Magnitude of Change in Velocity in Uniform Circular Motion

In uniform circular motion, the magnitude of the change in velocity is equal to the speed of the particle. This can be explained by considering the definition of acceleration in circular motion.

Acceleration in Uniform Circular Motion:
In uniform circular motion, the particle moves along a circular path with a constant speed. However, its direction of motion is continuously changing. This change in direction indicates that the particle is experiencing acceleration.

The acceleration of a particle in uniform circular motion is directed towards the center of the circular path and is called centripetal acceleration. It is given by the formula:

a = v^2 / r

where:
a = centripetal acceleration
v = speed of the particle
r = radius of the circular path

Change in Velocity:
The change in velocity of the particle is the difference between the final velocity and the initial velocity. In uniform circular motion, the initial and final velocities have the same magnitude but different directions.

Since the speed of the particle remains constant, the magnitude of the change in velocity is equal to the speed of the particle, which is denoted by v.

Magnitude of Average Force on the Particle:
According to Newton's second law of motion, the magnitude of the average force acting on a particle is equal to the product of the mass of the particle and its acceleration.

F = m * a

In uniform circular motion, the centripetal acceleration can be related to the speed of the particle and the radius of the circular path. Substituting the value of acceleration in the above equation, we get:

F = m * (v^2 / r)

Therefore, the magnitude of the average force on the particle is proportional to the square of the speed of the particle and inversely proportional to the radius of the circular path.

Conclusion:
In uniform circular motion, the magnitude of the change in velocity of a particle is equal to its speed. The magnitude of the average force acting on the particle is proportional to the square of the speed of the particle and inversely proportional to the radius of the circular path. This relationship can be derived from the definition of acceleration in circular motion and Newton's second law of motion.