the contacts of each set. The contacts should be secured in this position by means of the lock nuts.

To increase the amplitude of vibration the electromagnet adjustment screw is turned so as to bring the electromagnet closer to the bar, and to reduce the amplitude it is moved away from the bar. A further reduction in amplitude, if necessary, may be made by increasing the gaps between the bar and the stationary contacts. Connect the high potential leads to the binding posts at the base of the fall apparatus, one through the series gap to the outer wire, the other to the frame of the apparatus. The frames of the spark timer and the free fall apparatus should be grounded.

For the greatest precision in spark timing, adjust the series spark gap so that the sparks occur at peak voltage. This may be attained by increasing the series spark gap to the point which will just allow the sparks to jump from the outer wire through the body to the inner wire while the body is falling. To effect this adjustment, when the body is not in falling position B Fig. 4, temporarily place a gap across the binding posts of the fall apparatus equal to the total clearance between the wires W1 and the body, and W2 and the body. The temporary gap may conveniently be made from a small length of copper wire fastened to one of the binding posts of the fall apparatus and bent toward the other until it is separated the proper distance. Increase the length of the series gap to the maximum length which will allow the sparks to jump consistently. This adjustment simulates the conditions which obtain at the time the body is falling. After removal of the temporary gap, experimental runs may be taken.

Suspend the body from the electromagnet and lightly touch the body until it hangs motionless. Start the spark timer. Observe that the sparks now jump from the outer wire through the body to the electromagnet and grounded support. When these conditions are realized, release the body by opening the switch in the electromagnet circuit. When the body has fallen, stop the spark timer and examine the record of the dots on the paper. If any of the spots are missing, shift the paper to one side and repeat. A slight decrease in the series spark gap may be necessary to produce consistent results. Remove the record from the apparatus by drawing it upward, which at the same time puts afresh strip in place. Repeat the process until each individual observer has one good trace.

The impulse counter should be left in the circuit while the spark timer is in operation, so that the time interval may be determined without any changes in the electrical circuit. Also, the contact and electromagnet adjustments on the timer should be left unchanged until the time interval has been determined. Measure with a stopwatch or clock the time corresponding to a suitably large number of impulses indicated on the counter. A time of at least one or two minutes should be used. By dividing this time, in seconds, by the number of impulses during the time, the time interval T between consecutive sparks is obtained.

Tabulation of Data: With a good-quality metric scale, measure the total distances S₁, S₂, S₃, etc., between the first clear spot and succeeding spots. To insure accuracy in these measurements, place the strip on a flat surface where there is good light. Place the meter stick edgewise on the trace in such a manner that the graduations are directly touching the dots. Leave the meter stick stationary and read the positions of the dots, estimating readings to fractions of a millimeter.

Subtract each position reading from the one immediately following it. This difference gives the distances s₁, s₂, s₃, etc., fallen during successive equal time intervals. By dividing these distances by the time interval the average velocity for that interval may be calculated. These values of average velocities are also the instantaneous velocities for the mid-point of the interval considered.

The data may be tabulated as in Table I.

Interpretation of Data:
Required Analysis: Plot a curve showing the relation of average velocity to time, using velocity as ordinates and time as abscissas, and plotting the points in the first quadrant. Locate each average velocity at the mid-point of the corresponding time interval, since for uniform acceleration the average velocity is identical with the actual velocity at the middle of the interval. Place the zero of abscissas (time intervals) somewhat to the right of the left-hand edge of the graph paper, since at “zero” time (the first spot) the body already had a small initial velocity. From the slope of this velocity-time curve determine the acceleration g of the falling body. Calculate the percentage difference between this value of g and the standard value.

On the same graph sheet plot a curve to show the total distance S fallen against the time. The times are simply the product of the ordinal numbers by the sparking interval T.

Thoroughly interpret the graphs in the report of the experiment. This interpretation should include conclusions to be drawn from the shapes of the curves, their slopes, and their intercepts. Careful explanations of the reasons for all conclusions should be given.

Optional Analyses:

1. Compute the value of g by applying the method of equal intervals to the last column of your table.
2. By taking corresponding values of distances and velocities for particular times from curves as in (a) and (b) of Fig. 2, plot a velocity-distance curve. Explain its shape.
3. Select some point on the distance-time curve and draw a tangent to the curve. From the slope of the tangent determine the velocity at that instant and compare it with the computed value.
4. Using the data for any two points on the record, compute the initial velocity by the use of Eq. (5). Compare this value with the initial velocity determined from the intercept of the velocity-time curve.

