Question

A constant external torque $\tau$ acts for a very brief period $\Delta t$ on a rotating system having moment of inertia $I$, then

• A. the angular momentum of the system will change by $\tau \Delta t$
• B. the angular velocity of the system will change by $\tau \Delta t/I$
• C. if the system was initially at rest, it will acquire rotational kinetic energy ${\left(\tau \Delta t\right)}^2/2I$
• D. the kinetic energy of the system will change by ${\left(\tau \Delta t\right)}^2/2I$

Answer 1

Answer: (A), (B), (C)

the angular velocity of the system will change by $\tau \Delta t/I$
the angular momentum of the system will change by $\tau \Delta t$
if the system was initially at rest, it will acquire rotational kinetic energy ${\left(\tau \Delta t\right)}^2/2I$

Using $\frac{dL}{dt}=\tau$

Thus $\Delta L=\tau \left(\Delta t\right)$

Also, $L=Iw$

$⟹\Delta L=I\Delta w$

$\Delta w=\frac{\tau \left(\Delta t\right)}{I}$

If $w_i=0$, thus $w_f=\Delta w$

Now $K{E}_R=\frac{1}{2}I{w}_f^2=\frac{(\tau \Delta t{)}^2}{2I}$

$\Delta K{E}_R=\frac{(\tau \Delta t{)}^2}{2I}$ only when $w_i=0$.