Physics 120 Lab 3 Report PDF

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Physics 120 Laboratory ReportIntroductionThe task in this experiment is to measure the velocity v of a ball rolling down a ramp with angle θ to the horizontal, to ultimately investigate the acceleration a of the ball using a multitude of different height increments of the slope. The acceleration val...

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Physics 120 Laboratory Report Introduction The task in this experiment is to measure the velocity v of a ball rolling down a ramp with angle θ to the horizontal, to ultimately investigate the acceleration a of the ball using a multitude of different height increments of the slope. The acceleration values reached by measurement and derivation will be compared against expected acceleration values found using independent formula to ultimately check for consistency of values found of a. Critical relationships investigated is the final velocity of the ball at the base of the slope (found using two methods for interest) and the acceleration of the ball as it passes this point, also found through various different methods. Quantities comprised of physical constants, control variables, measured quantities, and derived quantities are explored further in the experiment to let us observe the principles of acceleration of a rolling mass. The fundamental physics concept of rolling motion helps us to investigate what force is causing the acceleration down the slope and how the alteration of the angle θ to the horizonal affects the motion and acceleration of the body down the incline.

Method The experimental setup included gathering the equipment; a table, a retort stand, a meter ruler, a metal ball, a photogate timer, a measuring tape, a plumb bob, a mat, a piece of carbon paper and a piece of plain paper.

the ball left a marking on the carbon paper, also checking and noting the value of the final velocity on the photogate timer. We changed the height of the ruler against the retort stand 5 times to examine how the changes of the angle of the track relative to horizontal θ will impact the velocities and the acceleration, and took 5 trial runs to find the mean final velocity and the mean distance fallen (for a more accurate measured value). The ultimate aim of the experiment is to discover the acceleration of the ball down the slope which will be attainable once the velocity at the bottom of the slope is investigated. This is done two ways, by using the photogate timer to simply give us the value, and using kinematic equations to calculate it based on the principles of projectile motion. The experiment covers both ways and compares results. Out of this experiment we are expecting to find the acceleration values increase as we increase the height of the ruler and therefore increase the angle θ of the slope relative to horizontal. When the slope of the incline is larger, there is a bigger component of the gravitational force acting upon the ball, causing to it accelerate at a higher rate. Although rolling motion without slipping implies there is a rolling friction on the surface of the track, we will not be analysing this in this experiment as we are focussing our attention on other measurements.

Results Length of track L - 0.980 0.0005 m Height of the table y - 0.978 0.0005 m Height increments to investigate the effect of on the acceleration readingsh=0.10 0.0005 m, 0.15 0.0005 m, 0.20 0.0005 m, 0.25 0.0005 m, 0.30 0.0005 m Uncertainties were attained for lengths from the measuring tape.

Figure 1- A diagram of the experimental setup

Figure 1 expresses the setup of the experiment displaying the variables h, L, x, y, θ, and other quantities of interest while conducting our tests. The experiment was run so that one partner released the ball on the meter ruler slope at a specific marked spot (for consistency with trials) and the other was at the other side labelling where

Tables below present data for the horizontal distance the ball fell of the table, and the velocity readings from the bottom of the slope. Each of 5 height increments were tried 5 times to gain a mean value for more accuracy. Horizontal distance ball travelled off table as calculated by carbon paperHeight of track, m

0.1 0.0005

0.15 0.0005

0.2 0.0005

0.25 0.0005

0.3 0.0005

Trial 1, m Trial 2, m Trial 3, m Trial 4, m Trial 5, m

0.474 0.474 0.474 0.470 0.478

0.597 0.593 0.590 0.596 0.597

0.690 0.690 0.692 0.690 0.690

0.790 0.786 0.784 0.783 0.784

0.836 0.845 0.842 0.842 0.843

Velocity of the ball at the bottom of the slope as calculated by the photogateHeight of track, m Trial 1, ms-1 Trial 2, ms-1 Trial 3, ms-1 Trial 4, ms-1 Trial 5, ms-1

