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  4. A 'T' shaped object with dimensions shown in the figure, is lying on a smooth floor. A force 'F' 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 to C :

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Q. A '$T$' shaped object with dimensions shown in the figure, is lying on a smooth floor. A force '$\overrightarrow{F}$' 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 to $C$ :Physics Question Image

AIEEEAIEEE 2005System of Particles and Rotational Motion

Solution:

For pure translatory motion, net torque about centre of mass should be zero.
Thus, $\overrightarrow{F}$ is applied at centre of mass of system.
$OP=\frac{m_{1}\times O+m_{1}\times l}{m_{1}+m_{2}}$
where, $m_{1}$ and $m_{2}$ are masses of horizontal and vertical section of the object. Assuming object is uniform,
$m_{2}=2m_{1} \Rightarrow OP=\frac{2l}{3}$
$\therefore PC=\left(l-\frac{2l}{3}+l\right)$
$=\left(2l-\frac{2l}{3}\right)=\frac{4l}{3}$

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