Question

Three-point masses each of mass 'm' are kept at the corners of an equilateral triangle of side 'L' The system rotates about the center of the triangle without any change in the separation of masses during rotation. The period of rotation is directly proportional to ______.

`(cos 30^circ = sin 60^circ = sqrt3/2)`

Options

• `sqrt"L"`

• L^3/2

• L

• None of the above

Answer 1

None of the above

Explanation:

The situation can be shown below,

where, O is the centroid of equivalent triangle. The length AO can be found by

AO = `sqrt("OD"^2 + "AD"^2)`

= `sqrt((1/3 "BD")^2 + "AD"^2)`

= `sqrt(1/9 xx (sqrt3/2 "L")^2 + "L"^2/4)`

`= sqrt(1/12 "L"^2 + "L"^2/4)`

`= "L"/sqrt3`

The moment of inertia about O is given by

I = MR²

= `"m" xx ("L"/sqrt3)^2`

`"mL"^2/3`    ....(i)

According to law of conservation of angular momentum,

lω = constant

`"l" xx (2pi)/"T"` = constant

⇒ T ∝ l

From Eq. (i), we get

T ∝ L²