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 and the normal reaction force on the bead, we can use Newton's laws of motion.
a. Using Newton's laws of motion:
Let's denote the length of the spring when it is stretched as x. The gravitational force acting on the bead is mg, where m is the mass of the bead and g is the acceleration due to gravity.
1. Forces acting on the bead:
- Gravity: mg (acting downward)
- Spring force: -kx (acting upward, due to Hooke's Law)
- Normal reaction: N (acting upward)
2. Equation of motion:
According to Newton's second law, the sum of the forces acting on the bead is equal to the mass of the bead times its acceleration:
mg - kx - N = ma
Simplifying the equation, we have:
ma + kx + N = mg
3. Normal reaction force:
The normal reaction force acts perpendicular to the surface and prevents the bead from passing through the ring. In this case, since the bead is hanging vertically, the normal reaction force is equal to the weight of the bead, which is m * g:
N = mg
Substituting this into the equation of motion, we get:
ma + kx + mg = mg
ma + kx = 0
This is the equation of motion for the bead.
b. Using the work-energy principle:
The work-energy principle states that the change in the kinetic energy of an object is equal to the net work done on it.
1. Consider the bead at position x, with a velocity of v. The total mechanical energy of the system is given by:
E = 1/2 * mv^2 + 1/2 * kx^2 - mgx
The first term represents the kinetic energy of the bead, the second term represents the potential energy stored in the spring, and the third term represents the gravitational potential energy.
2. When the bead moves from position x1 to x2, the work done by the spring force is given by the change in potential energy stored in the spring:
W_spring = 1/2 * k * (x2^2 - x1^2)
3. The net work done on the bead is the sum of the work done by the spring force and the work done by gravity:
W_net = W_spring + W_gravity
W_gravity = mg * (x2 - x1)
4. According to the work-energy principle, the change in kinetic energy is equal to the net work done:
Delta KE = W_net
1/2 * m * (v2^2 - v1^2) = W_net
5. Substituting the expressions for W_spring and W_gravity, we have:
1/2 * m * (v2^2 - v1^2) = 1/2 * k * (x2^2 - x1^2) + mg * (x2 - x1)
6. Simplifying the equation, we get:
m * (v2^2 - v1^2) = k * (x2^2 - x1^2) + 2m * g * (x2 - x1)
This is the equation of motion for the bead using the work-energy principle.