Centripetal FORCE and centripetal ACCELERATION [formula derivation, definition and, example]

What is centripetal force?

When a body travels along a circular path its velocity changes continuously. Naturally and external forces always act on the body towards the center of the earth. The external force required to maintain the circular motion of the body is called centripetal force. if a body of mass M is moving on a circular path of radius r with uniform speed v, then the required centripetal force, F = mv2/r.

Centripetal force definition

Centripetal force is the force that acts on a body to maintain the circular path during the motion of the body.

Centripetal force formula derivation

As we know F = ma (newton's second law of motion)

a = F/m

v2/r = F/m (as acceleration a = v2/r)

Or,

F = mv2/r (where, F = centripetal force, m = mass of the object, v2/r = acceleration)

Centripetal Acceleration

If a particle moves in a circle of radius r with a constant speed v, its acceleration is v2 / r directed towards the center. This acceleration is called centripetal acceleration.

Note that the speed remains constant, the direction continuously changes, and hence the "velocity" changes and there is an acceleration during the motion.

Centripetal force explanation

If a particle moves in a circle as seen from an inertial frame, a resultant nonzero force must act on the particle. That is because a particle moving in a circle is accelerated and acceleration can be produced in an inertial frame only if a resultant force acts on it.

If the speed of the particle remains constant, the acceleration of the particle is towards the center and its magnitude is v2/r. Here v is the speed of the particle and r is the radius of the circle. The resultant force must act towards the center and its magnitude F must satisfy.

F = ma

a = F/m

v2/r = F/m

Or,

F = mv2/r

Since this resultant force is directed towards the center, it is called centripetal force. Thus, a centripetal force of magnitude mv2 / r is needed to keep the particle in uniform circular motion. It should be clearly understood that " centripetal force " is another word for " force towards the center ". This force must originate from some external source such as gravitation, tension, friction, Coulomb force, etc. Centripetal force is not a new kind of force, just as an " upward force " or a " downward force " is not a new kind of force.

Centripetal force Example

A small block of mass 100 g moves with uniform speed in a horizontal circular groove, with vertical sidewalls, of radius 25 cm. If the block takes 2.0 s to complete one round, find the normal contact force by the sidewall of the groove.

Solution:

The speed of the block is

v = 2πx(25 cm) / 2.0 s

= 0.785 m/s

The acceleration of the block is

a = v2 / r

v2 / r = (0.785 m/s)2 /0.25 m

= 2.5 m/ s2 towards the center,

The only force in this direction is the normal contact force due to the side walls.

thus, from Newton's second law, this force is

N = ma = (0.100 kg) (2.5 m/s2)

= 0.25 N.

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