Conservation of Momentum Lab 7 & Simple Pendulum Lab 8 Reports

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Your text should go through the derivation of the period of the pendulum, so we will not replicate it here. However, you should note that the form is of a general nature; 2πœ‹ multiplied by a ratio under a square root, of which is a dynamic variable over a measure of the acceleration of the system. In the case of the pendulum, the dynamic variable, or the thing that changes, is the length, and the acceleration is that of gravity.

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Lab 7 - Conservation of Momentum Background Collisions come in two general forms: elastic and inelastic. In both cases, momentum is conserved. However, kinetic energy is conserved in the elastic case, while it is not in the inelastic case. In practice few systems are perfectly elastic, but some that are very close include rubber balls bouncing off hard surfaces, billiard balls at low velocities, and atomic/nuclear collisions. In such cases, kinetic energy is very close to being conserved. We will consider only the one-dimensional case where two objects collide, one of which is stationary before the collision. In general, the conservation of momentum equation for the collision between two objects along the same direction of motion (along one axis) can be written as: π‘š1 𝑣⃗1,𝑖 + π‘š2 𝑣⃗2,𝑖 = π‘š1 𝑣⃗1,𝑓 + π‘š2 𝑣⃗2,𝑓 , where the subscripts indicate initial and final quantities. Let’s define the incoming object as m1, and the initially stationary object as m2. With that in mind, since m2 is initially stationary, we can write the equation as π‘š1 𝑣⃗1,𝑖 = π‘š1 𝑣⃗1,𝑓 + π‘š2 𝑣⃗2,𝑓 . This result is general for both elastic and inelastic collisions. However, for the inelastic case, since both objects stick together, they share the same final velocity. As a result, we modify the equation for the inelastic case as π‘š1 𝑣⃗1,𝑖 = (π‘š1 + π‘š2 )𝑣⃗𝑓 . It is these last two equations that you will verify. Remember, momentum is a vector and direction is important! Note: in the three videos below, two are where the masses are equal, and in the other the masses are unequal but differ by a factor of two. As a result, as long as you set up your equations correctly, the masses can cancel and you do not need to worry about their actual value. Procedure: You will use the following links: 1: https://www.youtube.com/watch?v=YgrlbHk486Q&t=58s - elastic collision between carts of equal mass. 2: https://www.youtube.com/watch?v=6HU7VIXHjYk&t=32s - elastic collision between carts of unequal mass 3: https://www.youtube.com/watch?v=nOIwDV26kbQ&t=100s - inelastic collision between carts of equal mass. In each of these videos, you should see a track with a ruler underneath, a cart (two if the incoming cart is already visible), and a timer underneath. You will use the timer to record the time measurements and the ruler to measure positions (you may convert to m or keep in cm). A note about the timer: it is in the form of a regular clock, except it goes from 0-10 instead of 012. Video was shot in slow motion, so you will see the second hand move very slowly. However, the timer is designed such that in real time as the hand passes each major number, for example from the 1 to the 2, represents one second. The least-count on the timer is 1/10th of a second. Also, use the shorter hand as the 10’s place. For example, at the beginning of the first video, the short hand is on the first tick, while the longer, faster moving hand is in between the 14th and 15th ticks (in between the 1 and 2). This time should be recorded as 11.45 seconds. Once you are comfortable reading the clock, on the first video, hit the play button and watch as the incoming cart comes into the screen. As the incoming cart is moving toward the stationary cart, pause the video a minimum of 8 different times, recording the time and position of the cart. Whatever point on the cart you use (the edge, one of the two knobs, etc) to measure the position, be sure to be consistent. Once the collision occurs, do the same, except this time for both carts. Plot all of your positions and times in Excel, determining velocities for each by fitting with linear trends and finding the slope. The slope of the first cart before the collision corresponds to v1,i while the slope of the first cart after the collision represents v1,f and the slope of the second cart after the collision is v2,f. Remember, VELOCITY (AND HENCE MOMENTUM) IS A VECTOR! Direction is important. So, lets define to the right in the videos as positive, and to the left as negative. Once your velocities are collected plug them into the appropriate equations as described below for each case and determine if momentum was conserved or not. Elastic Collision Between Carts of Equal Mass Before the collision: Times and Positions of Cart 1 Time (s) Position (cm) After the collision: Times and Positions of Cart 1: Time (s) Position (cm) After the collision: Times and Positions of Cart 2: Time (s) Position (cm) Elastic Collision Between Carts of Unequal Mass Before the collision: Times and Positions of Cart 1 Time (s) Position (cm) After the collision: Times and Positions of Cart 1: Time (s) Position (cm) After the collision: Times and Positions of Cart 2: Time (s) Position (cm) Inelastic Collision Between Carts of Equal Mass Before the collision: Times and Positions of Cart 1 Time (s) Position (cm) After the collision: Times and Positions of Cart 1: Time (s) Position (cm) After the collision: Times and Positions of Cart 2: Time (s) Position (cm) Discussion Questions 1. For each trial, calculate a % difference between the initial and final momenta. 2. What are the main sources of uncertainty that may play a role in causing the initial and final values to The Simple Pendulum PHY 202 and 241 Background β€’ Your text should go through the derivation of the period of the pendulum, so we will not replicate it here. However, you should note that the form is of a general nature; 2πœ‹ multiplied by a ratio under a square root, of which is a dynamic variable over a measure of the acceleration of the system. In the case of the pendulum, the dynamic variable, or the thing that changes, is the length, and the acceleration is that of gravity. β€’ In our case, it is given by: 𝑇 = 2πœ‹ 𝑙 . 𝑔 β€’ If we know the length in meters, and use the value of gravity as 9.8 m/s2, we can use that formula to predict the period of the pendulum. Procedure 1: β€’ You will be given seven links to YouTube videos with pendulums of different lengths, and the lengths will be given in the titles of the videos. β€’ Create a table in an Excel sheet with a table similar to the one on the following slide. β€’ Open each YouTube link; record the length of the pendulum in the first column. Start the video, and with a stopwatch/timer, time 10 full cycles of the swinging. Note that a full swing means coming back to its starting position. It is usually convenient to use one of the ends because it momentarily stops at that point and is easier to start and stop consistently. Divide the total time of the 10 cycles by 10 to get the time per cycle, which is the period. Using the length and the equation from the previous slide, predict what the period should be. Calculate a % error between the two numbers. Part 1: Prediction Length (meters) Time (10 cycles) Answer the questions in Blackboard Period (Time of 1 cycle) Predicted Period % Error Procedure Part 2: β€’ Next, we want to measure gravity. We do this in the following manor: β€’ Create a new column in your Excel sheet titled, β€˜Period Squared’. From your measured period values, create a formula that squares the value. Plot the square of the period vs. the length (this means length on x-axis and square of the period on the y-axis.) β€’ If the equation for the period is squared, we obtain β€’ 𝑇2 = 𝑙 2 4πœ‹ 𝑔 Procedure Part 2 Continued: β€’ We slightly re-arrange this equation to obtain 2 ‒𝑇 = 4πœ‹2 𝑔 𝑙 β€’ This now has the form y=mx+b, except b is zero. β€’ From your plot, fit the data with a linear trendline. Add the equation to the 2 4πœ‹ chart and record the slope. The slope should equal . Set the slope equal 𝑔 to this equation and solve for g. Discussion Questions β€’ 1: What is the average %error between the measured values and predictions of the periods of the different lengths? β€’ 2: What do you think would be the largest sources of error in the first part of the experiment? β€’ 3: What is the slope of the period-squared vs. length? β€’ 4: From the slope, what is the determined value of gravity? Compare the value of gravity you found from the slope with the known value of 9.8 m/s2?
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Explanation & Answer

