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 $C$:

For an object to undergo only translational motion without rotation, the force must be applied through its centre of mass (COM).Let the horizontal bar (AB) have mass $m_1$ and vertical bar (attached to C) have mass $m_2$.Assuming uniform density, if length of AB is $2l$ and vertical part is $2l$:$m_1$ is at $y_1=2l$. $m_2$ is at $y_2=l$ (midpoint of vertical bar).Assuming bars are identical per unit length: $m_1\propto 2l,m_2\propto 2l⟹m_1=m_2=m$.
$$y_{cm}=\frac{m(2l)+m(l)}{m+m}=\frac{3ml}{2m}=1.5l$$
Distance from C $=1.5l$ or $\frac{3}{2}l$.(Note: Based on exact image dimensions, $4/3l$ or other values might arise depending on the specific geometry/mass distribution described in the figure).