Angular Acceleration (last edited December 29, 2011)

Dr. Larry Bortner

 Learning Objectives

Use digital calipers to measure dimensions.

Explain how a three-beam balance can measure masses greater than one kilogram.

Define moment of inertia.

Discuss the rotational analog of Newton’s second law.

 Purpose

To verify Newton’s second law for rotational motion by finding the moments of inertia of a disk and a ring.

 Background

 

 Newton's Third Applied to Rotation

The rotational analog to F = ma  is

 

(1)      

 

where the torque τ is a rotation-producing quantity, the moment of inertia I is a measure of an object’s resistance to a change in rotational motion, and α is the angular acceleration. To talk about circular motion, one must indicate an axis about which this motion is occurring. At a point a perpendicular distance r from this axis, a force F applied at right angles to both the axis and to r results in a torque τ = rF . Likewise, the angular acceleration α is related to the linear acceleration a of the point by α = a/r. The mass is a defining property of an object that is a measure of its resistance to a change in linear motion and does not change (in the object’s rest frame). The moment of inertia of an object however, can vary, depending on the distance to the rotational axis as well as the orientation, size, and shape of the object.

 Theoretical Moments of Inertia

For an axis going through the center of mass and perpendicular to the circular shape, the theoretical moments of inertia for a disk and ring are:

 

(2)         

 

where the r’s are the appropriate radii.

 Forces, Friction, Acceleration, and Torque

In this experiment we hang a mass m with a string that is wrapped around a pulley of radius r on the vertical shaft of the rotational apparatus. When let go, the mass accelerates to the floor, causing the disk to rotate. The tension T in the string provides the torque that causes this rotation. Assuming a small frictional force f, Newton’s second law applied to the mass gives

 

(3)        

 

The torque applied to the shaft would be, after solving Eq. 3 for T and multiplying both sides by r,

 

(4)        

 

In a frictionless world the ideal torque is

 

(5)         

 

but no matter how well-designed a piece of mechanical equipment is, friction enters in. We could measure the friction and account for it, but that is not the main purpose of the experiment. We want to find an experimental value for the moment of inertia of the object rotating about the shaft.

 

The torque in Eq. (1) has to be the actual torque. This is related to the ideal torque in Eq. 4. Combining the two equations and solving for the ideal torque, we have

 

(6)          

 

This means that if we plot the ideal torque calculated from Eq. 5 (let’s just call it τ from here on) versus the measured angular acceleration, we would get a straight line with a slope that is the moment of inertia and a positive y-intercept that is the frictional force in the shaft times the radius of the pulley.

 Moments of Inertia Are Additive

As with multiple masses connected and moving together as a single object, the effective moment of inertia of multiple objects rotating about the same axis is the sum of the individual moments of inertia. In particular, the moment of inertia of the ring sitting on the disk is the moment of inertia of the ring plus the moment of inertia of the disk.

 Procedure

You need the following items:

*        digital calipers

*        3-beam balance with 1-kg add-on

*        computer with Science Workshop interface

*        rotating platform base

*        rotary motion sensor

*        9" plastic disk

*        4" steel ring

 

1.     Click on Start> Science Workshop Experiments> First Quarter> Angular Acceleration.


2.     Measure, record, and estimate the uncertainties in the following quantities:

a.     The mass of the disk mdisk in kg.

b.     The mass of the ring mring in kg.

Both of these masses are greater than 1 kg, the measuring limit of the standard balance. We have to use a 1 kg add-on weight which allows us to measure masses from 1 to 2 kg to the nearest 0.1 g. The actual weight of the add-on is less than 1 kg, but this is a case of the little skinny kid being able to balance out the big fat kid who is closer to the center by going way out at the end of the teeter-totter. Add 1 kg to the readings of the scale; you do not add the actual weight of the add-on.

 

c.     The diameter in mm of the middle pulley on the axle about which the string wraps dstring , with the string wrapped around it (see Figure 1).

 

a.     angular acceleration 1

Figure 1 Measuring the middle pulley diameter.

 

d.     The diameter in mm of the middle pulley dpulley , the bare plastic part without string.

e.     The diameter in mm of the circular part of the hole in the disk where the axle comes through dhole. (See Figure 2.)

 

b.     angular acceleration 2

Figure 2 Measuring the diameter of the hole in the disk.

 

 


f.      Since the disk has a hole in it, it’s really a big fat ring. Define the distance z as the wall thickness of this ring. Measure z.

g.     The inner diameter of the ring dinner in mm.

h.     The outer diameter of the ring douter in mm.

3.     Place the disk on the center shaft of the rotating platform base, with the groove facing up.  Place the ring in the groove. The first part of this experiment is to find the moment of inertia of the disk and ring combined, about the mutual axis of rotation.

