Question

Consider a solid sphere and a solid disk with the same radius and the same mass. Explain why the solid disk has a greater moment of inertia than the solid sphere, even though it has the same overall mass and radius.

Answer 1

Answer:

Explanation:

In a Solid sphere; the moment of inertia around its geometrical axis can be expressed by using the formula:

$\mathtt{I_s = \dfrac{2}{5} M_s R^2_s}$

For the solid disk; the moment of inertia around the central axis is:

$\mathtt{I_D= \dfrac{1}{2}M_DR_D^2}$

Suppose $M_D = M_S$; then we can say both to be equal to M

As well as $R_D = R_S$; then that too can be equal to R

Now;

$\mathtt{I_s = \dfrac{2}{5} M R^2} --- (1)$

$\mathtt{I_D= \dfrac{1}{2}MR^2}---(2)$

Multiplying equation (1) by 2, followed by dividing it by 2; we have:

$\mathtt{I_s= \dfrac{2}{5}MR^2} \times \dfrac{2}{2}$

$I_s = \dfrac{4}{5} \times \dfrac{1}{2}MR^2 \\ \\ I_s = \dfrac{4}{5}\times I_D \\ \\ I_s > I_D$