A ‘T’ shaped object with dimensions shown in the figure, is lying on a smooth floor. A force $\text{'}\overrightarrow{F}\text{'}$ 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.

 

To have translational motion without rotation, the force $\overrightarrow{F}$ has to be applied at centre of mass. i.e. the point ‘P’ has to be at the centre of mass. Taking point C at the origin position, positions of ${\text{y}}_1\text{,}$ and ${\text{y}}_{\text{2}}{\text{\hspace{0.17em}are\hspace{0.17em}r}}_{\text{1}}{\text{=2l,\hspace{0.17em}r}}_{\text{2}}{\text{=l\hspace{0.17em}and\hspace{0.17em}m}}_1{\text{=2\hspace{0.17em}and\hspace{0.17em}m}}_{\text{2}}\text{=2m}$

$\text{y=}\frac{m_1{y}_1+m_2{y}_2}{m_1+m_2}=\frac{2m\times 2\mathcal{l}+2m\times \mathcal{l}}{4m}=\frac{6\mathcal{l}}{4}=\frac{3\mathcal{l}}{2}$