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Centripetal Force

Consider a particle executing uniform circular motion. It has a constant speed and thus the velocity does not change in magnitude (Figure 4.10). From Figure 4.11(a) we see that the direction of velocity being tangential to the path changes continuously, i.e., the velocity vector is found to bend towards the centre constantly at every point.
Description: Description: 58489.png
Figure 4.10 Circular Motion
(a) Description: Description: 58525.png
(b) Description: Description: 58532.png
Figure 4.11 Centripetal Acceleration
Therefore, according to Newton’s law of motion, there must be an acceleration and a corresponding force. The acceleration is directed perpendicular to the path, i.e., towards the centre, as the velocity turns towards the centre at every instance. (See Figure 4.11(b) for a conceptual understanding. If points A and B are taken too close to each other, the direction of Description: 56872.png will be exactly towards the centre.)
The centre-seeking acceleration acting on a body in uniform circular motion is called centripetal acceleration or radial acceleration. The corresponding force is called centripetal force or radial force.
Let a particle of mass m move along the circumference of a circle of radius r with uniform speed v.
Then, centripetal acceleration (a) along the radius (r) is
Description: 56884.png
In Figure 4.11(b),
where v is the tangential velocity and r is the radius of the circle.
Force = Mass × Acceleration
∴ Centripetal force F = m × a
Description: 56959.png = 2r  [∴ v = ωr]

where m is the mass of the particle, v the linear velocity of the body (ms1), ω the angular velocity (rads1) and r the radius of the circular path, since the acceleration produced is directed towards the centre and along the radius.
Centripetal force is defined as the force acting on a particle in uniform circular motion and directed towards the centre along the radius.

Examples for Centripetal Force

  1. In railway tracks, the outer rail in a curve is raised a little above the inner. For a train negotiating a curve, the centripetal force is the force of friction between the wheels and rails. Generally, the force of friction is small and so the train has to negotiate the path slowly. Otherwise, it will skid and there are chances of the train going off the track.
Description: Description: FIG-4.12.tif
Figure 4.12 Truck Moving on a Banked Road
  1. For the same reason mentioned above, a curved road has to be inclined suitably by raising the outer edge over the inner for an automobile negotiating a curve. This is known as banking of roads (Figure 4.12).
    You might have observed race tracks of ‘formula 1’ car races in a TV show with a steep banking of the race tracks along bends.

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