Question

As shown in the figure a person is pulling a mass 'm' from ground on a fixed rough hemispherical surface upto the top of the hemisphere with the help of a light inextensible string. Find the work done (in Joules) by tension in the string if radius of hemisphere is R and friction coefficient is $\mu$ . Assume that the block is pulled with negligible velocity (take $\mu =0.1,m=1kg,g=10m/s^2,R=1m$).

Answer 1

Let at angle $\theta$, the work done in moving the object by angular distance $d\theta$ is given as,

$F\cdot dx=\left(\mu \sin \theta \right)mg\left(Rd\theta \right)$

The work done is given as,

$\mu Rmg{\int }_0^{\frac{\pi }{2}}\sin \theta d\theta =\mu Rmg$

The work done by tension in the string is given as,

$T=\mu mgR+mgR$

$=\left(\mu +1\right)mgR$

$=\left(0.1+1\right)\times 1\times 10\times 1$

$=11N$

Thus, the work done by tension in the string is $11N$.