7.2a - Rotational Inertia

10 min of Qualitative/ Quantitative Data Collection

Using the materials given, answer the following questions:
1. How does changing the distribution of the same total mass around the axis of rotation affect the angular acceleration of the system when subjected to a constant torque?
2. If you keep the shape of the rotating object the same, but change its mass, how does this affect the angular acceleration of the system under the influence of the same constant torque?
3. For a rotating object with a fixed mass and distribution, how does changing the radius at which the string is wound on the spindle of the apparatus affect the resulting angular acceleration?
Sketch a graph to represent the relationship for each of the relationships above.
How does this help with the creation of an equation to describe angular acceleration (α)?

AP Style Question:

A group of AP Physics 1 students is conducting the rotational inertia experiment described earlier, using a Vernier Rotational Motion Apparatus, a constant falling mass, and varying the rotating object. They perform two trials:

Trial 1: A solid aluminum disk of mass M and radius R is attached to the apparatus.
Trial 2: An aluminum hoop of the same mass M and the same outer radius R is attached to the apparatus.

Both trials use the same falling mass and the string is wound around the same spindle radius on the rotational motion sensor. The students collect angular velocity vs. time data for both trials and determine the angular acceleration (α) for each. They observe that the angular acceleration of the hoop is smaller than the angular acceleration of the disk.

Two students, Aisha and Omar, are discussing these results and trying to relate them to the concept of rotational inertia (I).

Aisha's Reasoning and Equation:

Aisha believes that since both objects have the same mass and the same outer radius, their resistance to rotational motion (rotational inertia) should be the same. She reasons that the total amount of "stuff" is the same, and the overall size is the same, so it should take the same effort (torque) to get them rotating at the same rate.

Aisha proposes the following equation to the right torelate the applied torque (τ), rotational inertia (I), and angular acceleration (α):

Where:

α is the angular acceleration.
τ is the constant torque provided by the falling mass (which Aisha assumes is the same for both trials).
M is the mass of the rotating object.
R is the outer radius of the rotating object.
k is a constant that Aisha believes should be the same for both the disk and the hoop since they have the same mass and radius.

Aisha argues that since M, R, and τ are the same for both trials, her equation predicts that α should also be the same, which contradicts the experimental results. She concludes that there must be an error in the data collection.

Omar's Reasoning and Equation:

Omar disagrees with Aisha. He recalls learning that rotational inertia depends not only on the total mass and size but also on how that mass is distributed relative to the axis of rotation. He believes the hoop has a larger rotational inertia than the disk because its mass is concentrated further away from the center.

Omar proposes the following equation to the right:

Where:

α is the angular acceleration.
τ is the constant torque provided by the falling mass (which Omar agrees is the same for both trials).
M is the mass of the rotating object.
R is the outer radius of the rotating object.
c is a dimensionless constant that depends on the mass distribution of the object. Omar believes that c will be different for the disk and the hoop.

Omar argues that for the solid disk, the mass is distributed closer to the axis of rotation, resulting in a smaller rotational inertia and therefore a larger value of c in his denominator (making α larger). For the hoop, the mass is distributed further from the axis, leading to a larger rotational inertia and a smaller value of c (making α smaller). This aligns with the experimental observation that the hoop has a smaller angular acceleration.

Questions to Consider:

Analyze Aisha's reasoning: What is the flaw in Aisha's assumption about rotational inertia? Why might her equation not accurately represent the relationship between torque, rotational inertia, and angular acceleration for different shapes?
Analyze Omar's reasoning: How does Omar's concept of mass distribution explain the difference in angular acceleration observed in the experiment? Why is his introduction of the constant 'c' a more appropriate way to model the rotational inertia in this scenario?
Relate to known formulas: How do the known formulas for the rotational inertia of a solid disk and a hoop support Omar's reasoning? What would be the values of 'c' in Omar's equation for the disk and the hoop based on these formulas (assuming τ=Iα)?
Qualitative to Quantitative Translation: Explain how the qualitative observation (the hoop accelerates slower) directly translates to a quantitative difference in their rotational inertias, as captured by Omar's equation with different values of 'c'.

