Nonuniform ball . In the figure, a ball of mass M and radius R rolls smoothly fr

Question

Nonuniform ball. In the figure, a ball of mass M and radius R rolls smoothly from rest down a ramp and onto a circular loop of radius 0.46 m. The initial height of the ball is h = 0.38 m. At the loop bottom, the magnitude of the normal force on the ball is 2.0 Mg. The ball consists of an outer spherical shell (of a certain uniform density) that is glued to a central sphere (of a different uniform density). The rotational inertia of the ball can be expressed in the general form I = ?MR2, but ? is not 0.4 as it is for a ball of uniform density. Determine ?.

Explanation / Answer

M,R, r=0.46m, h=0.38 m N=2.0 mg

the enrgey lost is converted inti knitic energy

Mgh = 1/2Mv2+1/2Iw2

Mgh= 1/2Mv2+1/2( MR2(V/R)2

Mgh=1/2Mv2+1/2Mv2.........................................(1)

the normal force provide centripetal force and weight

2Mg=1/rMv2+Mg

Mg=Mv2/r             v = sqrt gr

v=sqrt (9.8)(0.46)           v=2.123

Egn explain (1)

gh=v2/2+/2v2       gh= v2/2(1+)

(1+)=2gh/v2      1+=2(9.8(.38)/(2.123)2

1+=1.652     =0.65  

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