small mass hangs at rest from a vertical rope of length that is fixed to the ceiling. A force then pushes on the mass, perpendicular to the taut rope at all times, until the rope is oriented at an angle and the mass has been raised by a vertical distance . Assume the force's magnitude is adjusted so that the mass moves at constant speed along its curved trajectory.

There is no question here.

W=mgh =mgL(1-cosφ)

To find the force's magnitude F, we can use the concept of work-energy theorem. According to the work-energy theorem, the work done on an object is equal to the change in its kinetic energy.

In this scenario, the force applied is perpendicular to the rope at all times. Therefore, the only work done on the mass is due to the force's component along the displacement, which is in the vertical direction.

The work done on the mass is given by the formula: Work = Force * Distance * cos(θ)

Since the force is perpendicular to the rope, the angle between the force and the displacement is 90 degrees, so cos(90) = 0. Therefore, the work done on the object is 0, as the force doesn't do any work in the direction of the displacement.

This means that there is no change in the kinetic energy of the mass.

Hence, the magnitude of the force F can be adjusted to ensure that the mass moves at a constant speed along its curved trajectory without any change in kinetic energy.