D.E. Shaw . .

Conservation of Momentum and Impulse-Momentum Principle Spring 2009

Velcrotm fasteners on the carts should be facing each other so that the carts will stick together when a collision occurs. Cart A is given a push towards cart B and the motion sensor records the position of A before the collision and both carts after the collision. •Start collecting data, then give cart A a gentle push towards cart B. The carts must stick together after the collision. Examine the distance versus time plot. The distance should increase linearly with time before the collision and the slope of this part of the graph gives the speed of cart A before the Equipment: dual range Pasco motion sensor, Pasco force collision. After the collision the distance should also increase sensor with bracket and collision bumpers, Pasco Universal linearly with time but with a different slope that is the Table Clamp, two carts, wood plank, a set of masses with combined speed of both carts. A typical data set is shown in hanger and the Data Studio program. Fig. 2. If the data is not relatively smooth try changing the alignment of the motion sensor, its wide/narrow angle setting Part A: Conservation of Momentum (Totally Inelastic and reducing its trigger speed. In addition to the speed change Collision) caused by the collision the carts will slow down due to resistive forces. If you obtain the linear fit for all the data Theory: Suppose two objects are considered as a system in before the collision, the measured which the objects exert equal but cart B cart A motion sensor velocity will likely be larger than the opposite forces on each other and the velocity just before the collision. net external force on the system is zero. Similarly if you fit all of the data after Under these conditions the impulses the collision you will obtain a final associated with the internal forces ds wood plank velocity which is too small. Therefore change the momentum of the individual do a linear fit for the data just before the objects but the total momentum of the Equipment for Part A collision (these points are shown in a system cannot change. In a collision of Fig. 1 box in the figure) and for a set of points the two objects the total momentum just after the collision (also shown in the figure). Record the must be conserved and the total momentum just before the slopes (velocities) and the masses of both carts. collision must equal the total momentum just after the Do two more trials using different initial speeds. collision. The total kinetic energy of the carts after the •Place a .5 kg (500 gram) mass on top of cart A and do three collision is less than the total kinetic energy of the carts before trials with different initial speeds. Record the slopes and the collision. masses of the carts (including the added mass) in kilograms. •Place the .5 kg mass on top of cart B and do three trials with Data Collection: The equipment used to study momentum different initial speeds. Record the slopes and masses of the conservation is shown in Fig. 1. carts (including the added mass) in •Measure the mass of both carts. If the masses of 1 distance (m) kilograms. the carts are not the same you will need to mark collision 0.8 Analysis: the carts with a small piece of tape. •Open an Excel worksheet and enter the • Connect the wires from the motion sensor to the 0.6 masses of each cart (including any added digital channels 1 and 2. The wire with the yellow mass) and the initial and final speeds. 0.4 tape (or with the yellow plug) goes to channel 1. Compute the total momentum before the Set the switch on the motion sensor to the wide 0.2 collision and after the collision. Also angle position. compute the total kinetic energy before and 0 • Click on the digital input 1 icon. A window after the collision. 0 0.5 1 1.5 2 showing the available sensors will open. Select the time (s) • Find the percent change in momentum motion sensor in this list. Select only the position Collision of Two Carts 100(pf - pi)/pi . to record and choose a sample rate of 50 (Hz). Figure 2 • The theoretical fractional change in the •Construct a plot for the distance versus time. kinetic energy of the system caused by the totally inelastic • Place cart A about 0.20 (m) from the motion sensor and Cart collision is easily computed by using conservation of B about 0.20 (m) from cart A as shown in the diagram. The Introduction: The first objective of this experiment is to investigate the Conservation of Momentum Principle by examining a totally inelastic collision of two carts. The second objective is to study an almost elastic collision and apply the Impulse - Momentum Principle to one of the objects. A final objective is to observe how the maximum force exerted on a cart colliding with a fixed object is related to the duration of the collision.

momentum to find the final speed and is:

K f − Ki Ki

=−

mB m A + mB

Compute both the experimental and theoretical fractional changes in kinetic energy which are the left and right sides of the above equation respectively. Note that ma and mb include any added masses. • Plot a graph of the total momentum of both carts after the collision versus the total momentum before the collision. Do a linear fit through the origin. • Based on your calculations and the graph and allowing for reasonable experimental errors, was momentum conserved during the totally inelastic collision? Explain carefully. • Plot a graph of the experimental fractional energy change versus the theoretical fractional energy change. Do a linear fit through the origin. Comment on the significance of this graph. Part B: Conservation of Momentum, Energy and the Impulse – Momentum Principle Applied to an Almost Elastic Collision.