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1

Lab 7 - Conservation of Momentum
Name
University
July 5, 2020

2

INTRODUCTION
This experiment teaches us how to measure the amount of momentum before and after elastic
collisions. Using motion sensors and data studio we can measure the motion of the two carts
before and after the collision. When objects strike each other, whether engines, shopping carts, or
your foot and the sidewalk, the results can be complex. Yet even in the most chaotic of
collisions, as long as there are no net external forces acting on the colliding objects, one principle
always holds and provides an excellent tool for understanding the dynamics of the collision.
PROCEDURE LAB DATA
There will give 3 links of YouTube videos that can be observed in gathering the data of the
elastic collision between carts of equal mass, elastic collision between carts of unequal mass, and
for the inelastic collision between carts of equal mass. Thee three links are given below:
1: https://www.youtube.com/watch?v=YgrlbHk486Q&t=58s
2: https://www.youtube.com/watch?v=6HU7VIXHjYk&t=32s
3: https://www.youtube.com/watch?v=nOIwDV26kbQ&t=100s
After watching and recording the data, we calculate the % or error and analyzed the desired
results which record in the tables below.
Data:
Elastic Collision Between Carts of Equal Mass
Before the collision: Times and Positions of Cart 1

3

Time (s)
0
6.05

Position (cm)
0
15

8.35
10.12
11.23
11.69
10.36
6.45

20
25
35
40
45
47
Total: 64.55

After the collision: Times and Positions of Cart 1...


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