A T shape object with dimensions shown in 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. 

The object will have translation motion without rotation, when force F is applied at the centre of mass of system. If m is mass per unit length, the mass of AB = $\mathrm{m}\times \mathrm{l}=\mathrm{ml}$ at O and mass of OC = $\mathrm{m}\times 2\mathrm{l}=2\mathrm{ml}$. Here, CD = l.

Let DP = x

As P is the center of mas, therefore,

ml (l - x) = 2mlx $\Rightarrow \mathrm{x}=\frac{\mathrm{l}}{3}$

$\mathrm{CP}=\mathrm{CD}+\mathrm{DP}=\mathrm{l}+\frac{\mathrm{l}}{3}=\frac{4\mathrm{l}}{3}$