A small particle of mass $m$ is pulled to the top of a frictionless half-cylinder (of radius $R )$ by a cord that passes over the top of the cylinder as illustrated in Figure $\mathrm{P} 7.25$ . (a) Assuming the particle moves at a constant speed, show that $F=m g \cos \theta .$ 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 $W=\int \overrightarrow{\mathbf{F}} \cdot d \overrightarrow{\mathbf{r}},$ find the work done in moving the particle at constant speed from the bottom to the top of the half-cylinder.