A bead of mass m slides without friction on a vertical hoop of radius R . The bead moves under the combined action of gravity and a spring, with spring constant k , attached to the bottom of the hoop. Assume that the equilibrium (relaxed) length of the spring is R. The bead is released from rest at θ = 0 with a non-zero but negligible speed to the right. The bead starts on the top of the circle opposing gravitational pull of the earth

(a) What is the speed v of the bead when θ = 90∘ ? Express your answer in terms of m, R, k, and g.

(b) What is the magnitude of the force the hoop exerts on the bead when θ = 90∘ ? Express your answer in terms of m, R, k, and g.

To solve this problem, we'll use the principles of energy conservation and forces.

(a) To find the speed v of the bead when θ = 90∘, we can use the conservation of energy. At the top of the circle and when θ = 90∘, the bead is at its highest point.

Initially, the bead only has potential energy due to its position. As it descends, it converts potential energy into both kinetic energy and potential energy of the spring. At the top, all the potential energy is converted into kinetic energy. We can write this as:

Potential energy at θ = 0 = Kinetic energy at θ = 90

For the potential energy at θ = 0, we have:
Potential energy = m * g * (2R - R) = m * g * R

For the kinetic energy at θ = 90, we have:
Kinetic energy = (1/2) * m * v^2

Equating the two, we get:
m * g * R = (1/2) * m * v^2

Simplifying and solving for v, we find:
v = sqrt(2 * g * R)

Therefore, the speed v of the bead when θ = 90∘ is sqrt(2 * g * R).

(b) To find the magnitude of the force the hoop exerts on the bead when θ = 90∘, we need to balance the net force acting on the bead. The net force at the top equals the centripetal force required to keep the object moving in a circular path. This centripetal force is provided by two forces: gravity and the spring force.

At θ = 90∘, the net force is given by:
Net force = F_gravity + F_spring

The force of gravity is given by:
F_gravity = m * g

Now, let's calculate the spring force at θ = 90∘. The spring is compressed by a distance of (2R - R) = R, causing a spring potential energy at the top. The spring force is the derivative of this potential energy with respect to displacement:
F_spring = -k * x

Substituting x = R, we get:
F_spring = -k * R

Summing up the forces at θ = 90∘, we get:
Net force = F_gravity + F_spring
Net force = m * g - k * R

The hoop exerts an equal and opposite force on the bead, so the magnitude of the force the hoop exerts is the same as the net force:
Magnitude of the force = |Net force| = |m * g - k * R|

Therefore, the magnitude of the force the hoop exerts on the bead when θ = 90∘ is |m * g - k * R|.