Question

If the rotational inertia of a sphere about an axis through the center of the sphere is I, what is the rotational inertia of another sphere that has the same density, but has twice the radius?

Answer 1

32I

Step-by-step explanation:

Data provided in the question:

Rotational inertia of a sphere about an axis through the center = I

Now,

Let the radius of the sphere be 'R'

also,

Rotational inertia = MR²

Here,

M is the mass

Mass = Density ÷ Volume

Volume of sphere =
`(4)/(3)\pi R^3`

Therefore,

M = Density ×
`(4)/(3)\pi R^3`

Thus,

I =
`\text{Density}*{(4)/(3)\pi R^3}* R^2`

Now for the sphere of radius twice the radius i.e 2R

Volume =
`(4)/(3)\pi (2R^3)=(4)/(3)\pi*8R^3`

Since the density is same

Mass =
`\text{Density}*(4)/(3)\pi8R^3`

Thus,

I' =
`\text{Density}*{(4)/(3)\pi 8R^3}* (2R)^2`

or

I' = 8 × 4 ×
`\text{Density}*{(4)/(3)\pi R^3}* R^2`

or

I' = 32I