Centripetal Force

Centripetal Force

Recall that the centripetal acceleration of a body is give by a = rω², or a = v²/2, where r is the radius of the circular path, ω is the angular velocity, and v the linear velocity.

According to Newton's second law, the force F = ma, where m is the mass of the body. So the force needed to produce the centripetal acceleration is F = mrω², or F = mv²/2.

This force is called the centripetal force.

Exercise. I tie a stone to a string of length 1 m. I swing the stone round in a horizontal circle, once every 1 s. If the mass of the stone is 0.1 kg, find the tension in the string. Answer. The tension in the string provides the centripetal force, and also support the weight of the stone. Since the stone is in a horizontal circle, the centripetal force is horizontal. The angular velocity ω = 2π rad/s, radius r = 1 m, mass m = 0.1 kg. So the centripetal force F = mrω² = 0.1 x 1 x (2π)² = 0.4π² N. The weight of the stone is W = mg = 0.1 x 9.81 = 0.981 N. The resultant of weight and centripetal force is the tension in the string. Since one force is vertical and the other horizontal, we can use Pythagoras theorem to find the resultant: [0.981² + (0.4π²)²]^1/2 = ___ N.

Exercise. I sit in a car travelling at 10 m/s. When it turns left, I find myself pushing at the right door with a force of 100 N. If I am 50 kg, what is the radius of the bend? Answer. Velocity v = 10 m/s, mass m = 50 kg, force F = 100 N. Since F = mv²/r, radius r = mv²/F = 60 x 10² / 100 = 60 m.

In the above exercise, when a car turns left, I push to the right - involuntarily. This force is commonly called the centrifugal force, which pushes away from the centre of the circle. Note that this force comes from the body. It is different from the centripetal force, which comes from the car, pushing on the body. So the centrifugal force is the equal and opposite reaction to the centripetal force.

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