PURPOSE: To study the torque and moment of inertia of a rotating disk that is accelerated by a falling mass. The same data used for the constant angular acceleration laboratory can be used here.

THEORY: Torque if defined by

and Newton's second law for rotational motion is

where t is torque ( in units of Newton sec or kg m/s), F is the force (in Newtons), R is the distance from the point of rotation to the point where the force is applied, I is the moment of inertia (the angular equivalent of mass, expressed in kg m²), and a in the angular acceleration (in radians/sec²). The torque in this experiment is provided by the tension in the cord holding the falling mass.

The force acting on the falling mass is T - M_f g = - M_F a where a = R_S a . The torque acting on the disk is t = I a = RT. (I = I_s + I_D, I_s = 1/2 M_s R_s², and I_D = 1/2 M_D R_D² assuming both are disk) Solving the two equations for the angular acceleration a , the following results.

This value of angular, a will be compared with that obtained form the angular velocity equations.

EQUIPMENT: Computer, VideoPoint Software, graph paper and straight edge.

PROCEDURE: There are several video files from which to chose. From the "DSON" subdirectory chose the one named dosn014. With the VideoPoint software mark the position of the object as it accelerates. There is a step-by step procedure at the end of the laboratory for using VideoPoint for this experiment. You should have reviewed the introduction provided with the VideoPoint CD and the Walkthrough on the CD or the Web sight. This data can be copied into the provided data table. Record the angular position q and time t on the provided table and calculate the angular velocity and angular acceleration for each interval possible. Use this data for the following graphs.

Graph the results. Make two graphs q verses t, and w verses t. Use the graphs and information in the data table to find the following. Make only one graph per page and make the graph take up at least half of the page both horizontally and vertically. Make sure you label the graph, the axis, and indicate the units the numbers on the graph represent.

1. The slope of the angular velocity verses time graph. Draw the best straight line through the data (do not play connect the dots). The slope of the line is the rise over the run for the line drawn through the data. Remember that the numbers on the axis have units and so will the slope.

2. Find the average angular acceleration for the car for the entire experiment. Add up all of the accelerations and divide by the number of accelerations added.

3. The slope of the angular velocity verses time line should be the average angular acceleration. Find the percent error in agreement of the angular acceleration calculated from the slope of this line and the average of the angular accelerations from the table. Treat the average angular acceleration as the expected average

QUESTIONS:

1. Are the equations for angular motion and linear motion the same except using different symbols?

2. State Newton's three laws in terms of rotational quantities

3. What are the angular equivalent (analogue) of the linear terms Velocity, Position, Acceleration, Force, Momentum, and Mass?

4. What factors might cause the accelerations calculated form the forces and torque not to agree with that form the slope of the line?

5. How would the results of the experiment been changed if the mass of string connecting the falling mass to the spool had not been negligible?

Start VideoPoint.

Click left mouse button while the pointer is anywhere inside the "about VideoPoint" dialogue box.

In the "open" dialogue box do the following.

Select the drive letter of the CD drive in which VideoPoint is located.

Select movies folder and the appropriate folder under movies (here the "dson" folder).

Under filename select the appropriate file to be loaded.

Press "OK".

The "Number of points" dialogue box will appear. Make sure "1" is selected and press "OK". The movie and software should now be loaded.

On the menu bar select "Movie" and "Full Screen" using the left mouse button.

Record data -When the cursor is inside the movie screen it appears as a target. Place this target cursor on a white dot on the rotating wheel and press the left mouse button, the movie will advance one frame and the data is recorded. Repeat this process until all frames of the movie have been used.

Calibrate the movie.

Move the cursor to the "meter stick" tool button at the left of the screen (6th button from top) and click the left mouse button.

The "Scale Movie" dialogue box appears. Change the "known length" to 0.10 and Select continue with the left mouse button.

Put the target cursor on the top end of the meter stick in the first movie frame and press left mouse button - place the target cursor at the 10 cm mark of the meter stick and press the left mouse button. Place the target cursor at the 10 cm mark of the meter stick and press the left mouse button.

Relocate the coordinate axis by placing the pointer on the axis and pressing and holding the left mouse button and dragging the axis to the center of the rotating wheel.

On the menu bar select "Movie" and "Normal Size".

Graph the data.

Select "Graph" tool button on tool bar at left side of the window with left mouse button. (8th button from top) - "Plot Series" dialogue box will appear.

The horizontal axes should be "time".

The vertical axis should be "Point S1" and "angle".

The position, velocity, and acceleration options should be highlighted by clicking the left mouse button when the cursor is on the word.

Press "OK" and the graphs should appear.

Fit equations to the graphs.

With cursor on each graph click left mouse button then select the "F" flag and click left mouse button again.

By using the left mouse button select the following.

For angular position choose" type of fit" = "polynomial" and "order of fit" = "2" and "OK".

For the angular velocity select "type of fit" = "linear" and "OK".

For the angular acceleration select "type of fit" = "Average" and "OK".

For each graph the equation or average is displayed with the plots.

For position Ax² + Bx + C (A = 1/2 a ; B = w _o ; C = q _o ; and x represents to t).

For velocity Ax + B (A = a ; B = w _o ; and x corresponds to t).

For acceleration the average is the average acceleration.

These values should be compared to the corresponding values from your plots

Transferring the data

With cursor inside table area click left mouse button to select table.
Put mouse in title area of table and click and hold left mouse button and drag the

table to top of window.

Place the pointer on lower right corner of table (cursor should be double arrow) and drag the corner of the table to make the table larger so all the data can be seem.

Record the time and the angular position of data in data table.

Record the masses and radii listed it the first frame of the video.

Disk Mass M_D = __________ I_D = 1/2 M_D R_D² = _________

Disk Radius R_D = __________

Falling Mass M_F = __________ I_S = 1/2 M_S R_S² = _________

Spool Mass M_S = __________

Spool Radius R_S = __________ I = I_D + I_S = __________
 

Average angular acceleration a = ________

Angular acceleration from slope a = ________ % error __________

Angular acceleration from torque and force calculations

a = M_Fg/[I/R_S +M_FR_S] = __________ % error __________