Circular Motion

M. JonesPisgah High School (Ret.)11/09/12

A derivation of the equations for Centripetal Acceleration and Centripetal Force

movement of an object at a constant a velocity in a straight line

a force is exerted on the object at right angles

centripetal force

Centripetal force is defined as a force acting at right angles to the direction of motion.

The centripetal force will cause a change in direction, resulting in circular motion.

centripetal force

The centripetal force will cause a change in direction, resulting in circular motion.

vi

vf

centripetal force

A force acting on an object produces acceleration according to Newton's second law. F = ma

The centripetal force produces a centripetal acceleration.

While the magnitude of the tangential velocities of the object remains constant, the direction is constantly changing.

Acceleration is defined as a change in velocity over time.

Since velocity is a vector composed of both direction and magnitude, a change in either quantity over time is a change in velocity. Therefore, the object must be accelerating even though its tangential speed remains constant.

r

mvF

r

va

2

c

2

c

t

va

vi

vf

r

r

Acceleration is defined as a change in velocity over a change in time.

The first step in finding the centripetal acceleration is to determine the change in velocity, v.

v = vf - vi

t

va

vi

vf-vi

v

r

r

v = vf - vi

We can find v graphically by adding the vectors tip-to-tail. Since v is the difference, we can add the inverse of the initial velocity. That simply means that we will reverse the direction. v is the vector from the tail of the first vector to the tip of the last vector.

- vi

vfv

Since vi and vf have the same magnitude, v, the valuefor v is determined from the Pythagorean formula.

2vvvv 22

vi

vf-vi

v

midpoint of arc

vr

r

If we slide v up into the circle and start it at the midpoint of the arc between vi and vf, we see that the change in velocity is directed to the center of the circle.

This means that the centripetal acceleration and the centripetal force are also directed toward the center of the circle.

But that still leaves us with the question of how do we determine the equation for centripetal acceleration.

r

mvF

r

va

2

c

2

c

vi

vf

sr

r

The distance the object travels between the initial and final points on the circle is given by s, the displacement.

We can determine s from the Pythagorean formula.

2rrrs 22

vi

vf-vi

v

midpoint of arc

v sr

r

2rs 2vv

2r

s

r

s

v

v

2v

v

Since s = vT, we can substituter

tv

v

v

Rearrange v and tr

v

t

v 2

Since a=v/t and F=ma, we get the equations for centripetal acceleration and centripetal force

r

mvF

r

va

2

c

2

c

Since both terms are equal to 2, we can set them equal.

vi

vf-vi

v

midpoint of arc

v sr

r