Two solid uniform spheres, Sphere A and Sphere B

PHYSICS · Multi-part · AP · Physics 1: Algebra-based · Energy and Momentum of Rotating Systems · Energy and Momentum of Rotating Systems

Problem

Two solid uniform spheres, Sphere A and Sphere B, have equal mass

M

but different radii such that

R_{A}=2R_{B}

. A thin spherical shell, Sphere C, has the same radius as Sphere A and the same mass

M

. The rotational inertias about the centers of mass are

I_{\text{solid}}=\tfrac{2}{5}MR^{2}

for a solid sphere and

I_{\text{shell}}=\tfrac{2}{3}MR^{2}

for a thin shell. All three spheres are released from rest at the same height

h

above the bottom of a rough incline that allows rolling without slipping. (a) Derive a general algebraic expression for the magnitude of the translational speed

v

of any object that rolls without slipping from rest down a vertical distance

h

, expressing your result in terms of

g

h

, and the dimensionless constant

k

I=kMR^{2}

. (b) Use your result from part (a) to calculate the numerical value of

v

for Spheres A, B, and C. Rank the speeds and justify your ranking. (c) After reaching the bottom, each sphere continues without slipping up a second identical incline until it momentarily stops. Compare the maximum vertical height reached by each sphere on the second incline to the original release height

h

, and explain any differences among the three spheres.

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