⎛m−M ⎞ vf = ⎜ ⎟vi ...(4 ) ⎝m+M ⎠ ⎛ 2m ⎞ V =⎜ ⎟vi ...(5) ⎝m+M ⎠ Since “M” is much larger than “m” our elastic collision model predicts that: lim ⎛ m − M ⎞ ⎜ ⎟vi = −vi ...(6 ) M >> m ⎝ m + M ⎠ lim ⎛ 2m ⎞ V= ⎜ ⎟vi = 0 ...(7 ) M >> m ⎝ m + M ⎠ vf =

According to our elastic model the cart should be moving in the opposite direction with the same speed it had just before the collision provided that the force sensor is securely fastened to the table so that the mass “M” is very large.

The Impulse-Momentum Principle: The impulse of a force is the integral of the force over a specific time interval. The impulse momentum principle states that the change in momentum of the object during this time interval equals the Theory: impulse of the net applied force. We test this principle by Momentum and Energy Conservation: An almost elastic measuring the momentum change of the cart shown in Fig. 3. collision is studied by having the cart of mass “m” collide with The force sensor exerts a force on the cart during the collision a spring that is attached to the force sensor which is clamped time. This force produces an impulse which causes the firmly to the table as shown in Fig. 3. The mass “M” which momentum of the cart to change as a result of the collision. collides with the cart not only includes the force sensor but However, the momentum of the cart does change even if the also the table and floor to which force sensor Pasco speed does not change since the directions of the cart motion sensor the table is attached. Therefore clamp velocity vector and momentum vectors change as the mass “M” is very much a result of the collision. For our real spring not m M larger than “m”. We take the all of the stored elastic potential energy is spring positive direction to be to the ds plank returned to the cart so the speeds before and after right and let “vi” and “vf” be the the collision will not be exactly the same. The Equipment for Part B speeds of the cart just before force sensor measures the force exerted on the Fig. 3 and just after the collision and cart by the spring during the collision. By “V” be the speed of “M” just integrating this force over the collision time we can find the after the collision. We apply the Conservation of Momentum impulse which is then compared with the momentum change Principle: of the cart. m vi − v f mvi = mv f + MV ⇒ V = ...(1) Experimental Details and Analysis: M • The Pasco Universal Table Clamp is used to connect the Our model assumes that the spring and the force sensor are force sensor to one end of the wood plank to prevent the force ideal so that no kinetic energy is transferred into the spring as sensor from moving during the collision. The cart is initially “internal energy”. During the collision kinetic energy of the given a speed towards the force sensor, a collision occurs and cart is converted into elastic potential energy stored in the the cart then moves in the opposite direction. The force sensor spring and the flexible bar of the force sensor. For the ideal measures the force exerted on it by the cart during the spring this energy is completely returned as kinetic energy collision and the motion sensor measures the cart's velocity after the collision. Therefore, for this elastic collision model just before and just after the collision. the kinetic energy does not change: We choose the direction of the positive X axis is to the right in Fig. 1. If the mass of the cart is “m” and the speeds before and 1 2 1 2 1 mvi = mv f + MV 2 ...(2 ) after the collision are “vi” and “vf”, the change in momentum 2 2 2 of the cart caused by the collision is:

(

)

We can find the speeds after the collision by substituting Eq. (1) into (2) and solving. We find that:

) (

)

Note that vf and vi in this formula are both positive since they are the magnitudes of the velocities. It is important to note that the momentum of the cart itself is definitely not conserved since its momentum change is not zero. By Newton's III Law the force exerted on the cart by the transducer is opposite to the force exerted on the transducer by the cart. If the force exerted on the cart by the transducer during the collision has a magnitude Ft then the magnitude of the impulse associated with this force is: t2

