Science
Projectile Motion and Atwood Machine Lab 5 & 6 Report

Question Description

Introduction

In this lab, you are going to explore the motion of a ball set into projectile motion after falling off a ramp and bouncing on the floor. Classic projectile motion carries one simplifying assumption, which is that air resistance is negligible. Under that condition, the motion of the projectile can easily be analyzed by separating velocities and positions into their horizontal and vertical components.

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Lab 5 - Projectile Motion Introduction In this lab, you are going to explore the motion of a ball set into projectile motion after falling off a ramp and bouncing on the floor. Classic projectile motion carries one simplifying assumption, which is that air resistance is negligible. Under that condition, the motion of the projectile can easily be analyzed by separating velocities and positions into their horizontal and vertical components. The two quantities of interest that can be used to then predict the entire trajectory are the initial velocity, vo, and the angle that the initial velocity makes with the x-axis, ΞΈ. If these two quantities are known, then other quantities of interest, such as the total time of flight, maximum height, and total horizontal distance, or the range, can be predicted. Conversely, if we can physically measure the total time, maximum height, and range, we can calculate the initial velocity, the initial angle, and the acceleration of the system. Remember that the only acceleration in classic projectile motion is that of gravity, downward, in the vertical (the y) direction. There is no acceleration in the horizontal (the x) direction. The equations governing the motion are as follows: π‘₯ = π‘£π‘œ cos πœƒ 𝑑 1 𝑦 = π‘£π‘œ sin πœƒ 𝑑 βˆ’ 𝑔𝑑 2 2 𝑣𝑦 = π‘£π‘œ sin πœƒ βˆ’ 𝑔𝑑 horizontal position as a function of time vertical position as a function of time, it is assumed up is positive and down is negative vertical velocity as a function of time One way to confirm projectile motion is to measure the vertical positions at various times. Once the data is plotted (time on the x-axis and position on the y), if the best trend that fits the data is a 2nd-order polynomial, then projectile motion is confirmed since the vertical position equation is quadratic. If air resistance acted, the flight times up and down would not be symmetric and therefore not parabolic. There is an added bonus that, because the acceleration appears in the coefficient of the quadratic term, we can use the quadratic fit to attempt to measure the acceleration. One of the special points in the trajectory of a projectile is at the highest point, or π‘¦π‘šπ‘Žπ‘₯ . First, the path of the projectile is symmetric on the way up, and on the way down, about this point. This means it takes just as long to rise as it does to fall, and the actual path it travels on the way up is mirrored on the way back down. Secondly, the vertical velocity, 𝑣𝑦 , is zero at this point. We can exploit this fact to find the total time of flight. If we set 𝑣𝑦 equal to zero and solve for t, the result is π‘‘π‘π‘’π‘Žπ‘˜ = π‘£π‘œ sin πœƒ . 𝑔 To find the total time, we double this, since it takes the projectile as long to fall as it does to rise. Therefore, the total time of flight is given by: π‘‘π‘‘π‘œπ‘‘π‘Žπ‘™ = 2π‘£π‘œ sin πœƒ . 𝑔 Next, the maximum height is found by taking the time it takes to reach the peak and plugging it in for t in the equation for the vertical position. After simplification, the result is π‘¦π‘šπ‘Žπ‘₯ = π‘£π‘œ2 sin2 πœƒ . 2𝑔 In the first part of the experiment as outlined below, you will watch a slow motion video of a bouncing ball set into projectile motion and record times/positions. Plotting the data and confirming that a second order polynomial is the best simple fit to the data will confirm the ball obeys the vertical position equation from the previous page. Secondly, you will watch the video, timing the entire path from launch to landing, as well as noting the maximum height. You will externally measure these with a stopwatch and the ruler in the video. We don’t know the initial velocity or the angle of launch, but we can eliminate them if we solve the total time equation for π‘£π‘œ sin πœƒ, and square the result, and set it equal to the rearrangement of the π‘¦π‘šπ‘Žπ‘₯ equation after solving for π‘£π‘œ2 sin2 πœƒ. The result is 2 𝑔2 π‘‘π‘‘π‘œπ‘‘π‘Žπ‘™ = 2π‘”π‘¦π‘šπ‘Žπ‘₯ . 4 If we solve this equation for g, we obtain 𝑔= 8π‘¦π‘šπ‘Žπ‘₯ . 𝑑2 We can obtain an estimate for the gravitational acceleration if we can measure the maximum height and the total time. Procedure: Materials: Stopwatch YouTube video found here: https://www.youtube.com/watch?v=lC_s-MsFf3c&list=PLD3LFNpL8oRSJ5jwNI-JJYyEDQn5qyxi&index=3 Once the link is opened, you will see a video of a ball falling off a ramp, bouncing off the floor, and being set into projectile motion. The video was shot in 2000 frames per second and encoded at 24 frames per second. This gives a correction factor of 83.333, meaning that for any time measurement made with a stopwatch will need to be divided by 83.333 (actually, as a matter of fact, the factor is 83. 