Question

Question #25b65

Answer 1

Option (3) only if `omega_o=omega_s\ i.e.,`
Orbital angular velocity is equal to spin angular velocity.

Explanation:

Let the spherical satellite of mass `m` be spinning about an axis passing through its center with angular velocity `omega_s`.
Its spin angular momentum `L_s` is is given as

`L_s=Iomega`
where `I` is its moment of inertia and for a sphere `=2/5ma^2`

`:.L_s=2/5ma^2omega_s` .........(1)

Orbital angular momentum `vecL_o` of satellite, considered a point mass, revolving in a circle of radius `r` is given as the cross product of the position vector `vecr` and its linear momentum `vecp = m vecv`.

`vecL = vecr × vecp`

Let `omega_o` be orbital angular velocity. We have `v=bomega_o`.
Orbital velocity is always perpendicular to the radius vector hence `sin theta` of the cross product is `=1`, and `r=b`, we get

`L_o = b^2 momega_o` .........(2)

Ratio of orbital angular momentum to spin angular momentum is

`L_o/L_s=(b^2 momega_o)/(2/5ma^2omega_s)`
`L_o/L_s=(5b^2 omega_o)/(2a^2omega_s)`