Question

A particle moves around a semicircle of radius R, from one end A of a diameter to the other B. It is attracted to its starting point A by a force proportional to its distance from A. When the particle is at B, the force towards A is F0. Calculate the work done against this force when the particle moves around the semicircle from A to B.

Answer 1

The work done against the force when the particle moves around the semicircle from A to B can be calculated using the formula W = ∫(F ⋅ dr). The total work done is equal to the work done along the radial portion, which is equal to ∫(-kx ⋅ dr).

Step-by-step explanation:

The work done against the force when the particle moves around the semicircle from A to B can be calculated using the formula W = ∫(F ⋅ dr). In this case, the force is proportional to the distance from A, so we can write F = -kx, where k is the proportionality constant.

Since the particle moves in a semicircle from A to B, we can divide the path into two parts: the radial portion and the arc. Along the radial portion, F is opposite to the direction of motion, so the work done is negative.

On the arc, the force is perpendicular to the displacement, so the dot product is zero and no work is done. Therefore, the total work done against the force is equal to the work done along the radial portion, which is equal to

∫(F ⋅ dr) = ∫(-kx ⋅ dr).

Answer 2

work = 2 F₀ R

Step-by-step explanation:

As the particle moves along the circumference of semicircle . It reaches from A to B . The force at B is F₀. The displacement from A to B is 2 R .

Because the displacement is defined as the shortest path between two points . Here it is 2 R .

The work done can be defined as the product of force and displacement .

Thus the work done in moving from A to B = F₀ x 2 R = 2 F₀ R