Question

A $T$ shaped object with dimensions shown in fig, is lying on smooth floor. A force $F$ is applied at point $P$ parallel to AB such that the object has only translation motion without rotation. Find location of $P$ from $C$ :

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

Answer 1

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

Torque about centre of mass = ${\overrightarrow{T}}_{CM}$

Angular momentum about centre of mass = ${\overrightarrow{L}}_{CM}$

If a body is undergoing only translation, then the torque about its centre of mass should be 0.

Reason:

If a body is undergoing only translation, then every point in the body will have the same velocity

$\overrightarrow{L_{CM}}=\sum \overrightarrow{r}\times m\overrightarrow{v}$

where the position vector of each point is measured with respect to the centre of mass. Since the velocity of every point is the same, we can take it out of the summation.

$\overrightarrow{L_{CM}}=(\sum m\overrightarrow{r})\times \overrightarrow{v}$

Since $\sum m\overrightarrow{r}$ about the centre of mass = 0,

$\overrightarrow{L_{CM}}=0$

${\overrightarrow{T}}_{CM}=\frac{d\overrightarrow{L_{CM}}}{dt}$ = 0

In order for the torque about the centre of mass to be 0, $\overrightarrow{F}$ should act at the centre of mass.

Taking C as the origin,

$\overrightarrow{R_{CM}}=\frac{2m\times l\hat{j}+m\times 2l\hat{j}}{3m}$ = $\frac{4}{3}l\hat{j}$

=> distance of the point of application of the force from P = $\frac{4}{3}l$