3Μ…, so you may take as many decimal places in converting your times, but your final result after the conversion should contain the same number of significant figures as your stopwatch allows. The region of interest in the video is right when the ball bounces the first time, rises, and then falls back and hits the floor. This is approximately from the 37 second mark to the 1:35 mark on the time ribbon along the bottom of the video in YouTube. Procedure 1: - Tracking the Trajectory 1.1 Position vs. Time While carefully watching the video, start your stopwatch right as the ball leaves the surface, and measure how long it takes to reach the following positions on both the way up and down. If using the lap button on your smartphone, you will need to add each previous lap to obtain the total for that measurement. The first position should start at 0.0 s and 0.0 m. Note that the measuring stick is in cm, so these need to be converted to meters. The positions you should measure the times for are symmetric; from 0.00 m to 0.55 m is the ball rising, and then from 0.55 m to 0.00 m is the ball falling. Once your final times have been obtained, plot your positions vs. time in Excel. Fit your data with a 2nd order polynomial and display the equation on your chart. Be sure to include labels on your axes and include your plot in your report. Times (s) 0.0 Times/83.333 (s) 0.0 Positions (m) 0.00 0.10 0.20 0.30 0.40 0.50 0.55 0.55 0.50 0.40 0.30 0.20 0.10 0.00 From the equation for vertical position as a function of time, the leading coefficient on the 1 quadratic term represents 2 𝑔. Therefore, multiply the quadratic coefficient by 2, and this provides an estimate for 𝑔. Coefficient: ___________ Estimate for 𝑔:____________ Compare your estimate for 𝑔 with the accepted value of 9.8 m/s2 using a percent error calculation. 1.2 - Velocity vs. Time Using the same procedure to calculate velocities from position/time data as the previous lab, calculate an estimate for the instantaneous velocities, and plot the velocity vs. the time for these values. The result should be a linear trend with a negative slope, according to the equation for vertical velocity as a function of time, and the slope should be the acceleration due to gravity. Note: there may some outliers, where do they occur, and are they outlier enough to discard? Fit the data with a linear trendline and record the slope. Compare this with the accepted value of gravity, 9.8 m/s2 using a percent error. Next, find the uncertainty on the slope of the trend using a LINEST function. Procedure 2 - Measuring 𝑔 from Time of Flight and Maximum Height For this procedure, rewind the video to just before the ball bounces the first time. As the ball bounces, start your stopwatch, timing the entire flight, stopping the stopwatch as the ball lands. Now, we want to assign a value for the uncertainty on the total time. Typical reaction time for a human is around 0.1 - 0.2 seconds; it takes about that amount of time for your finger to press the stopwatch button once your brain realizes it needs to. This is complicated further by the fact that there isn’t a specific instant at which you β€˜know’ when the button is to be pressed, at either end. The ball is in contact with the floor for some length of time in the slow motion regime and knowing exactly when the button is to be pressed is not trivial. As a result, it is probably not unfair to assert that there is about 0.3 seconds of reaction time at both ends of the flight. As a result, the total uncertainty on your time measurement may be somewhere around 0.6 seconds. If you are extremely confident that the start and end times were performed very precisely, this may be around 0.4 or so. It is up to you to estimate your uncertainty, and there is not a standard procedure, such as for using a ruler with the 20% rule as used previously. It is entirely your estimate as to what uncertainty to assign, within reason. Afterall, uncertainty is your estimate for how well you know your measurement. Once you have your measurement and determined your uncertainty (remember the measurement and uncertainty must have the same number of decimal places), divide both by the 83.333 factor to obtain the actual total time of flight and associated uncertainty (yes, its valid to divide both by the conversion). Denote the total time by t. Time (s): __________ + _________ seconds t = actual time (s): __________ + _________ seconds Next to obtain a measurement for the maximum height, watch the video again, carefully attempting to note the location of the maximum height. Fortunately, this occurs very close to where the ball crosses the meter stick. However, we cannot be 100% precise about this. So, watching the ball carefully, and attempting to note its precise vertical position relative to the meter stick, obtain an estimate for the maximum height and record this in meters. Estimate the uncertainty (note, this should be larger than the 20% of the least count as explained in the power point and used previously, due to the very small size of the least count relative to the perception of the ball as it crosses the meter stick, and the fact that we cannot be 100% certain where the exact maximum height occurs.) Record your estimate for the maximum height and your estimate for its uncertainty below: π‘¦π‘šπ‘Žπ‘₯ : ________ + ________ m From your values for the actual time and the maximum height, calculate an estimate for the acceleration due to gravity from the formula 𝑔= 8π‘¦π‘šπ‘Žπ‘₯ . 