Question

A uniform ${}^{\prime }{T}^{\prime }$ shaped object with dimensions as shown in the figure, is lying on a smooth floor. A force $\overrightarrow{F}$ is applied at the point P parallel to limb $AB$, such that the object has only the translational motion without rotation. Find the distance of point $P$ from $C$

• A. $\frac{2l}{3}$
• B. $\frac{3l}{2}$
• C. $\frac{4l}{3}$
• D. $l$

Answer 1

The correct option is C $\frac{4l}{3}$

$\overrightarrow{F}$ must be directed towards the centre of mass of system, so that it passes through $COM$ and ${\overrightarrow{\tau }}_{ext}=0$
$\Rightarrow$ If ${\tau }_{ext}=0$ then angular acceleration of body $\alpha =0$, which means that rotational aspect of motion will be absent, and body will be in pure translational motion.
$\because T$ shape is symmetrical about $y-$ axis hence, $COM$ will lie along $y-$ axis only.

Considering origin $O$ at the centre of horizontal limb of length $l$ and axis as shown in figure:


since (length $\propto$ mass), for uniform rod
$\therefore m_1=m,m_2=2m$
$m_2\to$vertical limb of length $2l$

Replacing the individual limbs with a point mass placed at their respective $COM$:
$y_1=0,y_2=l$
$\Rightarrow y_{\text{CM}}=\frac{m_1{y}_1+m_2{y}_2}{m_1+m_2}$
$y_{\text{CM}}=\frac{(m\times 0)+(2m\times l)}{m+2m}=\frac{2ml}{3m}$
$\therefore y_{\text{CM}}=\frac{2l}{3}$
$\Rightarrow PC=CO-PO=(2l-\frac{2l}{3})$
Distance of point $P$ with respect to $C$,
$\therefore PC=\frac{4l}{3}$