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Rotational Inertia and Rotational Motions Second Law – mrsmithphsyics.com
 Openstax Textbook 10.3
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
DefinitionSI UnitsSymbol and EquationRotational InertiaSometimes 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 TorqueA 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. 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. Openstax Sample Problem 10.3 #14Suppose you exert a force of 180 N tangential to a 0.280-m-radius 75.0-kg grindstone (a solid disk).(a)What torque is exerted?(b) What is the angular acceleration assuming negligible opposing friction?(c) What is the angular acceleration if there is an opposing frictional force of 20.0 N exerted 1.50 cm from the axis?Video Solution Bicycle Tire And Mass ProblemWatch the video below and use the graph to answer the following questions.(a) What is the rotational inertia of the bicycle wheel?(b) If you decreased the radius of the wheel, but left the wheels mass and the hanging mass exactly the same, how would the velocity time graph below be different. Explain your reasoning in a paragraph length response. 
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