A solid uniform sphere of mass and negligible radius compared with the track is…

PHYSICS · Multi-part · AP · Physics 1: Algebra-based · Force and Translational Dynamics · Force and Translational Dynamics

Problem

A solid uniform sphere of mass

m

and negligible radius compared with the track is released from rest at a vertical height

h

above the bottom of a smooth track that includes a vertical circular loop of radius

R=1.0\,\text{m}

. The sphere rolls without slipping at all times, and energy losses are negligible. (a) Derive an expression for the speed

v_{\text{top}}

of the sphere’s center of mass at the top of the loop in terms of

h

g

, and

R

. (b) Using the condition that the normal force on the sphere is zero at the top of the loop when it is on the verge of losing contact, determine the minimum release height

h_{\text{min}}

required for the sphere to complete the loop without losing contact. (c) Express

h_{\text{min}}

as a multiple of the loop radius

R

, and state the numerical value for

R=1.0\,\text{m}

. Give your answer to two significant figures.

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