A 'T' shaped object with dimensions shown in the figure, is lying on a smooth floor. A force `'vecF'` is applied at the point P parallel to AB, such that the object has only the translational motion without rotation. Find the location of P with respect C.

Options

A. `(2)/(3)l`
B. `(3)/(2)l`
C. `(4)/(3)l`
D. l

Answer

C

Explanation

Total torque about the point P is zero and the object is having pure translatory motion. So the point P is centre of mass of the system.
Thus, the force `vecF` is applied at the centre of mass of system.

Mass of horizontal part =m
Mass of vertical part =2m
Taking origin at O
We can assume that there are two particles of masses m and 2m, placed at 0 (0,0) and Of(0, -1) respectively.
As the masses of the rods are uniformly distributed over their lengths, hence the centre of mass of each part lies at its geometric centre.
`X_(cm)=0`
`Y_(cm)=((0xxm)+(-lxx2m))/(m+2m)`
`=(-2ml)/(3m)=(-2l)/(3)`
Centre of mass lies at point P `(0,(-2l)/(3))`
`OC=OP+PC=2l`
Distance of the centre
`impliesPC=2l-OP`
`=2l-(-2l)/(3)=(4l)/(3)`