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

Answer:

Part a)

$I = 17.4 \times 10^{-3} kg m^2$

Part b)

$I = 14.9 \times 10^{-3} kg m^2$

Explanation:

Part a)

Moment of inertia of the core of the ball

$I_1 = \frac{2}{5}m_1r_1^2$

$I_1 = \frac{2}{5}(1.6)((\frac{0.196}{2})^2)$

$I_1 = 6.14 \times 10^{-3} kg m^2$

now the moment of inertia for thin shell

$I_2 = \frac{2}{3} m_2r_2^2$

$I_2 = \frac{2}{3}(1.6)((\frac{0.206}{2})^2)$

$I_2 = 11.3 \times 10^{-3} kg m^2$

now total inertia of the ball is given as

$I = I_1 + I_2$

$I = 17.4 \times 10^{-3} kg m^2$

Part b)

Moment of inertia of uniform ball of mass 3.2 kg

$I = \frac{2}{5} mr^2$

$I = \frac{2}{5}(3.2)((\frac{0.216}{2})^2)$

$I = 14.9 \times 10^{-3} kg m^2$