How Do You Know if Kinetic Energy and Momentum Are Conserved

Conservation of Momentum and Free energy

NOTE

This manual describes the laboratory experiment used during the 1996 - 1997 bookish twelvemonth. Meaning changes have been made since then, and the transmission used during the current academic twelvemonth is in NOT available nevertheless on the Web. Hardcopies can be purchased at the bookstore.

Conservation of Momentum and Energy
- Purpose
- Introduction
- Prelab Homework
- Function A: The Track:
  - Functioning of the Timers:
  - Operation and Setup of the Rails:
  - Overview
  - Procedure for experiment A.one: Newtons Start Law
  - Procedure for Experiment A.2: Rubberband Collisions
  - Procedure for Experiment A.3: Inelastic Collisions
  - Procedure for Experiment A.4: optional
  - Data Analysis for Part A.i
  - Data Analysis for Part A.two
  - Data Analysis for Part A.3
  - Error Analysis for Part A
- Role B: Velocity of a Projectile
  - Part B1 Measuring velocity with a Ballistic Pendulum
  - Procedure for Function B1
  - Office B2: Measuring velocity with projectile motility
  - Process for Office B2
  - Information analysis for Part B1
  - Data Analysis for Part B2

Conservation of Momentum and Energy

Purpose

To experimentally verify the laws of conservation of momentum and free energy. In particular you will do:

Part A. The Linear Runway (set-upwards in room 266)
- Newton'due south Starting time Police force
- Rubberband Collisions
- Inelastic Collisions
- (optional). Test of Newton's First Law using the Kinesthetic Cart.
Role B. The Ballistic Pendulum (fix-upwards in room 267)

Introduction

The laws of conservation of energy and momentum are among the most fundamental and useful laws of physics. They aid in the solution of many mechanics problems and come upwards frequently in many fields of scientific discipline. What these laws say is that if there are no cyberspace forces on a system, then that arrangement will accept the same momentum, p = mv, at all times. In addition, if there are no external or internal forces interim in or on a organization, so the energy of that system will remain constant. Newton's First Law is hidden in these conservation laws. Newton's First Law states that bodies at rest will remain at rest as long as no forces deed upon them, and bodies in movement volition remain in motion as long as no forces deed upon them. As one tin can run across, Newton's First Law is a argument almost conservation of momentum and energy. Things stay the aforementioned, as long equally they are left solitary.

Despite their fundamental nature, the conservation laws are often difficult to detect in ordinary experiences, primarily because of the presence of friction. Friction between moving bodies and their surroundings ways at that place are external forces acting on the system, therefore, the conservation laws practise not apply. And so, to observe the conservation laws, friction must exist eliminated as much as possible.

This lab will deal primarily with the conservation laws as they apply to collisions betwixt material objects. These collisions can be divided into two different classes; elastic collisions and inelastic collisions. If the kinetic energy of a particle is the same earlier and later the collision, so the collision is said to be elastic. Discover the reference to particles. Solid bodies are non particles, but have structure. If the collision, however, leaves at that place structure unchanged, they can be treated equally particles. For the other type of standoff, energy will flow between the ii objects, and the kinetic energy volition not exist conserved. In this example, the collision is said to exist inelastic. Note that in the absence of friction, the momentum will exist conserved in both types of collisions.

In one dimension, the conditions earlier and after an rubberband standoff between ii bodies of masses k₁ and one thousand₂ , initial velocities v_1i and v_2i ,and final velocities five_1f and v_2f, are given past

(Kinetic Free energy)

(Momentum)

For an inelastic collision only the momentum equation is valid.

Prelab Homework

The prelab homework must exist done at home and handed to the lab TA before you lot showtime the lab.

ane) A small ball of mass m₁ and velocity v_1i has an elastic collision with a large, stationary object of mass m₂ . Show that the velocity five_1f of the ball and five_2f of the large object after the collision in terms of the ii masses and v_1i are

2) For the previous problem, evidence that the velocity of the ball is reversed later the collision if the stationary object is extremely big, one thousand_ii going to infinity. Bespeak the limits of the velocities.

Function A: The Track:

Fig. 1

For these experiments you lot will exist using a rail with carts that have very low friction wheels. The rails setup with a cart used in the experiments is shown in figure 1. Yous will be able to mensurate the fourth dimension intervals of a cart using the photo gate timers (encounter handout on photograph gate timers), from which y'all tin calculate the velocity of the cart. This is an electronic timer controlled by the interruption of an invisible infrared light axle when the cart passes the gate. If an object of length L interrupts the axle for a time interval t while passing through the gate, then the average velocity of the object during that fourth dimension is given by

(3.one)

This is the basic measurement you volition make in this experiment. By measuring the velocity of the carts before and after various collisions, too as weighing the carts, yous will be able to calculate the energy and momentum and test the conservation laws.

