# D-5 BALLISTIC PENDULUM

1. PURPOSE:

To use the principles of conservation of energy and momentum applied to a ballistic pendulum, to find the speed of a projectile.

2. BACKGROUND PHYSICS:

This experiment makes use of the following principles:

1. The total momentum of a closed system does not change.

2. The total energy of a closed system does not change.

An elastic impact is one in which the kinetic energy of the entire system remains unchanged. All other collisions are called inelastic.

The commonest kind of inelastic collision in mechanics is one in which part of the system's kinetic energy is converted to thermal energy. The thermal energy produced in a collision is generally difficult to measure, for it soon dissipates to the surroundings. Thus, in an inelastic collision, even though total energy is conserved, application of the energy conservation equation may be impractical. In these cases it may be possible use the conservation of momentum, for momentum is not at all affected by thermal energy production.

3. APPARATUS:

Blackwood ballistic pendulum, metric balance, meter stick, carbon paper, wire restraint loop, heavy wire or cable, high range spring balances.

The Blackwood pendulum (Fig. 1) consists of a spring gun (G) which fires a small steel ball (B) at a catcher (C) on the end of a rigid pendulum arm (PC). The target is designed to capture the ball in a totally inelastic interaction. The momentum of the ball, transferred to the pendulum causes it to swing upward, where it is stopped at the highest point of its arc by a pawl on the pendulum which engages a ratchet (S) on the frame of the apparatus.

You will use this apparatus to make the following measurements:

1) The firing velocity of the ball, by two independent methods.

2) The efficiency of the spring gun.

3) The kinetic energy lost in the ball-pendulum impact.

4) The mechanical energy dissipated in the firing mechanism.

4. NOTES ON THE APPARATUS:

(1) PROPER REMOVAL OF BALL FROM THE PENDULUM CATCHER: The Cenco apparatus has a piece of spring steel (1) in the catcher (C) to retain the ball after impact. The spring may break if it is bent backward. To remove the ball, push the spring slightly upward, and the ball will roll out.

(2) RECOIL RESTRAINT: A simple recoil restraint may be fashioned from a loop of strong wire or rope. Loop the center of the wire around a table leg, bring its two end loops together and loop them over the ratchet post of the apparatus (See Fig. 2). Then move the whole apparatus back from the table edge until the wire is taut. This simple method avoids tying knots each time the loop is used. When firing the gun, have someone hold the base of the apparatus firmly in place with their hands (keeping them out of the way of the projectile!). Do not use any kind of clamp to hold down the apparatus, for the base of the apparatus is made of pot metal which can easily break or crack.

(3) REMOVAL OF THE PENDULUM ARM: You must remove the pendulum arm to weigh it. Do this before or after the rest of the experiment. The upper end of the arm is secured by two cone bearings which are slotted for a screwdriver, and each has a locking nut. Use a small wrench to loosen the nut, then unscrew one bearing until the pendulum can easily be detached. When reattaching it, be sure the cone bearings are not too tight. There should be a slight play for free pendulum movement.

(4) MEASURING THE WORK DONE IN COCKING THE GUN: Cenco apparatus. A hole has been drilled in the back end of the gun's firing plunger. Use a short strong wire through this hole to attach a spring balance. Then use the spring balance to pull the plunger back to cocked position. While doing this, have someone hold the trigger back out of the way. Pull back until you just reach the position the plunger had when cocked. If this force exceeds the range of the available spring balances, use two balances of identical range together. Think about how they should be used "together," and if in doubt, consult your instructor.

Beck apparatus. This apparatus doesn't provide a convenient method for measuring the force required to cock the gun. We have devised a method which can do this, which your instructor will demonstrate and supervise.

5. PROCEDURE:

(1) FIRING VELOCITY, BALLISTIC METHOD: Fire the ball into the pendulum catcher 5 to 10 times, recording which ratchet tooth the pawl stops at each time. Average these and find the tooth corresponding to the average stopping position. A pointer on the pendulum indicates its center of gravity. Measure carefully the vertical distance the center of gravity rises from its lower rest position to the average stopping position. This change of height is used to calculate the change in potential energy of the pendulum and ball. The change in potential energy is equal to the kinetic energy of pendulum and ball at the instant after impact. Application of conservation of momentum to the impact itself allows the calculation of the ball's velocity just before impact, which is essentially its firing velocity.

(2) FIRING VELOCITY, TRAJECTORY METHOD: Move the pendulum up out of the way so that the ball may be fired horizontally to the floor. Measurement of the height (H) of the firing point, and the horizontal range (R) allows calculation of the velocity. Calculate the position on the floor at which you predict the ball will land, using the value of velocity you determined in part (1). At that place put a sheet of cardboard, then white paper, a face-down sheet of carbon paper and another sheet of white paper. When the steel ball lands on this sandwich of materials it will leave a black imprint on the bottom sheet of white paper. Fire the ball. How accurate was your prediction of the range.