∫ F (t )dt T

...(9 )

t1

where t2-t1 is the time interval of the collision. The normal and gravitational forces exerted on the cart cancel and therefore produce no net impulse. The impulse momentum theorem states that the impulse of the force exerted on the cart is equal to the cart's change in momentum. • Forces to be measured are applied to the force sensor's hook or to other adapters such as springs that can be used in place of the hook. The force sensor is connected to the Pasco interface box via analog channel A. The Data Studio program, records the position of the cart with the motion sensor and simultaneously records the force exerted on the force sensor by the colliding cart. Connect the force sensor to channel A. Click on the icon for Analog Channel A and select the force sensor (not the student force sensor) from the list of available analog sensors and set the sample rate to 1000 (hz). Leave the motion sensor connected to the digital channels #1 and #2. Change the data collection time to 10 seconds. • In previous experiments we exerted forces that pulled on the sensor. However, in this experiment we push on the sensor and consequently it produces negative values if we use the same calibration method used before. By Newton’s III Law the magnitudes of the force exerted by the cart on the sensor and the force exerted on the cart by the sensor are equal in magnitude and opposite in direction. Since the direction of the force on the cart is in the negative “X” direction we can use the negative force reading returned by the sensor to represent the force on the cart. • Click the calibrate sensors tab and select the two point calibration. In the Calibration Point 1 area enter the value of zero in the Standard Value 0.8 Box. In the Calibration distance (m) Point 2 area enter the value 0.6 of 10.29 in the Standard Value Box. 0.4 collision • In the following steps you must hold the force 0.2 sensor at rest in a vertical position. With nothing 0 connected to the force 0 0.5 1 1.5 2 sensor press the TARE time (s) button on the sensor and Fig. 4 Collision with Force Sensor then click on the Read

from Sensor Button in the Calibration Point 1 area. • Press the TARE button again with nothing connected to the force sensor. Add an additional mass of 1.00 (kg) to the mass hanger for a total mass of 1.05 kg (corresponding to a total gravitational force of 10.29 N). Holding the force sensor vertically, connect the mass hanger to the force sensor’s hook and then click on the Read from Sensor Button in the Calibration Point 2 area. Finally click OK to exit the calibration procedure. • It is important to verify that the force sensor is working correctly. Holding the force sensor vertically with nothing connected to it, press the TARE button and collect data for a few seconds and then connect the hanger with the 1.00 (kg) added mass to the force sensor. The graph should indicate a net force of zero while nothing was connected and 10.3 (N) with the hanger and added 1.00 kg mass connected. If you do not obtain these values you need to repeat the calibration process. Include a copy of the calibration plot with your report. • For this part of the Time (s) experiment use the spring 0 1 1.05 1.1 1.15 1.2 adapter with the smaller force -1 constant rather than the hook. -2 • Use the Options button to select a new data collection -3 time of 2.0 seconds. -4 • The following operation will -5 take a little practice since a -6 significant time delay occurs between your clicking on the Fig. 5 Start icon and the commencement of data collection. Give the cart a gentle push towards the force sensor and click on the Start icon before the collision occurs. Even though the force sensor is clamped you should use your hand to provide extra support to prevent the force sensor from moving during the collision. The objective is to obtain about the same number of data points before and after the collision. If your timing is good you should see a triangular plot for the distance as shown in Figure 4. Try a few more runs until you think you have perfected your timing. The linear range of the force sensor is from 0 to 40 Newtons. A larger force could possible damage the sensor. Therefore it is important to limit the maximum applied force to 40 N. Actually for reasonable speeds it will probably be much less than this value in Part B of the experiment. Keep practicing until you have obtained a satisfactory data set. Check with your instructor if you are not sure. • The velocity before the collision can be found from the slope of the linear fit of the distance versus time graph. Select a range of data points before the collision and obtain the linear fit and record the slope which is the magnitude of the velocity before the collision. In the same way determine the magnitude of the velocity after the collision. In order to minimize errors due to resistive forces, use a reasonably small range of data points before and after the collision as was done in Part A. The slope is negative after the collision but just record the absolute value of this slope since we are going to use the Force (N)

