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

Step-by-step explanation:

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

`\mathtt{I_s = (2)/(5) M_s R^2_s}`

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

`\mathtt{I_D= (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 = (2)/(5) M R^2} --- (1)`

`\mathtt{I_D= (1)/(2)MR^2}---(2)`

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

`\mathtt{I_s= (2)/(5)MR^2} * (2)/(2)`

`I_s = (4)/(5) * (1)/(2)MR^2 \\ \\ I_s = (4)/(5)* I_D \\ \\ I_s > I_D`