0.1 0.0005 1.12 1.12 1.12 1.12 1.12

0.15 0.0005 1.39 1.39 1.37 1.39 1.40

0.2 0.0005 1.60 1.60 1.60 1.60 1.60

0.25 0.0005 1.80 1.80 1.80 1.80 1.79

0.3 0.0005 1.94 1.93 1.94 1.94 1.93

In order to be as accurate as possible we can now express our measured time trials with the mean of the data the standard error. For h = 0.1 0.0005 m Distance travelled by ball- 0.4740 ± 0.0013 m Velocity at bottom of slope- 1.120 ± 0.0000 ms-1 For h= 0.15 0.0005 m Distance travelled by ball- 0.5946 ± 0.0014 m Velocity at bottom of slope- 1.388 ± 0.0049 ms-1

Mean

Horizontal distance ball travelled off table as calculated by carbon paperHeight of track, m Mean, m

0.1 0.0005 0.4740

0.15 0.0005 0.5946

0.2 0.0005 0.6904

0.25 0.0005 0.7854

0.3 0.0005 0.8416

Velocity of the ball at the bottom of the slope as calculated by the photogateHeight of track, m Mean, ms-1

0.1 0.0005 1.120

0.15 0.0005 1.388

0.2 0.0005 1.600

0.25 0.0005 1.798

0.3 0.0005 1.936

Standard Deviation

For h = 0.2 0.0005 m Distance travelled by ball- 0.6904 ± 0.0004 m Velocity at bottom of slope- 1.600 ± 0.0000 ms-1 For h = 0.25 0.0005 m Distance travelled by ball- 0.7854 ± 0.0013 m Velocity at bottom of slope- 1.798 ± 0.0020 ms-1 For h = 0.3 0.0005 m Distance travelled by ball- 0.8416 ± 0.0015 m Velocity at bottom of slope- 1.936 ± 0.0024 ms-1 Other calculations

Horizontal distance ball travelled off table as calculated by carbon paperHeight of track, m Standard Deviation, m

0.1 0.0005 0.00282 8

0.15 0.0005 0.0030496

0.2 0.0005 0.000894

0.25 0.0005 0.002879

0.3 0.0005 0.003349 6

When h=0.10m-

Velocity of the ball at the bottom of the slope as calculated by the photogateHeight of track, m Standard Deviation, ms-1

0.1 0.0005 0

0.15 0.0005 0.0109545

0.2 0.0005 0

0.25 0.0005 0.00447

0.3 0.0005 0.005477

Standard Error

0.1 0.0005 0.00126 5

0.15 0.0005 0.0013638

0.2 0.0005 0.0004

0.25 0.0005 0.001288

0.3 0.0005 0.001498

Velocity of the ball at the bottom of the slope as calculated by the photogateHeight of track, m Standard Error, ms-1

0.1 0.0005 0

,

When h=0.15mWhen h=0.20mWhen h=0.25mWhen h=0.30mUncertainty-

Horizontal distance ball travelled off table as calculated by carbon paperHeight of track, m Standard Error, m

Angle of the string relative to horizontal, . Using relationship sin

0.15 0.0005 0.004899

0.2 0.0005 0

Best estimate uncertainty

0.25 0.0005 0.001999

0.3 0.0005 0.002449

When h=0.10m, = 0.0006 When h=0.15mWhen h=0.20mWhen h=0.25mWhen h=0.30m-

= 0.0006 = 0.0006 = 0.0006 = 0.0007

Found through uncertainty propagation, understanding that the values h and L are dependent quantities so we add their relative uncertainties to attain the relative uncertainty for sin.

