A bead on a fixed ring and a spring

A ring fixed to a support at the top is hanging vertically. A bead of mass m slides through the ring and is connected to the support through a spring of stiffness k as shown in the figure-1. The length of the unstretched spring is almost negligible. Note that gravity acts.

a. Find the equation of motion of the bead using Newton’s laws and find the normal reaction force on the bead.
b. Find the equation of motion of the bead using the work-energy principle.

To find the equation of motion of the bead using Newton's laws, we need to analyze the forces acting on the bead.

a. Equation of motion using Newton's laws:

1. Gravitational Force (mg): The bead experiences a downward force due to gravity, given by the product of its mass (m) and the acceleration due to gravity (g).

2. Normal Reaction Force (N): The bead also experiences an upward normal reaction force due to the contact with the ring. The magnitude of this force will vary depending on the motion of the bead.

3. Spring Force: The spring exerts a force on the bead that is proportional to its displacement from its equilibrium position. The magnitude of the spring force (F_s) can be calculated using Hooke's Law: F_s = -kx, where k is the spring stiffness and x is the displacement of the bead from the equilibrium position.

Since the bead is sliding through the ring, there is no constraint in the horizontal direction. Thus, we only need to consider the forces acting on the bead in the vertical direction.

Applying Newton's second law (F = ma) to the vertical direction:

Sum of forces = ma

mg - N - kx = ma

Simplifying the equation, we get:

N = mg - kx - ma

This equation gives us the normal reaction force on the bead in terms of the bead's mass, acceleration due to gravity, spring stiffness, displacement, and acceleration.

b. Equation of motion using the work-energy principle:

The work-energy principle states that the work done on an object equals the change in its kinetic energy. In this case, we can calculate the work done by the spring force on the bead.

The work done (W) by the spring force is given by the equation: W = (1/2)kx^2, where x is the displacement of the bead from the equilibrium position.

Since the bead is initially at rest, its initial kinetic energy (K_i) is zero. When the bead is displaced from its equilibrium position, it gains potential energy that is equal to the work done by the spring force.

Therefore, the potential energy (U) gained by the bead is given by: U = (1/2)kx^2

Using the principle of conservation of mechanical energy, the total mechanical energy (E) of the system (bead and spring) is conserved, which means that the sum of the kinetic energy and potential energy remains constant.

At any point during the motion, the total mechanical energy (E) can be expressed as:

E = K + U

Since the bead is initially at rest, its initial total mechanical energy (E_i) is zero. When the bead is displaced from its equilibrium position, it gains potential energy.

Therefore, the equation of motion using the work-energy principle is:

0 = K + (1/2)kx^2

This equation relates the kinetic energy of the bead and the potential energy stored in the spring to the displacement of the bead.

By manipulating and simplifying these equations, we can solve for the motion of the bead and determine the normal reaction force exerted on it.