A crate of mass m hangs from a mass less rope of length L magnitude pushes horizontal force F of variable the crate at a constant speed (that is, there is no net force on the crate at L. A any point along its path) from the initial position shown in the figure to the final position shown in the figure (displaced by a distance D in the horizontal direction)

(a) Derive an expression for the force F as a function of θ, the angle between the rope and the vertical direction. Your expression may contain some or all of the following quantities: m, g, L and θ. (nh, The angle θ varies as the crate is pushed from the initial position to the final position. The distance D in the figure indicates the final position.

(b) Calculate (that is, do the line integral) the work done W _g by this variable force in moving from the bottom position to the position displaced by the horizontal position D as shown in the figure. (n.b., the crate follows the circular segment shown in the figure.) Your expression for W _g may contain some r all of the following quantities: m, g, L and D.

(c) What is the work done by the weight W _g of the crate in moving from the bottom position to the position displaced by the horizontal position D as shown in the figure. Your expression for W _g may contain some or all of the following quantities: m, g, L and D

(d) How does your expression in (c) compare to the change in gravitational potential energy of the crate?

(e) What is the work done by the tension W _T in moving from the bottom position to the position displaced by the horizontal position D as shown in the figure. Your expression for W _T may contain some or all of the following quantities: m, g, L and D

 (f) What is the total change in kinetic energy of the crate? How can use use the results in (d) and (e) to infer the answer in (b) without actually calculating the integral.

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