QUESTIONS:

1. If by some suitable mechanism the falling body had been given an initial downward push instead of being just released, would the resulting observed value of g have been different? Explain.
2. Classify the following as to whether they would introduce systematic or random errors in this experiment: (a) air friction, (b) estimations of fractional parts of millimeters on the scale, (c) zero error of stop watch, (d) time lag of observer in starting and stopping watch.
3. Neglecting friction, which of the following statements properly characterizes the motion of a heavy object thrown violently downward from a tall building: (a) uniform speed, (b) uniform deceleration, (c) constant acceleration, (d) uniformly increasing acceleration, or (e) a non-uniformly changing acceleration?
4. When the switch is opened the electromagnet does not instantly lose quite all of its magnetism. What effect does this have on your results?
5. What would be the appearance of the velocity-time curve if the falling body were so light that the effect of air friction could not be neglected?

Equipment List

• 6V Zinc-Carbon Batteries
• C Clamp
• Free Fall Apparatus, Digital Photogate
• Decade Resistance Box
• Ohaus CS Series Scale
• Knife Switches
• Digital Stopwatch
• Tabletop Acceleration Timer

From: Cenco Physics Selected Experiments in Physics (No. 71990-124), Copyright, 2003, Sargent-Welch Scientific Company.

Materials

• Battery (D or C cell)
• Insulated wire, about 18-24 gauge
• Small Magnet
• 2 Large Uncoated Paper Clips
• Masking tape
• Wire Cutters
• Sand paper

Procedure

1. Have students cut about one meter of insulated wire off the spool
2. Wrap the wire around the circle part of the battery at least ten times
3. Leave about 5 cm of wire on both ends of the coil
4. Take one five centimeter end of the wire and stick it through the center of all ten loops, then bring the end around to the outside and wrap all the loops with the end.
5. Do the same with the other end of the wire
   
6. Lay the coil flat on a table surface
7. Sand the insulation off of the half of wire facing you on the 5 cm ends that are sticking out
   
8. Tape a paper clip on each end of a battery and bend the top of the paper clip out so it can be bent into a holder for the metal coil
   
9. Place the magnet on the battery between the two paper clips that are sticking out
10. Place the coil on top of the two paper clips and give a little spin, if it doesn’t start rotating on it’s own, try spinning it the other way
11. Johnson motors need a little fine tuning, try adjusting the paper clips, sanding off the contact leads or making the coils around the motor more uniform
12. Be sure not to sand away too much insulation or your Johnson Motor won’t work.

Explanation
When the coil of wire is placed on the paper clips so that the uninsulated part of the wire is touching the metal of the paper clips, a complete circuit is formed. As the current through the wire changes it induces a magnetic field. This magnetic field repels the magnetic field of the permanent magnet attached to the battery. Therefore the coil turns to align itself properly with the magnetic field. The coil gets halfway through its rotation and then the insulated part of the coil is not in contact with the paper clips. Since there is no current, there is no magnetic field, so the coil’s inertia carries the coil the rest of the way through the rotation until the coil rotates enough so that there is current going through the coil again. The current causes a magnetic field that repels the permanent magnet and the process continues causing the coil to rotate very quickly.

Reinforcement Activities
The Ring Thrower is an excellent way to demonstrate that a changing electric field produces a changing magnetic field and vise versa. This device will throw a metal ring up to 12 feet in the air using principles of electromagnetism. It can also be used to demonstrate that current is indeed being produced around the ring launcher. Students can light a light bulb around the launcher without using a battery.

Don’t want to hunt down all the supplies needed for this kit yourself? The World’s Simplest Motor Kit comes with everything you need to make the motor described above except for a D cell battery.

Make many different types of electromagnetic motors, several different kits are listed below to suit your varying needs.

Related Products

• World’s Simplest Motor Kit
• DC Motor
• DC Motor Kit
• Demonstration Motor and Demonstrator
• Hand-Held DC Generator
• Genecon Hand Electric Generator
• Genecon Activity Set with Manual
• Ring Thrower

Materials

• A ramp about one meter long
• A can of cream of broccoli soup (98% fat free if able)
• A can of beef broth
• Stopwatch
• Meter Stick

Procedure

1. If you look in the store you can usually find a broth type soup and a cream type soup in the grocery store whose cans are the same size and whose mass is also the same. This works the best for this experiment.
2. Suggestion to have on the front board when walking in:Ladies and Gentlemen, you are about to witness a pivotal event. The annual Great Soup Can Race!Please allow me to introduce our contestants:The reigning champion, Beef Broth, weighing at a spectacular 12.5 fluid ounces has won the competition numerous times and is confident of demolishing the competition in our race today.

   Introducing also, Cream of Broccoli soup, she’s meaner and leaner. Now appearing 98% fat free and also weighing in at 12.5 fluid ounces. Cream of Broccoli has been training all year and is convinced that this is the year she will take home the championship. Who do you think will win?

   (Write who you think will reach the bottom of the ramp first if both soup cans are released at the same time and allowed to roll down the ramp on their sides. Be sure to include the reason you think the way you do.)

