Three particles A, B and C each of mass m, are connected to each other by three massless rigid rods to form a rigid, equilateral triangular body of side l. This body is placed on a horizonta frictionsess table (x-y plane) and is hinged to it at the point A so that it can move without friction about the vertical axis through A . the body is set into rotational motion on the table about A with a constant angular velocity `omega`.

(a) Find the magnitude of the horizontal force exerted by the hinge on the body.
(b) At time T, when the side BC is parallel to the x-axis, a force F is applied on B along BC (as shown). Obtain the x-component and the y-component of the force exerted by the hinge on the body, immediately after time T.

Explanation

(a). The distance of centre of mass (COM) of the system about point A will be
Therefore, the magnitude of horizotal force exerted by the hinge on the body is
`F=` centripetal force
or `F=(3m)romega^(2)`
or `F=(3m)((l)/(sqrt(3))omega^(2)`
or `F=sqrt(3)mlomega^(2)`
(b). Angualr acceleration of system about point A is
`alpha=(tau_(A))/(I_(A))`
`=((F)((sqrt(3))/(2)l))/(2ml^(2))`
`=(sqrt(3)F)/(4ml)`
Now acceleration of COM along x-axis is
`a_(x)=ralpha=((l)/(sqrt(3)))((sqrt(3)F)/(4ml))` or `a_(x)=(F)/(4m)`
Now, let `F_(x)` be the force applied by the hinge along x-axis
then, `F_(S)+F=(3m)a_(s)`
or `F_(S)+F=(3m)((F)/(4m))`
or `F_(S)+F=(3)/(4)F` or `F_(x)=-(F)/(4)`
Further if `F_(y)` be the force applied by the hinge along y-axis then
`F_(y)=` centripetal force
or `F_(y)=sqrt(3)mlomega^(2)`