 

angular acceleration 3

Figure 3 Falling mass that rotates the disk and ring.

 

4.     Place the string with the 50-g mass hanger over the edge of the pulley on the horizontal shaft as in Figure 3.

a.     Record the value of the total mass hanging over the edge.

b.     If necessary, rotate the disk counterclockwise to bring the top of the hanger just below the pulley.

c.     Make sure that the string is wrapped around the middle pulley only and that the ring is centered on the disk.

d.     Hold on to the disc to keep it from rotating The mass should not be swinging.

5.     Let go of the disk. 

a.     It should start spinning in a clockwise direction, as viewed from the top.

b.     If it is spinning counterclockwise, you wound the string in the wrong direction around the pulley. If this is the case, let the mass fall all the way to the bottom and allow the disk to keep spinning so that it winds the string in the correct direction.

6.     Click on the Start button at the top of the DataStudio window.

a.     This DataStudio program takes five data points a second. It includes a plot of angular velocity vs. time as the mass falls. This should be a straight line with a positive slope (going up).

i.      If the line is going down, your rotating object is spinning counterclockwise. See 5.b.

b.     In addition to the plot, there are four digital displays that show the following:

i.      The last measured value of the angular acceleration. (We ignore this number but it is needed so that the other numbers will display.)

ii.     A running average of these angular accelerations for a particular run.

iii.    The standard deviation of these values.

iv.    The number of measurements.

c.     Click on the Stop button then grab the disk to stop its rotation, before the weight hits the floor.

7.     Record the run number, the mean and the standard deviation of the angular acceleration, and the number of trials.

8.     Repeat steps 4 − 7 for the following masses in g: 60, 70, 80, 90, 100, 110, 120, 130, 140, & 150. This gives you at least 11 mass-angular acceleration pairs.

9.     Remove the ring. Repeat steps 4 − 8 to determine the moment of inertia of the disk alone. Because of the limited data buffer, you may have to delete some runs in DataStudio.

10.  Minimize the DataStudio window. You may need to refer to the raw data, despite what you may have written down.

11.  Close DataStudio at the end of class. Don’t save the activity.

 Analysis

You need to propagate the uncertainties in these values:

*        rinner

*        rdisk

*        τideal (assume (g-rα) is constant)

*        Iring

 

 


 

1.   Click on Excel Templates> First Quarter> Angular Acceleration.


 

2.   Save the empty template with the standard file name.

3.   Enter in all of your data and uncertainties into the shaded cells.

4.   Calculate the following radii in meters:

a. The effective radius for the torque provided by the falling mass.

b. The radius of the disk.

c. The inner radius of the ring.

d. The outer radius of the ring.

5.   Propagate the uncertainties of the calculations in steps 4.b through 4.d.

6.   Calculate these uncertainties. Use the standard error of dstring and dpulley to find u{r}.

7.   Enter in the appropriate Excel formulas for the masses that went over the edge (unit conversion) and the torques (using Eq. (5)).

a. Use units of N·m.

b. Use the scientific number format.

8.   Calculate the standard error u{α} for each mass.

9.   A good approximation to the uncertainty in the torque is u{τ}= τ·u{r}/r . Calculate this in the appropriate column.

10.The plots of torque vs. angular acceleration for both objects are automatically drawn on the same graph, with trendlines and error bars. You need to include the equations.

11.  Calculate the theoretical moments of inertia of the two objects (the disk and the ring) and their sum.

a.    The disk has a hole in it, so it’s really a fat ring. Use the ring formula to find its moment of inertia.

b.    Propagate the errors of these three values then calculate them.

12.  For the experimental moments of inertia, do a least squares fit with errors for both sets of data (τ vs. α). Format the slope and its error so that there is only one significant digit in the error and the two quantities have the same number of decimal places

13.  Compare the experimental values with their theoretical counterparts of the moments of inertia of the disk and the ring.

14.  Report the values in your experiment of the frictional force in gram-weights retarding the disk and ring and the frictional force in gram-weights retarding the disk alone.

15.  In the Systematic Error section of your Error Analysis, be sure to comment on how taking these things into account would change your results:

a.    The groove in the disk.

b.    The metal axis and the pulley fixture that also rotated.

 Questions

1.     A ramp is placed on a table top so that the bottom is hanging over the edge. A marble and a golf ball are released from rest at the same time on the ramp, at the same height. (The contact point for each is at the same height.) Assuming they roll straight down the ramp without slipping, following parallel paths, which reaches the bottom first? Or do they both reach the bottom at the same time? Why?

2.     A filled cylindrical can and an empty can the same size are released from rest on the same ramp at the same height. Assuming they roll straight down the ramp without slipping, following parallel paths, which reaches the bottom first? Or do they both reach the bottom at the same time? Why?