To complete a verticle circular motion of a particle of mass m the minimum value of h

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Question

To complete a verticle circular motion of a particle of mass m the minimum value of h?

Answer

Vertical Circular Motion of a Particle

In vertical circular motion, a particle moves in a circular path with a constant speed, but the direction of its velocity changes continuously. The motion can be divided into two phases:

1. Bottom-most Point of the Circle (Lowest Position)

At the bottom-most point of the circle, the particle experiences the maximum force of gravity acting on it.

The force of gravity provides the centripetal force required to keep the particle moving in a circular path.

The centripetal force (Fc) is given by the equation: Fc = m * v^2 / r, where m is the mass of the particle, v is its velocity, and r is the radius of the circular path.

Since the velocity is constant, the centripetal force is constant as well.

At this point, the net force acting on the particle is the difference between the force of gravity and the centripetal force: Fnet = mg - Fc.

The net force provides the necessary acceleration (a) towards the center of the circle: Fnet = ma.

2. Top-most Point of the Circle (Highest Position)

At the top-most point of the circle, the force of gravity acts in the opposite direction to the velocity of the particle.

The centripetal force still points towards the center of the circle.

The net force is now the sum of the force of gravity and the centripetal force: Fnet = mg + Fc.

The net force provides the necessary acceleration towards the center of the circle: Fnet = ma.

Minimum Height (h) for Complete Circular Motion

In order for the particle to complete a full vertical circular motion, it must not lose contact with the circular path at any point.

This means that the net force at the top-most point should be greater than or equal to zero, ensuring that the particle remains in contact with the circular path.

Therefore, we equate the net force at the top-most point to zero: Fnet = mg + Fc = 0.

Solving for h, the minimum height required for complete circular motion can be found:

mg + Fc = 0

mg + m * v^2 / r = 0

mg = - m * v^2 / r

g = - v^2 / r

h = r - r * cos(theta)

h = r * (1 - cos(theta))

Therefore, the minimum height (h) required for complete vertical circular motion of a particle of mass m is given by the formula h = r * (1 - cos(theta)).

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