Operation of the Timers:

For a complete description of the timers, see the handout.

Operation and Setup of the Rails:

At that place are several things that must be done in training for the experiments. Most importantly is to bank check if the runway is level. Levels are provided for this purpose. Accept i, and place information technology over the rails's feet at one cease of the track. Outset, identify it along the length of the track and observe the level'due south "bubble". You want this bubble to exist exactly in the middle. If it is non, arrange the feet of the track by turning the screws at the base of the feet. When the bubble is in the middle of the level, repeat this process at the opposite finish of the rails with the level once again length wise. Once both ends take been adjusted, repeat this again at both ends with the level along the width of the rail. Make sure that 1 of the end stops is on the right end of the track and the launcher is on the left side of the track (see Fig. 2). For a clarification of the launchers see the handout. To move the end stop and the launcher, loosen the screws on the side and slide them into position. Lastly, align the 2 photo gates equidistant from each other, making sure that the display is facing you. The track is equipped with a measuring tape so that distances tin can be measured. Identify the first photo gate at 50cm from the launcher. Place the second photo gate at virtually 100cm from the launcher so that the ii photo gates are near 50cm apart.

Fig. 2

Let u.s.a. turn our attending to the carts. One finish of the cart has a magnet within and the other end has a plunger and Velcro strips labeled in figure as Hook-and-pile Pads (come across fig. 3). You volition use the plunger for elastic collisions and the Velcro for inelastic collisions. It might be the case that the Velcro will not stick together well. To test this, push the plunger in all the way on both carts. Put the ii carts together by their Velcro ends. If they are non sticking well, you can roughen the Velcro by repeatedly putting the carts together and pulling them apart. Bank check that the carts have no irregularities past testing them on the rail. Identify each cart on the track, making sure that the wheels are in the grooves. Give each cart a small push button and see how well information technology moves on the track. There should be minimal loss in velocity along the track. If there is noticeable loss in velocity, check the level of the track again. If that is fine, check the wheels of the cart and ask the TA for help.

Fig. iii

Overview

The runway will exist used to create various collisions between the carts. The collisions volition be arranged so that an initially moving cart will pass through a photo gate before hitting some other object, so as to measure the initial velocity. Later the collision, the cart will again pass through a photo gate and then as to measure the final velocity. With these measured velocities, also as the measured masses, the momentum and kinetic energy of the carts before and after the collisions can exist calculated and compared.

Steps volition be taken to bargain with systematic errors in this experiment. The approximate effects of friction volition be measured. Launchers are used so the initial weather condition of consecutive trials can be reproduced.

Procedure for experiment A.1: Newtons Kickoff Law

The object of this experiment is to test Newton's First Police force, which states that an object will remain in its state of movement so long as no forces act upon the object. In this instance, a cart moving with no friction would not modify its velocity equally information technology moves along the track.

i) Place two of the rectangular masses on peak of one of the carts. Place a cardboard strip between the ii masses length wise so that it fits firmly.

2) You must measure the length of the cardboard and weigh the cart with the mass and the paper-thin. Record these measurements.

3) Fix the top of the photo gates so that the beam is interrupted past the cardboard. Do this by bringing the cart up to each of the photo gates and adjusting the acme of bracket and so that it is the paper-thin that sets the timer off. You tin adjust the height of the photograph gate by turning the screw on the post and moving the bracket upwards or down. Make sure you suit both of the photo gates.

iv) Arrange the cart launcher so the scale reads three.2cm when it is cocked. You can accommodate the compression by loosening the screw on the latching clamp and sliding information technology into position. Take 1 of the carts and bring it upward to the launcher. Make sure the launcher is aimed at the middle of the cart. Make certain that the cart'due south wheels are in the grooves of the track. Launch the cart by pulling the string on the launcher. Once it has passed through both of the photo gates, stop the cart and record the times displayed on the photo gates.

five) Repeat step 4 ii more times for a total of three trials.

6) Echo steps 4 and 5, this fourth dimension with a friction cake instead of the masses. Identify a piece of cardboard into the groove on the friction cake, (y'all might need 2 to fit tightly). Make sure you lot record the new mass of the cart.

Procedure for Experiment A.2: Elastic Collisions

For this experiment, yous will create elastic collisions with the two carts. Yous will explore all the possible scenarios; ii carts of the same mass colliding, a cart of lower mass colliding with a cart of a higher mass, a cart of a higher mass colliding with one of a lower mass, and a cart colliding with an infinite mass (the end stops). It is of import that yous fix the launchers to the settings described. These settings demonstrate each scenario best.