Vi of the projectilewhen h=0.1m: a= a= When h=0.15mWhen h=0.20mWhen h=0.25mWhen h=0.30mFigure 2- diagram of the projectile motion of the ball as it falls off the table

Using relationships derived from Knight, 2016. As projectiles have independent horizontal components and vertical components, except for the time taken which can unite the two equations. The value g will be taken from the University of Auckland, 2021. We will use g as 9.7994 0.00002 ms-2 .

a= 0.574 ms-2 ms-2 ms-2 ms-2 ms-2

Uncertainty-

when h=0.1m= 0.002 ms-2 When h=0.15mWhen h=0.20mWhen h=0.25mWhen h=0.30m-

ms-2 ms-2 ms-2 ms-2

Acceleration using the photogate timer, a. Using the relationship a=as derived in Knight, 2016.

when h=0.10mms-1 When h=0.15mWhen h=0.20mWhen h=0.25mWhen h=0.30m-

ms-1 ms-1 ms-1 ms-1

when h=0.1m: a= a= When h=0.15mWhen h=0.20mWhen h=0.25mWhen h=0.30mUncertainty-

UncertaintyBecause independent quantities, add in quadrature.

when h=0.1m:

For tf,

= 0.00033 ms-2

2

When h=0.15mWhen h=0.20mWhen h=0.25mWhen h=0.30m-

when h=0.1m: ms-1 When h=0.15mWhen h=0.20mWhen h=0.25mWhen h=0.30m-

ms-1 ms-1 ms-1 ms-1

Acceleration using projectile motion, a. Using the relationship a=as derived in Knight, 2016.

a= 0.640 ms-2 ms-2 ms-2 ms-2 ms-2

ms-2 ms-2 ms-2 ms-2

Acceleration using the conceptual formula, a. Using where c is a constant- for a sphere it is 2/5. when h=0.1m: a= 0.7143 ms-2 When h=0.15mms-2 When h=0.20mms-2

When h=0.25mWhen h=0.30m-

ms-2 ms-2

show that our experimental results were somewhat in the right direction, as our outcomes do align with the hypothesis made at the start of the lab.

Uncertainty Figure 3- acceleration vs. sin theta (projectile motion)

when h=0.1m:

Figure 4- acceleration vs. sin theta (photogate timer)

-2

= 0.00394 ms

When h=0.15mWhen h=0.20mWhen h=0.25mWhen h=0.30m-

ms-2 ms-2 ms-2 ms-2

Figure 5- acceleration vs. sin theta (conceptual formula)

Discussion Fundamental theory surrounding physical experiments assumes that measured values for a quantity should be accurate when investigated in a perfect experiment. In a realistic environment however, lots of errors have to be accounted for when taking measurements. Random errors relate to the instrument’s limitations or human errors from the measurer. Random errors were prevalent in our experiment as they can be hard to minimise, an example of the possibility of random errors occurring is the fact that my partner and I only took 5 trials for our experiment. To reduce the amount of random errors effecting a single value, we should have taken more trials. The environmental conditions (as we were in a busy classroom) could have also been revised in a future experimental model. In our lab, we aimed to gather data in order to investigate the acceleration on a small metal ball as it rolled down an incline. We used a multitude of different methods to get there, and yet we can

We can see from these 3 graphs using the attained acceleration values and that the values have a linear relationship. The acceleration of the body down the slope is directly proportional to the sin of the angle relative to horizontal as the relationship is linear. The values that we attained for a in all instances were quite close to each other, which leads us to believe that they will be accepted. If we use our conceptual formula acceleration values as our reference values, we can notice that the graphs are very similar and they should be within 2 of each other, as 95.44% of data falls between 2 of the mean value (Yang, 2021). There are still some values that lie outside this range and this can be graphed exponentially, but our values are so close that we can assume we’ve minimised as many errors as we can during the experiment to reach as accurate a result as possible. Conclusion Our experiment has allowed us to observe the physics fundamental theory of the acceleration of a rolling body down an incline. By finding acceleration values through 3 different methods we were able to graph them against sin theta and observe their linear relationship.

Reference List: Knight, R. D. (2016). Physics for scientists and engineers: A strategic approach with modern physics. Pearson Higher Ed.

University of Auckland. (2021). Mechanics 13: Lab2, PHYSICS 120 Advancing Physics. [PowerPoint Slides]. Canvas. https://canvas.auckland.ac.nz/courses/62113/files/6378978?wrap=1 Yang, A., & Edwards, F. (2021). Measurement and Uncertainty, (Physics 120 Laboratory 1). University of Auckland....