A circular object of radius R starts from rest and rolls down an incline of height H without slipping. The object has a mass M, a moment of inertia / and a radius R. We go through the steps in finding the speed of the object at the bottom of the incline below.

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**Educational Text Transcription:**

**1. A circular object of radius \( R \) starts from rest and rolls down an incline of height \( H \) without slipping. The object has a mass \( M \), a moment of inertia \( I \), and a radius \( R \). We go through the steps in finding the speed of the object at the bottom of the incline below.**

**a.** What is the total mechanical energy of the object at the top of the incline? Write your answer in terms of \( I, M, R, H \) and/or \( g \).

**b.** What is the total mechanical energy at the bottom of the incline in terms of the speed \( v \) and the givens? Use the relation \( v = \omega R \) to eliminate the angular speed from your expression. Check that your expression has the correct dimensions. Your final expression should only contain \( v, I, M, R \) and/or \( g \).

**c.** When an object rolls without slipping, the point in contact with the ground is at rest. Therefore, the friction force has no displacement, does no work, and the mechanical energy is conserved. Use conservation of mechanical energy to find the speed \( v \) of the object at the bottom of the hill in terms of \( I, R, M, H \) and/or \( g \).

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**Physics 211**  
**The Great Race - Rolling and Rotational Kinetic Energy**  
**Rotation 6 - 2**

**d.** Assume the object is a hoop of radius \( R \). Determine the speed at the bottom in terms of \( M, H, \) and \( g \).

**e.** Assume the object is a solid disk of radius \( R \). Determine the speed at the bottom in terms of \( M, H, \) and \( g \).

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**Explanation of Diagram:**

The diagram shows a circular object on an inclined plane. The object has a radius \( R \) and is moving along the incline with a mass \( M \) and moment of inertia \( I \). The incline has a height \( H \). The object rolls without slipping down the incline.
Transcribed Image Text:**Educational Text Transcription:** **1. A circular object of radius \( R \) starts from rest and rolls down an incline of height \( H \) without slipping. The object has a mass \( M \), a moment of inertia \( I \), and a radius \( R \). We go through the steps in finding the speed of the object at the bottom of the incline below.** **a.** What is the total mechanical energy of the object at the top of the incline? Write your answer in terms of \( I, M, R, H \) and/or \( g \). **b.** What is the total mechanical energy at the bottom of the incline in terms of the speed \( v \) and the givens? Use the relation \( v = \omega R \) to eliminate the angular speed from your expression. Check that your expression has the correct dimensions. Your final expression should only contain \( v, I, M, R \) and/or \( g \). **c.** When an object rolls without slipping, the point in contact with the ground is at rest. Therefore, the friction force has no displacement, does no work, and the mechanical energy is conserved. Use conservation of mechanical energy to find the speed \( v \) of the object at the bottom of the hill in terms of \( I, R, M, H \) and/or \( g \). --- **Physics 211** **The Great Race - Rolling and Rotational Kinetic Energy** **Rotation 6 - 2** **d.** Assume the object is a hoop of radius \( R \). Determine the speed at the bottom in terms of \( M, H, \) and \( g \). **e.** Assume the object is a solid disk of radius \( R \). Determine the speed at the bottom in terms of \( M, H, \) and \( g \). --- **Explanation of Diagram:** The diagram shows a circular object on an inclined plane. The object has a radius \( R \) and is moving along the incline with a mass \( M \) and moment of inertia \( I \). The incline has a height \( H \). The object rolls without slipping down the incline.
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