1) Place one of the carts with two of the rectangular masses on information technology and the cardboard placed every bit before, betwixt the two photo gates, nearly three quarters of the manner to the 2nd photograph gate. Set the launcher to 3.5cm. Place the other cart with the friction block and the cardboard in position to be launched. Make sure that the carts' plungers are facing each other and that the plungers are all the mode out.

2) Launch the cart toward the stationary cart that is in the heart. The cart will laissez passer through the photograph gate and repel off of the stationary cart. The two carts will so pass through their respective photograph gates. Record the initial time that the cart took to pass through the photo gate and the time the carts took to pass through the gates after the standoff. Remember that the time displayed in memory is the total fourth dimension. Yous must subtract to become the actual second time.

3) Repeat this process ii more than times.

4) Echo steps ane-3, just this time both carts should have the friction blocks on them so their masses are equal. Set the launcher to 2cm.

5) Echo steps ane-three, but this time the cart that you launch should have the two masses and the cart in the middle should have the friction block. Suit the launcher to 3cm.

6) Repeat steps 1-3 once again, merely this time, use only one cart colliding with an end stop. Movement the photo gates to the opposite end of the track spaced the same mode. Slide the launcher to about 100cm from information technology's current position. Ready the launcher to 3.5cm. Movement the cart into position and launch it. Record the time displayed on both the photograph gates before and after the standoff.

Procedure for Experiment A.3: Inelastic Collisions

This last collision experiment will deal with inelastic collisions. Echo the process for part B, but this time the Velcro terminate of the carts should exist facing front end. In each standoff, the carts should collide and stick together. When the carts laissez passer through the photo gate together, the beam will exist cleaved only when the cardboard passes through. This means in that location will be a space where the beam is not severed, so make certain y'all measure the length of the 2 cardboard pieces and not the length of the two carts.

Procedure for Experiment A.4: optional

You and your partner with a tertiary person will repeat the experiment demonstrated past you lot TA. The instructions on how to do this experiment is in a hand-out which your TA will requite to you lot.

Data Analysis for Role A.1

The goal in all the trials is to compare the initial momentum of the carts with the final momentum, and as well the initial kinetic energy to the final. This volition exist done by dividing the final value of the momentum by the initial, giving the fraction of the momentum that is conserved. If the conservation laws concur, this should be equal to one. Results less than one signal that momentum was lost. In the aforementioned style, the fraction of kinetic energy that is conserved volition be establish.

Newton's First Law states that if at that place is no internet force acting on an object, its momentum will not change. Employ the data you have recorded to exam this by calculating the fraction of the momentum conserved in each trial. Because there is only i cart, its mass and length gene out of this ratio, which volition then be

(3.two)

where t_i and t_f are the earlier and subsequently time intervals, respectively. Kinetic energy should also be conserved. calculate the fraction that is conserved,

(3.3)

The presence of any external forces, such as friction or gravity, will create a systematic fault in all further measurements. To get a rough gauge of the effect this volition have, summate the average fraction of the initial energy and momentum lost for your data. To do this, beginning average all the momentum ratios and all the kinetic energy ratios calculated above, (to get the boilerplate fractions conserved), then subtract these results from i (to get the boilerplate fraction lost),

(iii.4)

(3.5)

These results will exist used in the fault analysis.

Data Analysis for Part A.2

Calculate the fraction of the momentum and kinetic energy that was conserved in all the collisions; that is calculate

(3.v)

(3.half dozen)

Velocity and momentum have directions, so requite them signs in your calculation (i.e. right is positive). You did three trials each, boilerplate these results.

Data Analysis for Part A.3

Again, discover the average fraction of momentum and energy conserved. In this case, v_1f and v_2f are the aforementioned (considering the carts are joined together).

Error Analysis for Part A

If the conservation laws are right, it is however unreasonable to expect the fractions calculated to accept values of exactly ane, because of experimental errors. Errors in length and time measurements volition have simply a small effect on the results. The systematic error acquired past friction Is more than of import. Once the losses are found, nosotros will need to know how much can exist attributed to friction before concluding the conservation laws were not followed. The losses due to friction were calculated for the trials in part A (with equation 3.4 and three.5). Although this volition not give the exact mistake due to friction, it should not be more than a few times larger. Then, as long equally the results differ from i past at most three or four times the losses found for part A, we can presume those differences are likely due to friction. Decide which conservation laws were verified in each function.

