Friday 2 March 2012
Class meeting
Topics
23
Torque
Textbook sections 10-10
Ponderables
Mini-labs (deliv.) Rotational inertia (data analysis)
Lab
Demonstrations
Mini-lectures
Quiz
Other
Ball hits block
PHYS 116 SCALE-UP

Mini-lab: Rotational inertia o 6 inertia wheels with string and hanging mass; for each group: meter stick, 2 stop watches, Vernier caliper

Quiz: Ball hits block
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Friday 2 March 2012
PHYS 116 SCALE-UP
1.
Derive the equation for the linear acceleration of a falling mass m suspended by a string from the rim of a uniform disk of mass M and radius R that is free to rotate without friction about its principal axis.
2.
If the hanging mass is doubled, what will happen to the angular acceleration of the wheel?
In your study of physics, you have already learned that objects in motion tend to maintain a constant velocity unless acted upon by an external force. Similarly, an object that can rotate will continue to rotate at a onstant angular velocity unless acted upon by an external torque. In this lab you will examine the concepts of rotational inertia ( I ), angular acceleration (

), and torque (

). The most fundamental relationship among these variables is Newton’s second law for rotation:
 
 net
The moment of inertia ( I ) of an object depends on the mass of the object and (most importantly) its distribution from the axis of rotation. The moment of inertia for discrete or continuous mass distributions can be calculated from:
I
 n  i

1 m i r i
2   r
2 dm where m is the mass and r is the distance of that mass from the rotational axis.
In this lab, we will be using a wheel that consists of 3 uniform disks that share the same principal axis.
From the above equation, it can be shown that the moment of inertia for a single uniform disk of mass M and radius R is
I disk

1
2
MR
2
The total moment of inertia of the wheel is simply the sum of the moments of each of the disks:
I wheel

1
2

M
1
R
1
2 
M
2
R
2
2 
M
3
R
3
2

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Friday 2 March 2012 PHYS 116 SCALE-UP
Since the wheel is constructed as a solid unit, we cannot directly measure the mass of each of the individual disks. However, we can find the density (

) of the wheel from the mass and the volume of the entire wheel, and then (assuming uniform density) the mass of each disk can be expressed as
i
 
i
Since the diameter (D) and length (L) of each cylindrical disk can be measured directly using a Vernier caliper, the volume of each disk is
i
 


i
2


2
i so in terms of the quantities that can be directly measured, the density of the wheel is
 
M
V

  i
4 M
L i
D i
2 therefore, the moment of inertia of the entire wheel can be expressed as:
I wheel

1
32
  i
L i
D i
4
or I wheel

M
8
 i
 i
L i
D i
4
L i
D i
2
In this experiment, we will use a mass hung from a string connected to the rim of the wheel to apply an external torque that will make the wheel rotate. The wheel rotates about its principal axis on metal points that allow it to spin freely but not completely without friction. The friction in these axles produces a torque that opposes the torque applied by the hanging mass. Newton’s second law for the wheel is then:
 string
  friction

I

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Friday 2 March 2012 PHYS 116 SCALE-UP
Since the string is tangent to the rim of the wheel, the tension in the string acts perpendicular to the radius of the wheel, so the torque produced by the hanging mass is simply the product of the tension times this radius.
 string

TR
Applying Newton’s second law to the hanging mass, we have
T
 mg
  ma
So the torque applied by the hanging mass via the string is
 string
 m ( g
 a ) R
With


, the moment of inertia can be expressed in terms of the measured parameters:
I
 m ( g
 a ) R
2 a
  friction
R
DO NOT REMOVE THE WHEEL FROM THE AXLE.
Doing so can damage the wheel’s bearings.
Part 1 – Direct calculation of rotational inertia
1. Use a Vernier caliper to measure the width L and diameter D of each of the three disks. Hint: Use the depth gauge on the caliper to measure L
1
and L
3
.
2.
The wheel used in this experiment is made of steel that has a density of 7.8 g/cm 3 . From your measurements of the wheel’s dimensions and the value of its mass that is stamped on the side of the wheel, calculate the density of your wheel and compare with this expected density (they should agree).
3.
Calculate the moment of inertia for each of the disks and find the total moment of inertia of the wheel.
Part 2 – Measuring acceleration
1.
The wheel that you will use in this experiment is designed to spin freely, but friction is still a factor that must not be ignored. Before you begin to take data, check to make sure your wheel does spin freely. When the axles are properly adjusted, the wheel should rotate for several seconds after gently spinning it with your hand. Most likely your wheel will not need adjustment, but if it does not spin freely, carefully adjust the axle bolts until the wheel has a slight amount of sideways movement but is still held securely by the axle points. When adjusting the axle bolts be careful not to damage the delicate axle points!
When you are satisfied that the wheel spins with a minimal amount of friction, hand-tighten the axle nuts to lock the axles. Once you begin taking
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Friday 2 March 2012 PHYS 116 SCALE-UP data, do not re-adjust the axles since this will change the frictional torque that we will assume to be constant throughout the experiment.
2.
Suspend a 50 g mass hanger from a string tied to the rim of the wheel (you will be instructed as to which of the disks to use) and align the bottom of the hanger with the bottom of the table top (or some other convenient and well-defined position). Use a stopwatch to measure the time for the hanging mass to fall a vertical distance h. Measure and record this height. The average acceleration during this time interval can be found from the equation for uniform acceleration: h
  y

1
2 at
2
3.
Repeat this measurement several times to find an average time of fall for a fixed height and applied torque. Use the standard error to estimate the uncertainty in the time. (Efficiency hint: If each lab partner uses a stopwatch, multiple time measurements can be made simultaneously for each trial.)
4.
Repeat the experiment using different hanging masses to obtain several different torque values in the range of 0.02 to 0.2 N-m.

Calculate
 string
and

for each of your measurements, together with their uncertainties.

Share your values with two other groups who used different-diameter disks so you have a
 complete set of measurements of all three disks for one wheel.
Plot a graph of
 string
versus

using all of the measurements (yours and those of the other two groups) and including error bars to find the moment of inertia of the wheel together with its uncertainty. What is the y-intercept of this graph, and what physical quantity does it represent?

Compare your experimental value of the moment of inertia of the wheel with the value you calculated in part 1 of the experiment.
The deliverable for this lab is the data analysis described above. Use Excel to plot the data and perform an appropriate linear regression, interpreting both the slope and the intercept. Use the techniques you learned in the Uniform Circular Motion lab to determine uncertainties in the quantities represented by the slope and intercept. Use the “comments” capability in Excel to lead the grader through the steps of your analysis, and include a brief summary discussion as well.
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Friday 2 March 2012
QUIZ
PHYS 116 SCALE-UP
A metal ball of mass M
A
hangs from a cord of length R. If the ball is held horizontal and released from rest, at the bottom of its swing it will strike a block of mass M
B
sitting on a frictionless surface. If the collision is totally elastic, what are the velocities of the ball and the block immediately after the collision?
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