Rotational Inertia and Rotational Motions Second Law

Learning Objectives

1. Be able to define, rotational inertia, calculate it for point masses and understand how it changes when mass is redistributed in different shapes.

2. Be able to apply the rotational second law, \( \tau = I \alpha\) to calculate simple problems involving rotational inertia, acceleration and torque.

Key Terms

	Definition	SI Units	Symbol and Equation
Rotational Inertia	Sometimes called the “moment of inertia” it is the resistance to change in rotational velocity, proportional to the mass times the radius squared for a point mass.	\( \frac{kg}{m^2}\)	\( I=\displaystyle \sum mr^2\)
Net Torque	A force that causes an object to rotationally accelerate.	Nm	\( \sum \tau=I \alpha\)

Rotational Inertia

Inertia, or mass, if you remember is an objects ability to resist a change in acceleration. Rotational inertia, or the moment of inertia, measures an objects resistance to an angular acceleration. The equation is \( I=\displaystyle \sum mr^2\). With something simple like a ball on a string or planet orbiting a star, this equation can be used to directly calculate the rotational inertia. In most cases though, we use it as a guide to see how the rotational inertia would change if we changed or redistributed the mass on an object. Increasing the mass or the distance the mass is from the point of rotation increases the rotational inertia. More complicated objects need calculus to calculate the moment of inertia independently so these equations will be provided for you if you need them. Here are some common examples.

Important Things to Remember about Rotational Inertia.

1. The equation is \( I=\displaystyle \sum mr^2\), but can only be used directly for simple point masses like a ball at the end of a string or a planet rotating around a sun.
2. You can increase the inertia of more complex objects by moving more mass further away from the axis of rotation.
3. Rotational Inertia takes the place of mass in rotational motion equation.

Angular Second Law

Newton’s second law states that \( \sum F=ma\). In a similar way the angular form is \( \sum \tau=I \alpha\). If you add up all the forces they equal the mass times the acceleration. If you add up all the torques, they equal the rotational inertia times the angular acceleration.

Bicycle Tire And Mass Problem

Watch the video below and use the graph to answer the following questions.
(a) What is the rotational inertia of the bicycle wheel?
(b) Imagine that you decreased the radius of the wheel, but left the wheels mass and the hanging mass exactly the same. In a paragraph length response, describe one argument that could be made that the acceleration of the mass would increase with a smaller radius, and one argument that could be made that the acceleration of the mass would decrease with a smaller radius.