Question 6 Nonuniform ball In the figure a ball of mass M an

Question 6

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 0.50 m. The initial height of the ball is h = 0.39 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 ?.

Solution

As the ball rolls down the circular loop at the bottom of the track ,it experiences centripetal force .

N-W=MV²/r

2Mg-Mg=MV²/r

=>V=sqrt[rg] =sqrt[0.5*9.8]

V=2.21 m/s

By COnseration of energy

(1/2)MV²+(1/2)IW² =Mgh

(1/2)MV²+(1/2)(BMR²)(V/R)² =Mgh

(1/2)*2.21²+(1/2)*B*2.21² =9.8*0.39

B=0.56