𝑑2 𝑔: _________ m/s2 Now for the uncertainty. It can be shown that the uncertainty on the gravitational acceleration is given by πœŽπ‘¦ π‘š 2 πœŽπ‘‘ 2 ) + 4( ) , πœŽπ‘” = π‘”βˆš( π‘¦π‘š 𝑑 where Οƒym is the uncertainty on the maximum height and Οƒt is the uncertainty on the time. Arranging the formula into this form simplifies the calculation, as well as puts the expression in the form of the relative uncertainty, or the ratio of an uncertainty to its measurement. This allows us to see which measurements are more precise. For the ones that are less precise, it can also serve as a starting point to re-evaluate our methodology and determine if a better measuring procedure exists for that quantity. Calculate the uncertainty for 𝑔, and record below: πœŽπ‘” : ________ m/s2. Discussion Questions 1. What are the measured values of gravity from fitting your data with the parabola and linear line from Part 1? What are the % errors of each with the accepted value? 2. Discuss how you might estimate the uncertainty for the positions of the ball as for some of your measurements the ball is not close to the meter stick? Discuss how the uncertainties in the position measurements may affect your data for Procedure 1.1. 3. From Procedure 1.2, after performing the LINEST function, does the accepted value of gravity fall within the uncertainty of the slope? 4. What is your value of gravity from Procedure 2? Include your uncertainty result in reporting this value. Does the accepted value for gravity fall within the bounds of the result/uncertainty? 5. Discuss your reasoning for assigning your uncertainties that you did in part 2. How accurate do you feel you were when pressing the stopwatch button relative to the ball launching/landing on the floor? How confident are you that you were able to determine and accurately measure where the maximum height occurred? Lab 6 - The Atwood Machine Background The Atwood machine is a device where two masses hang over a pulley (or system of pulley) and connected by a light string. Assume one mass is resting on the ground and the other is suspended in the air. If the mass on the ground is equal to or greater than the suspended mass, the system will not move unless disturbed by an external stimulus. However, if the suspended mass is larger, once the system is released from rest, the larger mass will fall, and the smaller mass will rise. Because they are connected by a string, the system will move with one magnitude of acceleration; one will be down and the other will be up. If we call the initially suspended mass m1 and the resting mass m2, using Newton’s 2nd law, the acceleration of the system can be shown to be: π‘Ž= π‘š1 βˆ’ π‘š2 𝑔. π‘š1 + π‘š2 It is this equation we will seek to verify. Procedure: Navigate to the following website: https://www.thephysicsaviary.com/Physics/singlepage.php?ID=17 Once you have navigated to the webpage, click anywhere inside the black rectangle. This will take you to a second screen with a β€˜Begin’ button. Click on Begin. This will open the simulation, and you should see two masses connected by a string suspended over two horizontally-aligned pulleys. Mass 1 is suspended, while Mass 2 is hanging just over the motion sensor which will be used to collect the data. Adjust the masses to the following values: Mass 1: 130 g (or 0.130 kg) Mass 2: 125 g (or 0.125 kg) Use the above equation to predict the value of the acceleration of the system (use 9.8 m/s2 for g). π‘Žπ‘π‘Ÿπ‘’π‘‘π‘–π‘π‘‘π‘–π‘œπ‘› : ___________ m/s2. Hit the Start button above Mass 1. This will set the system into motion. After doing so, scroll the page down and you will see two graphs; one a position vs time and the second a velocity vs. time. From the velocity vs. time graph, pick and record the time and velocity for 10 data points. Record your data in Excel and plot. Once plotted, fit with a linear trend line, and record the slope. The slope should also represent the acceleration of the system. Run a LINEST function on the data to then also obtain the uncertainty and record. t (seconds) velocity (m/s) Acceleration: _____________ + ___________ m/s2 Repeat the above procedure for the following values of the masses: Mass 1: 170 g Mass 2: 165 g and Mass 1: 200 g Mass 2: 195 g. Discussion Questions: 1. For the three trials, give a general statement about the agreement between your predicted and measured accelerations. 2. Perform a % error between the predictions and the measured values. State all three % errors. If these are more than 5%, it is likely that either there was a calculator error when using the formula to make the predictions, or there were mistakes in recording the data from the plot (be sure to pay attention to the units and axis labels.) 3. From the results of the LINEST functions, do the predicted accelerations fall within the range of the measured accelerations and their uncertainties? 4. What are the major sources of error in this procedure? Even though this is a simulation, there are a few possible sources that could affect the overall result negatively. ...
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Final Answer

