A small mass m hangs at rest from a vertical rope of length l that is fixed to the ceiling. A force F then pushes on the mass, perpendicular to the taut rope at all times, until the rope is oriented at an angle (theta)=(theta)0 and the mass has been raised by a vertical distance h . Assume the force's magnitude F is adjusted so that the mass moves at constant speed along its curved trajectory.

Find the work done by during this process. [Hint: When the angle is increased by d(theta) (in radians), the mass moves along an arc length ds=ld(theta)

W=mgh =mgL(1-cosφ)

To find the work done by the force F during this process, we need to find the displacement of the mass and then multiply it by the force applied to the mass.

Given:
Mass of the object, m
Length of the rope, l
Angle at which the rope is oriented, θ₀
Vertical distance raised by the mass, h

First, let's find the displacement of the mass. We can do this by finding the arc length s:

ds = l dθ

Now, since the force F is perpendicular to the taut rope at all times, the work done by F can be calculated as:

Work = Force * Displacement * cos(θ)

But since the mass moves at constant speed along its curved trajectory, the force applied must be equal to the gravitational force acting on the mass:

F = mg

Substituting this into the equation for work, we get:

Work = mg * l * dθ * cos(θ)

To find the total work done during the process, we need to integrate the above expression over the range of θ from 0 to θ₀:

∫Work = ∫[0 to θ₀] mg * l * cos(θ) dθ

Integrating with respect to θ, we get:

Work = mg * l * sin(θ₀)

Therefore, the work done by the force F during this process is mg * l * sin(θ₀).

To find the work done by the force F during this process, we need to calculate the integral of the dot product of the force and the displacement. However, since the mass moves along a curved trajectory, we need to consider the differential work done.

We know that the differential work done, dW, is equal to the dot product of the force, F, and the displacement, ds. Using the given hint, the displacement can be written as ds = l d(theta).

Therefore, we have:
dW = F · ds
dW = F · (l d(theta))

Since the force is perpendicular to the taut rope at all times, the angle between F and ds is 90 degrees, meaning the scalar product of F and ds is equal to the product of their magnitudes:
dW = F * l * d(theta)

Now we need to consider the work done while integrating this expression over the range of theta from 0 to theta_0. Since the force is adjusted so that the mass moves at constant speed along its curved trajectory, the magnitude of force F must be equal to the weight of the mass m, mg, to counteract the gravitational force.

Therefore, the work done by the force during this process is given by:
W = ∫ dW
W = ∫(0 to theta_0) F * l * d(theta)
W = ∫(0 to theta_0) (mg) * l * d(theta)
W = mg * l * ∫(0 to theta_0) d(theta)
W = mg * l * (theta_0 - 0)
W = mg * l * theta_0

So, the work done by the force during this process is given by W = mg * l * theta_0.