3. Have the students predict which can, if rolled down an incline, will reach the bottom of the hill first.
4. Have the students start both of the soup cans rolling at the same time.
5. A meter stick can be used to stop the cans from rolling originally and to release the cans at the same time.
6. Let the cans roll down the incline. Beef Broth will win every time.
7. This activity can be extended by having students time each can and calculate the amount of potential energy at the top of the hill compared to the kinetic energy at the bottom of the hill. The students will find in both cases that there is some energy lost, this energy is mostly rotational energy.

Explanation
Cream of Broccoli soup is more of a thick paste, so when this can is rolling down the ramp, the soup rotates along with the can as one solid object, like a solid cylinder. The Beef Broth is more of a liquid, so the can rolls, while the liquid stays oriented up and down, therefore the can rolls like a hollow cylinder. The rotational inertia of a solid cylinder is much greater than that of a hollow cylinder. Imagine trying to spin a 10 lb and a 15 lb bowling ball. It takes more energy to get the 15 lb bowling ball to rotate at the same rate as a 10 lb bowling ball.

Even though both soup cans have the same about of energy at the top of the ramp, the energy gets divided up differently as the soup cans roll down the ramp. The potential energy of the can at the top of the ramp gets converted into kinetic energy and rotational energy. Since it takes more energy to rotate the Cream of Broccoli soup, less energy is used for kinetic energy and the Cream of Broccoli soup rolls slower.

Reinforcement Activities
Have students feel the effects of angular momentum by using a gyroscope, once the gyroscope is spinning it will automatically right itself. Have the students hold the gyroscope in their hands and try to rotate the gyroscope to feel the force acting on it.

For an even more impressive demonstration, have the students use a bicycle wheel as a gyroscope. They can even use the gyroscope to spin themselves in a rotating chair. This shows the principle of conservation of angular momentum.

Have a contest of which student can create the slowest rolling spool down a ramp. Have students use what they learned about rotational inertia to their advantage and have fun doing it.

Related Products

• Simple Metal Gyroscope
• Gyroscope
• Gyroscope with Gimbal Cradle
• Bicycle Wheel Gyroscope
• Rotating Stool
• Rotating Platform
• Slow Rollers
• Inertia Wheel
• RotoDyne System
• Ball Bearing Rotating Support Kit

Materials

• Bottle of corn oil
• Two old Pyrex test tubes with the labels worn off of them
• Pyrex beaker
• Glass stirring rod that is NOT made out of Pyrex
• Two thick mailing envelopes
• Something to break glass with like a 500 g mass or rubber mallet
• Tongs or test tube holder

Procedure
There are several variations of this “magic” trick depending on what concept the students are or have learned:

1. Before the students enter the classroom take one of Pyrex test tubes and fill it with corn oil and place in a beaker, also filled with corn oil and the test tube will “magically” disappear
2. When the students enter the room you can tell them that scientist have discovered this magic liquid that will reverse entropy, so instead of increasing the disorder of thing, it increases the order
3. In front of the students take the other Pyrex test tube, put it in the mailing envelope, then put that envelope in the other envelope. (You should wear safety goggles and make sure you are not near the students when you do this next part) Smash the Pyrex test tube
4. Take the Pyrex shards and place it in the container of corn oil that already has the whole test tube in it
5. Say your favorite incantation while stirring the test tube with the glass stirring rod. (This is actually to help you locate the test tube and make it easier to pull it out.)
6. Then pull the whole test tube out of the beaker it will look like the container magically put itself back together
7. Of course don’t let the students actually believe that you can do magic, you’ll want to explain or have them help you explain what really happened

Another Variation

1. Fill the test tube with corn oil and have a beaker of corn oil ready to dip the test tube in
2. Tell the students that you have an acid that is so strong it will dissolve glass on contact
3. Put the test tube in the test tube holders and place the test tube in the corn oil, the part in the corn oil will appear to disappear
4. Your more observant students will notice that the beaker holding the corn oil is glass as well and will ask why the beaker doesn’t dissolve, now is a good time to admit you were fooling with them

Explanation
Corn oil and Pyrex have nearly the same index of refraction. We’ve all seen someone walk into a glass wall or door because the glass was so clean the person couldn’t tell it was there. The reason we can see glass beakers and test tubes is because they are curved and they bend the light that passes through them at different angles. If you look through the test tube everything on the other side will look really stretched out. The glass stirring rod and the corn oil have a different index of refraction, so the light gets bent going through the corn oil and then gets bent different going through the stirring rod. Since the corn oil and the Pyrex have the same index of refraction, the light only gets bent once since the light goes through the Pyrex at the same angle, we can’t see it.

Reinforcement Activities
Have students discover how the lens in their eye or a camera works. The ray optic demonstration set can be used to show refraction and reflection of light very clearly. It comes with plastic slides to show how the human eye works, explain what near sighted and far sighted is how glasses correct these conditions and well as show how a pinhole camera works.