Hi, here is your labs and the plagiarism results ;) Let me know if you have any concerns ;). Thank you for waiting ;)

1

Lab 6 - The Atwood Machine
Name
University
Date

2

INTRODUCTIONS:
This lab report contains information based on two masses being used in an apparatus known as
the Atwood’s Machine. The purpose was to measure the velocity of both masses they both move
in different directions due to their weight. There are objects that are heavier than others and
velocity changes depending on the masses of objects. This experiment we will be measuring the
velocity of two different types of masses. We will be utilizing an apparatus (known as the
Atwood’s machine) that contains a sensor measuring the velocity of both masses. We will be
adjusting the weights of both masses to see different results in velocity. We will be calculating a
theoretical acceleration and an experimental acceleration to compare both results to see if they
are identical. We hypothesize that we might get a percent error of at most 5% since the weights
may have minor air resistance that might have a slight interference with the force of gravity.
PROCEDURE LAB DATA:
Using the simulation that would be found in this link address:
https://www.thephysicsaviary.com/Physics/singlepage.php?ID=17
To set up the Atwood Machine, two hanging masses were changed to opposite ends of a string
on a pulley system. The two masses should be at least 40cm apart from one another to allow
adequate time for acceleration on both ends. A total of three trials were run. For trail 1, the mass
1 was bout130 g (or 0.130 kg) and 125 g (or 0.125 kg) for the mass 2. The total mass for the
system was 255g, with m1 always being heavier than m2. After the changes of tow masses, we
then start the simulation and record the result obtained. For trial 2, mass 1 is about 170 g and
165g for mass 2. For last trial (trial 3), mass 1 is about 200g and 195g for mass 2.

3

Trial 1:
Time
(s)
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1

Velocity
(m/s)
0.1
0.11
0.12
0.13
0.14
0.15
0.16
0.17
0.18
0.19

Table 1: Trial 1 Data

0.2
0.18

Velocity (m/s)

0.16
0.14
0.12
0.1
0.08
0.06
0.04
0.02
0
0

0.2

0.4

0.6

0.8

Time )s)

Figure 1: Trial 1: velocity vs time
Acceleration: 0.1 + 0.1 m/s2

1

1.2

4

Trial 2:
Time
Velocity
(s)
(m/s)
0.1
0.06
0.2
0.07
0.3
0.08
0.4
0.09
0.5
0.1
0.6
0.11
0.7
0.12
0.8
0.13
0.9
0.14
1
0.15
Table 2: Trial 2 Data
0.16
0.14

Velocity (m/s)

0.12
0.1
0.08
0.06
0.04
0.02

0
0

0.2

0.4

0.6

0.8

Time (s)

Figure 2: Trial 2: Velocity vs Time
Acceleration: 0.1 + 0.1 m/s2

1

1.2

5

Trial 3:
Time
(s)
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1

Velocity
(m/s)
0.04
0.05
0.06
0.07
0.08
0.09
0.1
0.11
0.12
0.13
...

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