A particle of mass m can slide without friction on the inside of a small tube bent in the form of a circle of radius a. The tube rotates about a vertical diameter at a constant rate of w rad/sec as shown in Fig. 1.79. Write the differential equation of motion. If the particle is disturbed slightly from its unstable equilibrium position at 0 = 0, find the position of maximum kinetic energy.

The problem involves a particle moving in a circular path due to the rotation of the tube. The forces acting on the particle are gravity and the centripetal force due to the rotation.

Step 1: Identify the forces acting on the particle

The gravitational force acting on the particle is mg, where m is the mass of the particle and g is the acceleration due to gravity. This force acts vertically downwards.

The centripetal force acting on the particle is mw^2r, where m is the mass of the particle, w is the angular velocity of the tube, and r is the distance of the particle from the center of the tube. This force acts towards the center of the tube.

Step 2: Write the differential equation of motion

The differential equation of motion can be obtained by applying Newton's second law in the radial direction. The net force acting on the particle is the difference between the centripetal force and the component of the gravitational force in the radial direction. This gives:

mw^2r - mgcosθ = mrθ''

where θ'' is the second derivative of θ with respect to time, representing the angular acceleration of the particle.

Step 3: Find the position of maximum kinetic energy

The kinetic energy of the particle is given by:

K = 1/2 m (rθ')^2

where rθ' is the speed of the particle. The kinetic energy is maximum when the speed is maximum. This occurs when the particle is at the bottom of the tube, where the gravitational potential energy is minimum. Therefore, the position of maximum kinetic energy is θ = π.