Do this at least ten times, and find the mean range. Use this to calculate the projectile velocity. Compare with the velocity found in part (1) and discuss the errors and discrepancy with reference to your data.

(3) STUDY OF THE SPRING GUN AS A SIMPLE MACHINE: When pulling back the plunger of the spring gun you do work on it. This work goes into potential energy of the compressed spring, which is later released and converted to kinetic energy when the gun is fired. However, there is considerable friction in the mechanism, and some of this energy is lost as thermal energy, making no contribution to the ball's velocity.

The efficiency of any mechanism is defined as the ratio of the useful energy you get out of it to the total energy you put into it. Here, the input energy is the work you do in cocking the gun. The "useful" output is the kinetic energy of the ball.

The work done in cocking the gun may be found from a measurement of the force required to compress the spring to its "cocked" position. If we assume that this force is a linear function of plunger displacement, the work done in cocking the is one half of the product of the maximum force and the distance the plunger moved.

Question 1: Derive this result, either by a geometric analysis of the area under the force-displacement curve (F = ks for a spring which has not exceeded its elastic limit), or by integrating the work dW = F ds.

7. THE SAGITTA FORMULA

In some forms of the apparatus it is easier to measure the angle of swing of the pendulum, or the horizontal displacement of the pendulum.

Fig. 3 shows the geometry of the pendulum swing. You need to determine the height of rise, S, of the pendulum as it swings through angle q. This may be determined from measurements of the radius R and of the horizontal distance of swing, X. The line S is called the sagitta.

The formula for S as a function of X and R is easily obtained.

A handy formula may be derived for approximate calculations, when S<<R. From the right triangle,

```
R2  = X2 + (R - S)2 = X2  + R2 - 2RS + S2 = X2 + R2 - 2RS

X2
S = —— , the Sagitta formula.
2R
```

(1) The energy loss during the pendulum upswing may be estimated. Disable the pawl with a rubber band and find how long the freely swinging pendulum takes to reduce its amplitude by half, and count the number of periods. Then calculate how much energy is lost in the first quarter period.

(2) The center of mass of the pendulum may be determined by finding its period for small oscillations.

7. ANALYSIS:

(1) Calculate and compare the values of firing velocity obtained in procedures (1) and (2). Report a single value of v as your best experimental determination.

(2) Calculate the efficiency of the spring gun.

(3) Calculate the kinetic energy lost in the impact between ball and pendulum as the ball is caught and held in the pendulum.

(4) Calculate the mechanical energy lost in the gun mechanism in the cocking and firing processes.

8. QUESTIONS:

(1) Did this experiment provide verification of the law of conservation of momentum? If not, why not, and what role did that law play in this experiment? If you say that the experiment did verify the law, specify which part of the experiment did this, and also state the limits of error within which the law was verified.

(2) Did this experiment provide verification of the law of conservation of energy? Justify your answer in the same manner as indicated in item (5).

(3*) Suppose you had not bothered to restrain the apparatus to prevent recoil. The mass of the Cenco apparatus is about 7 kilograms. The coefficient of kinetic friction between the rubber feet and varnished wood is probably about 1. Now if you had performed the calculations of this experiment without regard for the recoil, how much systematic error would this cause in the experimental value of vo? In each case indicate whether the error would make the result too high or too low in value.

(4) How much systematic error would be caused if the gun were misaligned and fired slightly upward, an angle α above the horizontal? Consider separately the systematic error this would cause in your calculated vo for each of the two methods, if the calculations had been done without knowledge of the misalignment.

(5) Kinetic energy is lost in the impact between the ball and the pendulum. Suggest what could have happened to the that energy. Did you observe anything to indicate into what form some of it might have been converted? What fraction of the ball's energy is lost in the impact with the pendulum if the ball has mass 75g, speed 60 cm/s and the pendulum has mass 150 g?

(6*) Suppose the ball hit the catcher a bit offside, so that it bounced off without being captured. This may have happened when you did the experiment. In this case, will the pendulum swing higher, less high, or the same height as when the ball is properly captured? Of course you must justify your answer by appeal to the physical laws and the details of your apparatus and procedure.

(7*) The speed of the pendulum after impact decreases smoothly to zero during its swing. Is the linear momentum of the loaded pendulum conserved during its swing? Is its angular momentum conserved during its swing? Discuss this, considering the changes of momentum of the pendulum during its swing, and the forces and torques which cause those changes.

Endnotes:

1. Some of the pendulums may have a springy plastic strip instead of the spring steel.

© 1997, 2004 by Donald E. Simanek