Question

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 kgcore, 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)Model a real bowling ball as a 0.196-m-diameter core with mass 1.6 kg plus a thin 1.6 kg shell with diameter 0.206 m (the average of the inner and outer diameters). What is the total moment of inertia? Express your answer with the appropriate units. (b)Find the moment of inertia of a uniform 3.2 kg ball with diameter 0.216 m. Express your answer with the appropriate units.

Answer 1

(a) The total moment of inertia which will be given as -

I_total = I_ball + I_shell

I_total = (2/5) M R² + (2/3) m r²

I_total = [(2/5) (1.6 kg) (0.098 m)²] + [(2/3) (1.6 kg) (0.103 m)²]

I_total = [(0.00614 kg.m²) + (0.01131 kg.m²)]

I_total = 0.01745 kg.m²

I_total = 1.74 x 10^-2 kg.m²

(b) The moment of inertia of a uniform ball which will b given by -

I = (2/5) M R²

where, M = mass of ball = 3.2 kg

R = radius of ball = 0.108 m

then, we get

I = (2/5) (3.2 kg) (0.108 m)²

I = 0.0149 kg.m²

I = 1.49 x 10^-2 kg.m²