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1 Rotational dynamics

The purpose of the experiment is to test Newton’s second law for rotational motion.

The main apparatus for this experiment consists of a large, wall-mounted disk that has a smaller radius hub around which a string is wrapped. The disk and hub assembly is free to rotate about an axle through its center.. The PASCO smart-pulley will be used to make digital acceleration measurements.

If a mass is hung from the end of the string and released, then the mass will accelerate downward and the tension in the string will cause the disk to experience an angular acceleration in the counter-clockwise sense ($\alpha$ points out of the paper). This is illustrated in the figure below.

Newton’s law is used to compute the acceleration of the hanging mass. A free body diagram for the disk includes only the tension in the string, because this is the only force acting on the disk that produces a torque about its axle.
If we apply Newton’s law

to the hanging mass we obtain the equation

If we apply

to the disk we get

where $r_2$ is the outer radius of the hub and $I$ is the moment of inertia of the disk and hub assembly about its axle. There are two contributions to the moment of inertia of the disk, one from the disk itself

and one from the hub. If the hub material has density $\rho$, then

in which $\ell$ is its thickness, and moment of inertia

for a total moment of inertia

Provided that the string does not slip on the hub, then the angular acceleration of the disk is related to the linear acceleration of the hanging mass by the following expression

If we eliminate $\alpha$ and $T$ from the equations of motion and solve for $a$, we find that Newton’s second law predicts that the acceleration of the hanging mass is given by

To test this prediction we need a method for measuring the acceleration of the hanging mass. If we release the mass from rest at a given height $h$ and time its fall to the floor with a stopwatch, then, according to one of the basic kinematic equations, the acceleration can be calculated from

Instead of this procedure, which would require making multiple runs and averaging in order to achieve any accuracy, we will use the PASCO smart-pulley to directly measure the acceleration.

1.1 Experimental procedure

Note the masses and measure the radii of the disk and hub, as well as the uncertainties in these quantities. Record this information as part of your data.
Compute the moment of inertia $I_{disk}$ of the disk, and compute the value of $\frac{I_{disk}}{r_2^2}$. Record this as well on your lab report.

Using your value for $\frac{I_{disk}}{r_2^2}$ and the accepted value of $g$, compute the theoretically predicted value of the acceleration $a$ from equation above, for masses $m=100,200,300,400,500,600$ grams and record these. Create a graph of these points, the computer graphing interface will connect your points with a smooth curve. This is your theoretical prediction curve.

Set up your computer interface with the smart-pulley. Take the following steps;
1. Click on the Science Workshop icon.
2. Click and drag the plug to digital channel 1.
3. A Digital Sensor window will pop up. Select Rotary Motion Sensor.
4. In the Linear Calibration window, select Large Pulley (groove) and set Divisions/Rotation to 360.
5. To display the sensor-acquired data, click and drag the Graph icon in the main window to digital channel 1.
6. A Calculations menu will pop up, select Acceleration, linAcc (cm/s/s), and click Display. Now a little graphics window will pop up and display acceleration versus time. You can adjust the scales with the Zoom and Autoscale buttons.
7. To begin recording data, hit REC button in the upper left of the main Workshop window. Hit STOP when you are done. Each time you do this, a new data set is created, and by clicking on them, you can display one or the other in the graphics window.

For each of the masses $m=100,200,300,400,500,600gms$ perform a data run, and determine the true acceleration of the system. Record this data on your lab report. Your acceleration should be constant once starting friction is overcome), so the PASCO acceleration versus time graph should be horizontal.

Put these data (actual, experimental accelerations versus $m$) on your graph. Be careful to make it clear which points are data and which are theoretical calculations.

1.2 Testing a hypothesis

We are making a hypothesis that the acceleration versus $m$ data conforms to

How is this hypothesis tested or verified? We will use the Chi-squared statistical test, which compares the standard deviation of the experimental $a$ versus $m$ with the theoretical.
The idea behind this is that the compliance of your data with the theoretical values is best measured by determining the probability that some other experimenter could do better. If that probability is low, then your data must fit the theoretical curve very well. To compute your so-called chi-squared statistic, let

for your $I_{disk}$, and let $a_{exp}\left(m\right)$ be the six actual experimental values. Calculate

We now look up in a table of the Chi-squared probability distribution function what the probability $\wp ({\chi }^2\ge {\chi }_{exp}^2)$ is that someone will do the experiment with your hardware and get a bigger standard deviation (a worse fit).

We will regard your data as a successful verification of Newton’s law of rotational dynamics if

meaning that the probability another experimenter would get a worse fit than yours is $95\%$.

After computing your ${\chi }_{exp}^2$ value, record it on the computerized lab report. The server will determine how well your data confirms Newton’s law, and will inform you of your experimental prowess! 

1.3 Pre-lab questions

1. In the figure below, a disk of mass $m_1=5.0kg$ and radius $\ell =0.2m$ is wall-mounted with an axle through its center. A $5.0kg$ mass hangs from a cord wrapped around its circumference.

a. Write down the $F=ma$ equation for the hanging mass $m_1$, and the $\tau =I\alpha$ equation for the disk. What is the relationship between $a$ and $\alpha$?

b. Solve these equations for $a$ and $\alpha$.

2. We will solve this problem in another way; Suppose that the hanging mass begins at rest, is released and allowed to fall distance $h=1m$. Determine the speed $v$ of the hanging mass and the angular speed $\omega$ of the disk at that time. How are $v$ and $\omega$ related?
Now use the formula for constant acceleration $v_f^2-v_i^2=2ah$ to determine the acceleration of the hanging mass.

3. Consider a yo-yo of mass $M=0.050kg$ and radius $\ell =0.05m$ with a string wrapped around its circumference.

If the yo-yo is released from rest, how fast is it moving ($v$) after falling $h=1.0m$? What is its angular speed $\omega$? Use this to find its acceleration $a$.

4. In an experiment like this one, with $\frac{I_{disk}}{r^2}=0.96kg$ a student gets the following set of data;

Calculate ${\chi }_{exp}^2$ for this data. According to statistical tables, the probability that someone’s ${\chi }^2$ will exceed yours is

So is this experimenter’s data in good agreement with the theoretical prediction? Explain.

1.4 Lab report