The moment of inertia of a uniform solid sphere (mass \( M \), radius \( R \)) about a diameter is \( \dfrac{2}{5}MR^2 \). The sphere is placed on an inclined plane (angle \( \theta \)) and released from rest.(a) Determine the minimum coefficient of friction \( \mu \) between the sphere and plane with which the sphere will roll down the incline without slipping. Write your equation in terms of \( \theta \) and any other fundamental constants as needed. , (b) If \( \mu \) were zero, would the speed of the sphere at the bottom be greater, smaller, or the same as in part a? Explain your answer. :

\(\mu = \frac{2}{7}\tan\theta\)\n\(\text{Greater}\)

The moment of inertia of a uniform solid sphere (mass \( M \), radius \( R \)) about a diameter is \( \dfrac{2}{5}MR^2 \). The sphere is placed on an inclined plane (angle \( \theta \)) and released from rest. • Nerd Notes

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AP Physics 1

6.5 Rolling

6.1 Rotational Kinetic Energy

5.6 Newton’s Second Law in Rotational Form

5.4 Rotational Inertia

3.4 Conservation of Energy

2.7 Kinetic and Static Friction

2.5 Newton’s Second Law

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The moment of inertia of a uniform solid sphere (mass \( M \), radius \( R \)) about a diameter is \( \dfrac{2}{5}MR^2 \). The sphere is placed on an inclined plane (angle \( \theta \)) and released from rest.

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