Aisha's Equation:

Omar's Equation:

Translational to Rotational Comparision

inertia: the tendency for an object to continue to do what it is doing (remain at rest or remain in motion).

rotational inertia (or angular inertia): the tendency for a rotating object to continue rotating. moment of inertia (I): a quantitative measure of the rotational inertia of an object.

Moment of inertia: is measured in units of kg·m2. Inertia in linear systems is a fairly easy concept to understand. The more mass an object has, the more it tends to remain at rest or in motion, and the more force is required to change its motion. I.e., in a linear system, inertia depends only on mass.

Translational Acceleration

Rotational Acceleration

The translational acceleration of an object is caused by the sum of the forces and limited by the mass of the system.

The rotational accleration of an object is caused by the sum of the torques and limited by the rotational inertial of the system.

Rotational inertia is somewhat more complicated than inertia in a non-rotating system. Suppose we have a mass that is being rotated at the end of a string. (Let’s imagine that we’re doing this in space, so we can neglect the effects of gravity.) The mass’s inertia keeps it moving around in a circle at the same speed. If you suddenly shorten the string, the mass continues moving at the same speed through the air, but because the radius is shorter, the mass makes more revolutions around the circle in a given amount of time.

In other words, the object has the same linear speed (not the same velocity because its direction is constantly changing), but its angular velocity (degrees per second) has increased.

This must mean that an object’s moment of inertia (its tendency to continue moving at a constant angular velocity) must depend on its distance from the center of rotation as well as its mass.

The rotational inertia of an object depends on its geometry. Typically, this is described as a fraction, some common shapes are show to the right. The derivation of the values is beyond the courses.

You do NOT need to memorize these relationships!

Find the moment of inertia of each of the following objects.

I = 0.36 kg m^2

I = 0.128 kg m^2

I = 0.5 kgm^2

I = 1 kg m^2

Sample Problem #1

Sample Problem #2

A solid brass cylinder has a density of 8500 kg m^-3, a radius of 0.10 m and a height of 0.20 m and is rotated about its center cylinder axis.

Determine the volume of the cylinder. Ans: 6.2x10^-3 m^3

Determine the mass of the cylinder. Ans: 53.4 kg

Using the table above determine the formula for the moment of inertia, determine its moment of inertia? Ans: 0.534 kg m^2

A solid sphere of mass m is fastened to another sphere of mass 2m by a thin rod with a length of 3x. The spheres have negligible size, and the rod has negligible mass. What is the moment of inertia of the system of spheres as the rod is rotated about the point located at position x, as shown?

a. 3 mx^2 b. 4 mx^2 c. 5 mx^2

d. 9 mx^2✓ e. 10 mx^2

Sample Problem #3

Shown below in a top view are six uniform rods that vary in mass (M) and length (L). Also shown are circles representing a vertical axis around which the rods are going to be rotated in a horizontal plane and arrows representing forces acting to rotate the rods. The forces change direction in order to always act perpendicular to the rods. Specific values for the lengths and masses of the rods and the magnitudes of the forces are given in each figure.

Rank these rods, from greatest to least, on the basis of the magnitude of their angular acceleration. That is, put first the rod that has the largest angular acceleration and put last the one that will have the smallest angular acceleration.

Ans: C=9, E=9/4, A=3/2, B=3/4, F = 9/16, D=0

Sample Problem #4:

Given the net torque of the system, find the angular acceleration for the system of three particle masses. The radius from the axis of rotation is 12 m and the masses are equidistant.

Determine the angular acceleration of the system. α=-1.08rads/s^2
Assuming a constant acceleration, what would the angular velocity be after 5 seconds? ω = - 5.4 rads/s
How many revolutions will be completed after 20 seconds with constant acceleration? θ = - 216 rads

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