Question

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

`(cos30^circ=sin60^circ=sqrt3/2,  cos60^circ=sin30^circ=1/2)`

विकल्प

• `sqrt"L"`

• L

• `"L"^(3/2)`

• L^-2

Answer 1

Three-point masses, each of mass 'm' are kept at the comers of an equilateral triangle of side 'L'. The system rotates about the centre of the triangle without any change in the eparation of masses during rotation. The period of rotation is directly proportional to `underline("L"^(3/2))`.

Explanation:

The force on any mass due to each of the other two masses will be of magnitude

`"F"="G"m^2/"L"^2`

The two forces are acting at an angle of 60° and their resultant is given by

`"F"^'=sqrt("F"^2+"F"^2+2"F"^2cos60^circ=sqrt3  "F")`

The height of the triangle h `="L"sin60^circ=sqrt3/2"L"`

The distance of the centroid from the vertices is `2/3"h"="L"/sqrt3`

∴ Radius of the circular motion of the masses `r="L"/sqrt3`

For circular motion the centripetal force in equal to the gravitational force due to the other masses.

`therefore m  r  omega^2=sqrt3  "F"=sqrt3"Gm"^2/"L"^2`

or `r  omega^2=sqrt3"Gm"/"L"^2`

`thereforeomega^2=(3"GM")/"L"^3`

`therefore(4pi^2)/T^2=(3"GM")/"L"^3`

`T^2=(4pi^2"L"^3)/(3"GM")`

`therefore T^2 prop "L"^3`

or `T prop "L"^(3/2)`