Question

A small particle of mass m is pulled to the top of a frictionless half cylinder (of radius R) by a light cord that passes over the top of the cylinder as illustrated in above figure. (a) Assuming the particle moves at a constant speed, show that F=mgcosθ. Note: If the particle moves at constant speed, the component of its acceleration tangent to the cylinder must be zero at all times. (b) By directly integrating W=∫F⋅dr, find the work done in moving the particle at constant speed from the bottom to the top of the half-cylinder.

(a) The radius to the object makes angle $\theta$ with the horizontal. Taking the $x$ axis in the direction of motion tangent to the cylinder, the object’s weight makes an angle $\theta$ with the $–x$ axis. Then,
$\sum F^x=m{a}^x$
$F-mg\cos \theta =0$
$F=mg\cos \theta$
(b) $W={\int }^{i}F\cdot dr$
We use radian measure to express the next bit of displacement as $dr=Rd\theta$ in terms of the next bit of angle moved through:
$W={\int }^{0}mg\cos \theta Rd\theta =mgR\sin {\int }^0=mgR(1-0)=mgR$