Question

The two ends a and b of a uniform rod of length l= 1m are moving with velocities va and vb . The angular velocity of rod is

Answer 1

Use concept of instantaneous centre of rotational motion.

Answer 2

The angular velocity of rod, ω= 25$\dfrac{rad}{s}$

Explanation:

Let the line joining A to B be X-axis and Y-axis is perpendicular to the X- axis. The velocity of point A with respect to B along the line joining A and B is zero(0).

To find, the angular velocity of rod, ω=

The velocity of A along  X-axis towards B,

$V_{AX}=20\cos53=20\times \dfrac{3}{5} =12\dfrac{m}{s}$

The velocity of B along  X-axis towards A,

$V_{BX}=V\cos37=V\times \dfrac{4}{5}\dfrac{m}{s}$

⇒ $12=V\times \dfrac{4}{5}$

⇒ $V=\dfrac{12\times 5}{4} =15\dfrac{m}{s}$

∴ The angular velocity of rod, ω=$\dfrac{The velocity of point A with respect to B in the direction perpendicular to the line joining A and B}{The separation between A and B}$The velocity of A  perpendicular to the line joining A and B is  $V_{AY}=20\sin53=20\timmes \dfrac{4}{5} =16\frac{m}{s}$

Also,

The velocity of B  perpendicular to the line joining A and B is $V_{BY}=15\sin37=15\times 35=9\dfrac{m}{s}$

∴ ω $=\dfrac{16+9}{1}$ = 25$\dfrac{rad}{s}$

Hence, the angular velocity of rod, ω= 25$\dfrac{rad}{s}$