Question

(1) A bowling ball is far from uniform. Lightweight bowling balls are made of a relatively low-density core surrounded by a thin shell with much higher density. A 7.0 lb (3.2 kg) bowling ball has a diameter of 0.216 m; 0.196 m of this is a 1.6 kg core, surrounded by a 1.6 kg shell. This composition gives the ball a higher moment of inertia than it would have if it were made of a uniform material. Given the importance of the angular motion of the ball as it moves down the alley, this has real consequences for the game.

(A) Compare the moments of inertia for these models by finding the ratio of Ir and Iu.

Answer 1

Moment of inertia of non uniform ball (I_r) = moment of inertia od solid + moment of inertia of hollow
= (2/5)mr² +(2/3)mR²
where r is radius of solid part = 0.196/ 2 = 0.098 m
R is radius of hollow part = 0.216/2 = 0.108 m
= [(2/5)*1.6*0.098²] +[(2/3)*1.6*0.108²]
= 0.0061 + 0.0124 = 0.01854 kg-m²
Now for uniform ball
Moment of inertia (I_u) = (2/5)MR² = (2/5)*3.2*0.108² = 0.01493 kg-m²  
Now the ratio
I_r/I_u =  0.01854 / 0.01493 = 1.242
Hence for the non uniform ball the moment of inertia will be 1.242 times the uniform ball.