LOOP, SPRING AND BEAD (14 points possible)
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.
(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 find the speed of the bead when θ = 90∘, we need to consider the forces acting on the bead at that point.
(a) The forces acting on the bead at θ = 90∘ are the spring force and the gravitational force. At this point, the gravitational force is acting vertically downward and the spring force is acting radially inward. The net force on the bead is the sum of these two forces.
The gravitational force can be calculated as mg, where m is the mass of the bead and g is the acceleration due to gravity.
The spring force can be calculated using Hooke's Law, which states that the force exerted by a spring is proportional to the displacement from its equilibrium position. In this case, the equilibrium position of the spring is at the bottom of the hoop, where θ = 0 and the length of the spring is R. At θ = 90∘, the length of the spring is R - R = 0, resulting in a displacement of 0. Therefore, the spring force at this point is also 0.
Since the net force is 0, the gravitational force mg must be equal to the spring force, which is 0. Therefore, there is no net force acting on the bead when θ = 90∘.
As a result, the speed of the bead at this point remains constant, which means the bead will have the same speed as the initial speed when it was released. Therefore, the speed v of the bead when θ = 90∘ is the same as the initial speed.
(b) Since there is no net force acting on the bead when θ = 90∘, the magnitude of the force the hoop exerts on the bead is zero. The only forces acting on the bead are the gravitational force and the spring force, and these forces cancel each other out, resulting in no net force on the bead. Therefore, the hoop exerts zero force on the bead at this point.