(

r r r Δp = p after − p before = m − v f iˆ − m + vi iˆ r Δp = −m(v f + vi )iˆ ...(8)

magnitudes of the velocities in Equation (8). • The impulse is obtained by finding the area under the force curve during the collision. Since the force applied to the force sensor is a push rather than the pull used for calibration the force curve will be negative. Since the direction of the force exerted by the force sensor on the cart is in the negative “X” direction the force should be negative. The calibration we have done should give negative values when the cart hits the force sensor. A typical force curve is shown in Fig. 5. Ideally the base line for the force versus time graph should be at zero. However, it is possible that a drift has changed the base line slightly. If the base line is not close to zero, significant errors will occur when the integration under the force curve is done. If your base line is not very close to zero we can use the Calculator to create a new corrected force by adding or subtracting the required amount to make the baseline zero. Click on the Calculate button and add or subtract the correction. Change the function name to Fc. Click on Variable button and select Data Measurement and then choose Force,CH A. The calculator window should display "Fc = x .01" assuming the correction is 0.01. Next to the Variables button you should see "x = Force,ChA". Click on Accept to enter the corrected force. Drag the corrected force icon to the plot icon to create a new plot to display this corrected force. • An examination of the Force versus time graph reveals that the collision time is very short. Select the region where F is non zero and use the Area option in the Statistics menu to obtain the integral of the force over time, or impulse. Also use the smart tool to measure the time t1 at the beginning of the collision and the time t2 at the end of the collision. • Do at least six more trials in each case varying the initial speed of the cart. Press the Tare button before each trial. For each trial be sure to record the initial velocity, final velocity, the times t1 and t2 and the impulse. For each new trial you must remember to select the area under the curve again otherwise the program will return the area under all of the data which may have a significant error. • All of the following calculations can be done conveniently in Excel. You can use the keys ALT + TAB to switch back and forth between Data Studio and Excel. Transfer your velocities before and after the collision and the impulses to Excel. The impulse of the force exerted by the force sensor on the cart is negative. • Our model predicted that the speed just before and after the collision should be the same and therefore the kinetic energy of the cart should not change. For each trial, compute the kinetic energy of the cart just before and just after the collision. Is the collision elastic? Explain clearly why the energies may not be exactly the same. Plot a correlation graph of the kinetic energy after the collision versus the kinetic energy before the collision and find its slope. How well are these quantities correlated? • Compute the change in momentum of the cart using Eq. (8). • Compare the net impulse with the change in momentum. How do they compare? Find the percent difference. Do your results support the impulse - momentum theorem? • Plot a correlation graph of the change in momentum versus

the impulse and find its slope. How well are these quantities correlated? •Explain clearly why the momentum of the cart itself was not conserved during the collision. Note that this has nothing to do with the elastic or non-elastic nature of the collision. •For the first trial find the change in momentum of the force sensor and attached table •Assuming an ideal spring determine the energy stored in the spring when it was compressed by its maximum amount during the collision. Do this for the first trial only. •For each trial calculate the collision time Δt = t2 – t1. Your intuition might suggest that the collision time would increase with the speed of the cart since the maximum spring compression will increase. However, in the next experiment we study the motion of a mass connected to a spring. We will find that the cart moves with Simple Harmonic Motion when it is in contact with the spring. An important property of this kind of motion is that the compression of the spring is a sine function of the time. The time (T) for one complete oscillation of the sine function is the period of the motion and is independent of the maximum spring compression which is also the amplitude. The motion shown in Fig. 5 represents one half of a complete oscillation of the sine function and as a result the period is: T = 2Δt. The plot shown in Fig.5 is the force versus time. However, according to Hooke’s Law the force exerted by a spring is proportional to its compression and so Fig. 5 also represents the spring compression versus time and should be one half of a complete oscillation of the sine function. Does the collision time, Δt, seem to be independent of the speed of the cart before the collision (which determines the maximum spring compression)? A more detailed study of Fig. 5 is described in the Optional Project section.

Part C: Comparison of the Collision of the Cart with the Spring and Other Objects Connected to the Force Sensor. The purpose of this part of the experiment is to compare the force versus time graphs for collisions with the hook, rubber stopper, putty and with springs. In order to obtain as many values of the force during the collision as possible the motion sensor is not used. The motion sensor should be disabled by clicking on its icon and hitting delete. • Using the Options button, set the data collection time to one second. Click on the force sensor icon and change its speed to 2000 hz. • Set the limits of the force axis to be -50 to 0 (N). • The absolute value of the maximum force that can be measured correctly by the force sensor is 40 (N). When the cart collides with the hook mounted in the force sensor, the maximum force applied to the hook can easily exceed 40 (N) even if the cart is moving slowly. To achieve a small cart speed we elevate one end of the ramp slightly and release the cart from rest from a location not far from the hook. Place the 200 gram mass under the end of the wood plank opposite the force sensor to make a ramp with a very small inclination angle. • Using the hook in the force sensor, press the TARE button,

