Question

Find the instantaneous axis of rotation of a rod of length $l$ from the end $A$ when it moves with a velocity $\overrightarrow{v_A}=v\hat{i}$ and the rod rotates with an angular velocity $\overrightarrow{\omega }=-\frac{v}{2l}\hat{k}$, shown in the figure.

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

Answer 1

The correct option is C $2l$

Let us choose the point $P$ as instantaneous center of rotation in the extended rod at a distance $r$ from point $A$ as shown below.
We can say ICR is point of zero velocity. So, we can write
$\overrightarrow{v_P}=\overrightarrow{v_{PA}}+\overrightarrow{v_A}$
Here, $\overrightarrow{v_P}=0,\overrightarrow{v_{PA}}=-\omega r\hat{i},\overrightarrow{v_A}=v\hat{i}$
$\Rightarrow -\omega r\hat{i}+v\hat{i}=0$
$\Rightarrow r=\frac{v}{\omega }=\frac{v}{\frac{v}{2l}}=2l$
Hence, ICR will be located at a distance $2l$ from $A$.