Function B: Velocity of a Projectile

In this experiment, the velocity of a projectile as it leaves a bound gun will exist measured using ii methods. Commencement, the conservation laws volition exist used in a somewhat subtle manner to determine the velocity. For comparison, it volition too be measured using projectile movement.

Office B1 Measuring velocity with a Ballistic Pendulum

Figure 3.3

A ball is shot by a spring gun into a catcher arranged to swing equally a pendulum (figure 3.iii). When the ball is caught, the combination of the brawl and catcher get the bob of the pendulum. Although the collision between brawl and catcher is inelastic and free energy is not conserved, momentum is. With the catcher at rest, the initial momentum of the system is provided by the ball, shot with velocity five_b . Only after the ball is defenseless, the momentum that is due to motion of the center of mass of the pendulum assembled from the catcher and ball, having velocity v_p . Conservation of momentum requires these be the same,

_(3.7)

where chiliad and Yard are the masses of the ball and pendulum respectively.

Subsequently the ball is defenseless, the energy of the combined ball and catcher arrangement is conserved. Initially the heart of mass is at a acme h₁, with velocity 5_p. Equally it moves upward against the forcefulness of gravity, the kinetic energy is converted into potential energy. A pawl machinery latches it at its highest bespeak, h₂, where all the initial kinetic energy of the pendulum has been used to produce a change in the potential free energy,

(3.eight)

which can be determined by measuring the increment in height h = h ₁ - h _two of the middle of mass of the assembly. The value of the initial velocity of the ball is,

(3.9)

Procedure for Part B1

Be certain your appliance is firmly fastened downwardly then it will not move during the experiment. Check to make sure the base of operations is level and adjust if necessary. The ball mounts on the gun's push button rod via a hole through its diameter. Make sure this fits smoothly on your apparatus. Make sure that the ball catcher is aligned with the jump gun.

Effigy three.4

1) Mensurate and tape the weight of your ball. To remove the ball from the catcher, lift the latch jump with your finger while pushing out from the rear. The ball should come out easily. Practise not bend the spring backwards, or it may break.

two) Weigh the pendulum. To do this, carefully unscrew the cone begetting spiral until the pendulum is released. Supercede the pendulum and tighten the cone begetting gently merely firmly back in place.

iii) To cock the gun, first residue the pendulum on the rack to get information technology out of the way. Then, with the ball in place on the rod, push on the brawl until the trigger latches.

4) With the pendulum hanging freely mensurate the height of the Middle of Mass (C.M.) indicator from the base of operations.

5) Starting with the pendulum at rest, fire the ball into the catcher nine times (be careful not to become your paw defenseless in the gun when you fire it) and measure out the elevation of the C.M. indicator from the base of operations each fourth dimension. It will not e'er be the aforementioned.

Part B2: Measuring velocity with projectile movement

Figure 3.5

With the catcher removed from its path, the ball will fall freely later it leaves the gun. Information technology will follow a parabolic arc until it hits the table top, traveling a horizontal altitude 50 while falling a vertical distance h. The equations of projectile motility can be used to analyze the motion in these directions, equally shown in figure 3.5. Combining the two equations to eliminate time, the initial velocity of the ball is and then

(3.10)

Procedure for Part B2

Brand sure the apparatus is securely attached and leveled on a block on the table. Latch the pendulum bob out of the way and practice firing the ball from the gun out over the table. Notation where the brawl lands. Exist careful of wild shots.

1) Tape a piece of white paper over the area of the table acme where the ball lands. Encompass it with carbon paper, carbon side downward, so that the impact point of the brawl will produce a marker.

ii) With the ball in identify and the gun released, find the position on the table top straight below the center of the brawl and marker it with a slice of tape.

three) Mensurate the height from the bottom of the ball to the table tiptop.

4) Fire five shots, marker and numbering each of the impacts.

5) Measure the range L of each of these shots from your tape mark.

Information analysis for Part B1

Calculate the average increase in tiptop of the heart of mass of the pendulum and its standard error (from the standard deviation of the measurements). Utilise this data to calculate the initial velocity of the ball and its error (meet Appendix on Error Assay). You may presume there is no mistake in the masses.

Data Analysis for Part B2

Calculate the average range L and its error [Delta]Fifty from the standard difference. Make a reasonable judge of [Delta]h, the possible fault in your height measurement. Summate the initial velocity of the ball from the measurements of h and L. Calculate the error in the initial velocity [Delta]v_b obtained with this method. The effect from propagation of errors (encounter Appendix on Error Analysis) is

(3.11)

Compare the results you obtained for the initial velocity of the ball by the two unlike methods. Are they consistent within your errors? Can you account for discrepancies?

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