release the cart from a distance of only 10 centimeters from the hook and begin collecting data just before the collision. As mentioned the collision with the hook produces large peak forces. Therefore press down firmly on the force sensor during the collision to minimize any possible movement of the sensor during the collision. The absolute value of the maximum force should a little less than 40 (N). If it is larger move the cart closer to the force sensor and try again. If it is too small move the cart a little farther from to the hook. • Change the limits of the time axis to display a range of 0.2 seconds surrounding the actual collision. Obtain the area under the curve during the collision, the maximum force, the average force (mean must be selected in the statistics option list) and the approximate duration of the collision. • Replace the hook by the spring and then obtain data for the collision of the cart with the spring. The cart should be released from the same point as before to produce the same momentum before the collision. • Plot this data using the same range for the force axis and the same time range of 0.2 seconds surrounding the collision. Obtain the area under the curve during the collision, the maximum force, the average force and the approximate duration of the collision. • Repeat the last two steps for the other spring, the rubber stopper and a chunk of putty in each case releasing the cart from the same initial position used before. • Compare the impulses, maximum forces, average forces and time duration for the collisions with the hook, springs and rubber stopper. • The magnitude of the impulse of a force having an average value Fave applied for a time interval Δt is:

I = ∫ Fdt = Fave Δt ⇒ Fave =

I Δt

...(10)

• Plot the average force versus “I/Δt” for each type of collision. Do a linear fit through the origin and compare your slope with the value predicted by Eq. (10). •Theoretically, the impulse for the collision with the putty should be half as large as the collision with the spring if the spring collision is perfectly elastic. Why? How do your impulses for the putty and spring collisions compare? •The maximum force exerted on the human body is the most important factor in determining possible injury in an automobile accident. Explain how your results for collisions with different types of objects may be significant in understanding the effectiveness of airbags in automobiles. Optional Project: Motion of the Cart while in Contact with the Spring.

The objective is to examine the Force versus Time graph shown as Fig. 5. For one trial, save the force data in a table and then copy it to an Excel worksheet. Plot the measured force versus time. Calculate a set of theoretical forces using the equation: F = Asin(ωt + φ). The best values of the model constants A, ω and φ are to be found using Solver to minimize the sum of the squares of the differences between the measured and calculated forces. The quantity “A” should be comparable to the maximum force and ω = 2π/T where “T” is the period. The period is related to the mass of the cart and the spring force constant “k” by: T = 2π(m/k)1/2. Find the spring force constant.

Check List: Minimal Requirements for Lab Notebook Report The significance of each graph must be discussed and the fitted values (such as the intercept and slope) must be compared with model values when possible. Part A: 9 A Data Studio plot of the position versus time for the motion of the carts for a single, typical trial. 9 An Excel worksheet with: the total momentum before and after the collision, the percent difference between these momenta, the theoretical and measured fractional changes in kinetic energy and the percent difference between these fractional energy changes. 9 A correlation plot of the total momentum after the collision versus the total momentum before the collision with a linear fit through the origin. The percent difference between the fitted slope and the ideal value for perfect data. 9 A correlation plot of the experimental versus the theoretical fractional energy change with a linear fit through the origin.. The percent difference between the slope of the fit and the ideal value for perfect data. 9 Discussion and conclusions. Part B: 9 Calibration plot for the Force Sensor. 9 An Excel worksheet with: the change in momentum of the cart, the impulse, the percent difference between the change in momentum and the impulse, the kinetic energy of the cart before the collision and the kinetic energy after the collision. 9 A correlation plot of the change in momentum versus the impulse with a linear fit through the origin. The percent difference between the slope of the fit and the ideal value for perfect data. 9 A correlation plot of the kinetic energy after and before the collision with a linear fit through the origin. The percent difference between the slope of the fit and the ideal value for a perfect elastic collision. 9 Comparison of the kinetic energy of the cart before and after the collision. Is the collision elastic? 9 Discussion and Conclusions Part C: 9 A table with: the maximum force, the average force, impulse and collision time. 9 Plot of the average force versus the reciprocal of the collision time. 9 Discussion and conclusions.