Have students extend their knowledge of lenses using an optical bench. Students can find focal points of lenses, project images of candles or light bulbs and reinforce drawing ray diagrams by showing real life examples.

Related Products

• Ray Optics Demonstration Set
• Laser Ray Box
• CENCO Laser Ray Box
• Complete Optical Bench Set
• Advanced Low Profile Optical Bench
• Air Track Optics Bench
• Complete Meter-Stick Optical Bench

Materials

Materials are per group:

• Slinky springs
• Pieces of brightly colored yarn
• Meter sticks
• Stop watches
• Spiral spring
• Masking tape

Procedure

• Several different relationships and wave characteristics can be investigated using slinky type springs. The following are just a few of many options.

Investigating the relationship between speed and amplitude:

1. Have the students stretch the slinky spring about 5 feet between them so that their hands are touching the floor and the slinky is laying sideways stretched out on the floor.
2. Have the students mark the floor with masking tape where the ends of the spring are
3. Have the students practice making pulses with different amplitudes from one length of the spring to the other
4. From the center of the masking tape on the floor, mark 10 centimeter intervals going one direction perpendicular to the length of the spring
5. Have students measure the amount of time it takes for the pulse to travel down the slinky and back making a 10 cm pulse
6. Repeat the last step three times to get three results to average
7. Then increase the amplitude to 20 cm and time three trials
8. Continue to increase the intervals until 60 centimeters
9. There should be no relationship between amplitude and speed

Investigating the relationship between tension of the spring and speed:

1. WARNING: it is easy to exceed the elastic limit of the slinky will performing this experiment make sure the students are aware of this and discuss that over stretching the spring will cause it never to go back to it’s original shape
2. Have the students stretch the slinky 2.0 meters between them
3. Start a pulse in the slinky and measure the time it takes for one pulse to travel the length of the spring, repeat measurements twice more for accuracy
4. Make sure to have the students record their data in a data table
5. Next stretch the slinky 2.5 meters between the students and time a pulse three times
6. Continue until students have reached 5.0 meters
7. Have students calculate speed by using v = distance pulse traveled/time it took for pulse to travel
8. Graph length vs. speed
9. As tension increases, speed increases

Does mass or energy get transferred in wave motion?

1. Have students tie some brightly colored string to various parts of the slinky
2. Stretch the slinky out between the two students on the floor
3. Have one student make a pulse and have the students observe what happens to the pieces of string, do they travel with the wave, or do they just jiggle back and forth?
4. The string jiggles back and forth, therefore mass, not energy is transferred

Verifying v = fλ; for a standing wave:

1. This experiment is tricky to do if students are sloppy with their measurements, but it is possible to get very accurate data
2. Have students stretch a slinky spring on the ground between them 4 meters apart
3. Mark the ends of the slinky with masking tape
4. Have one of the students holding the end of the slinky shake it so that a standing wave with one anti-node is produced
5. Have the students time 10 cycles and record their data in their data table
6. Now have the students produce a standing wave with two antinodes and record the time it take to complete 10 cycles
7. Have the students repeat this process, getting as many standing waves as possible set up on the spring. The high end of possibilities is usually around five antinodes
8. Have the students find the wavelength of each standing wave in the following way. If one anti-node was made, then ½ of a wavelength covered 4 meters, so the wavelength of the wave was twice that or 8.0 meters. For two anti-nodes the wavelength is 4 meters, for three antinodes the wavelength is 8/3 of a meter, four antinodes is 2 meters, etc.
9. The number of cycles in one second is the frequency of the wave, so have the students take the number of cycles (10) and divide that by the amount of time it took to complete those 10 cycles
10. Then have the students multiply each wavelength by it’s corresponding frequency for that trial. The students should get approximately the same speed.

Reinforcement Activities
Make easy to see, slower moving longitudinal and transverse waves using a Longitudinal Wave Model or a Transverse Wave Motion Model or A complete Wave Motion Demonstrator and Longitudinal Wave Demonstrator.

Have students study standing waves using a String Vibrator, or a Standing Wave Demonstrator.

Related Products

• Standing wave demonstrator
• Longitudinal Wave Model
• Transverse Wave Motion Model
• Complete Wave Motion Demonstrator
• Longitudinal Wave Demonstrator
• Spiral Spring
• String vibrator
• “Slinky” springs
• Differential and Sympathetic Tuning Fork Set
• Mechanical oscillator
• Resonating hoop
• Resonating reed
• Wave spring
• Ripple tank apparatus
• Overhead projection ripple tank set
• Dual string vibrator
• Myth Busters Force and Motion Video
• Myth Busters Circular Motion Video
• Heavy Duty Steel Support Rod