Question

A thin circular disc of mass M and radius R is rotating in a horizontal plane about an axis passing through its centre and perpendicular to its plane with angular velocity ω. If another disc of same dimensions but of mass M/2 is placed gently on the first disc co-axially, then the new angular velocity of the system is ______.

Options

• `4/5 omega`

• `5/4 omega`

• `2/3 omega`

• `3/2 omega`

Answer 1

A thin circular disc of mass M and radius R is rotating in a horizontal plane about an axis passing through its centre and perpendicular to its plane with angular velocity ω. If another disc of same dimensions but of mass M/2 is placed gently on the first disc co-axially, then the new angular velocity of the system is `bbunderline(2/3 omega)`.

Explanation:

Since the second disc is placed “gently” on the first, no external torque acts on the system about the rotation axis. Therefore, the total angular momentum (L) remains constant:

`L_"initial" = L_"final"`

I₁ω₁ = `I_"final" omega_2`    ...(i)

The first disc has mass M and radius R. Its moment of inertia about the central axis is:

I₁ = `1/2 MR^2`

When the second disc (mass M/2, radius R) is added co-axially, the new total moment of inertia is the sum of the two discs:

I₂ = `1/2 (M/2) R^2`

= `1/4 MR^2`

I_final = I₁ + I₂

= `1/2 MR^2 + 1/4 MR^2`

= `3/4 MR^2`

Substitute the values back into the equation (i):

`(1/2 MR^2) omega = (3/4 MR^2) omega_"new"`

`1/2 omega = 3/4 omega_"new"`

ω_new = `omega xx 4/(2 xx 3)`

= `2/3 omega`

∴ The new angular velocity of the system is `2/3 omega`.