An object moving in a circle of radius r
with constant speed v is accelerating. The
direction of its velocity vector is changing all the time, but the magnitude of
the velocity vector stays constant. The acceleration vector cannot have a
component in the direction of the velocity vector, since such a component would
cause a change in speed. The acceleration vector must therefore be
perpendicular to the velocity vector at any point on the circle. This
acceleration is called **radial** acceleration
or **centripetal** acceleration, and it points
towards the center of the circle. The magnitude of the centripetal
acceleration vector is a_{c} = v^{2}/r. (We skip the
derivation of this expression.)

Link: Uniform circular Motion

The orbit of the moon around
the earth is approximately circular, with a mean radius of 3.85*10^{8}
m. It takes 27.3 days for the moon to complete one revolution around the
earth. Find

(a) the mean orbital speed of the moon and

(b) its centripetal acceleration.

Solution:

- Reasoning:

The distance the moon travels in one orbital period T is d = 2πr.

Its speed is v = distance/time = d/T.

The centripetal acceleration of the moon is v^{2}/r. - Details of the calculation:

(a) The distance the moon travels in 27.3 days is d = 2πr = 2.41*10^{9 }m.

Its speed is v = d/(27.3 days) = (d/(2.36*10^{6 }s)) = 1023 m/s.

(b) The centripetal acceleration of the moon is v^{2}/r = 2.725*10^{-3 }m/s^{2}.

An object moving in a circle, either with uniform or non-uniform speed, is
accelerating. Since it is accelerating, it must be acted on by a force. Such a
force is called a **centripetal force**.

Let us solve some problems investigating this question.

3 kg mass attached to a light string rotates on a horizontal frictionless table. The radius of the circle is 0.8 m and the string can support a mass of 25 kg before breaking. What range of speeds can the mass have before the string breaks?

Solution:

- Reasoning:

A mass attached to a string rotates on a horizontal, frictionless table.

We assume that the mass rotates with uniform speed. It is accelerating. The direction of the acceleration is towards the center of the circle, and its magnitude is v^{2}/r. There is tension in the string. The string pulls on the mass with a force F directed towards the center of the circle. This force F is responsible for the centripetal acceleration, F = mv^{2}/r.

The string can support a mass of 25 kg before breaking, i.e. we can let a mass of up to 25 kg hang from the string near the surface of the earth. The maximum tension in the string therefore is F_{max}= mg = (25 kg)(9.8 m/s^{2})^{ }= 245 N.

Given F_{max }= 245 N and F = mv^{2}/r, we can find v_{max}. - Details of the calculation:

v_{max}^{2}= F_{max}r/m = (250 N)(0.8 m)/(3kg). v_{max}= 8.1 m/s.

A coin placed 30 cm from the center of a rotating, horizontal
turntable slips when its speed is 50 cm/s.

(a) What force provides the centripetal acceleration when coin is
stationary relative to the turntable?

(b) What is the coefficient of static friction between coin and
turntable?

Solution:

- Reasoning:

When the coin is at rest relative to the rotating turntable, the force of static friction between the coin and the turntable provides the centripetal acceleration.

The force of static friction has a maximum value f_{s }= μ_{s}N = μ_{s}mg.

Setting μ_{s}mg = mv^{2}/r, we can solve for μ_{s}. - Details of the calculation:

(a) The force of static friction between the coin and the turntable provides the centripetal acceleration.

(b) The magnitude of the maximum force of static friction is f_{s }= μ_{s}N. This maximum force of static friction is equal to mv^{2}/r when v = 0.5 m/s. We have μ_{s}N = μ_{s}mg = mv^{2}/r,

or μ_{s }= v^{2}/(rg) = (0.5m/s)^{2}/(0.3m 9.8m/s^{2}) = 0.085.

Do you feel yourself thrown to either side when you negotiate a curve that is ideally banked for your car's speed? What is the direction of the force exerted on you by the car seat?

**Discuss this with your fellow students in the discussion forum!**

For more information about uniform circular motion motion, study this material from "The Physics Classroom".

Link: Motion Characteristics for Circular Motion

- Speed and Velocity
- Acceleration
- The Centripetal Force Requirement
- The Forbidden F-Word
- Mathematics of Circular Motion

Link: Applications of Circular Motion

- Newton's Second law - Revisited
- Amusement Park Physics
- Athletics