Question

A time varying force $F=2t$ is applied on a spool rolling as shown in figure. The angular momentum of the spool at time $t$ about bottom-most point is :

• A. $\frac{r^2{t}^2}{R}$
• B. $\frac{(R+r{)}^2}{r}{t}^2$
• C. $(R+r)t^2$
• D. Data is insufficient

Answer 1

Answer: (C)

$(R+r)t^2$

Change in angular momentum(L)$=$Torque

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

$\Rightarrow dL=\tau dt$

Taking torque about lowest point distance of force applied from lowest point$=(R+r)$

$\tau =2t(R+r)$

$\Rightarrow dL=2t(R+r)dt$

at after time t, the angular momentum becomes

$L={\int }_0^{L}dL={\int }_0^{t}2t(R+r)$

$\Rightarrow L=(R+r)t^2$.