Uniform Circular Motion

When a velocity changes, it experiences an acceleration

Since velocity is a vector, acceleration can be experienced as …

A change in speed in the same directionA change in direction at the same speedA change in both speed and direction

• Consider a body travelling in a straight line:

• Now consider that it changes direction:

• We know from inertia that bodies tend to travel in a straight line at constant velocity

• If a body changes direction, we know there must have been a force acting upon it, and a corresponding acceleration.

• A body moving in a circle is constantly changing direction, and therefore constantly accelerating.

• In which direction does it accelerate?• Recall that a = … (vf – vi)/t

Vi

Vf

• By subtracting vi from vf …

• … dividing by a scalar: ‘t’• … and placing the resultant vector between the

initial and final vectors• … we find that our acceleration points towards

the centre of the circle• The force which causes this acceleration is

called CENTRIPETAL FORCE

Vf

ViVf – Vi

Vi

Vf

This makes sense because when you consider the forces here …

T

… and here

mg

What creates the centripetal force when a motorbike turns?

… or when an aeroplane turns?

‘Centrifugal Force’• It is often easy to believe that some force exists

to pull spinning objects outwards.• In Newtonian physics, no such force exists• This apparent ‘force’ is simply the body’s

tendency to continue travelling in a straight line, disallowed by the constant centripetal force.

• The effect is similar to the feeling of being pulled to the right when turning left in a car. Of course you are not being ‘pulled’ to the right, but trying to continue in your original direction.

• Relativity theory argues that such a centrifugal force exists from the spinning object’s frame of reference …

• … but that’s another story

Calculating Centripetal Force

In order to calculate centripetal force, we need to:

1. Understand angular velocity2. Understand the chain rule in differentiation3. Use angular velocity and the chain rule to

calculate centripetal acceleration4. Use acceleration to calculate centripetal

force.

r = radiusv = velocity= angle= angular velocity

v

r

1. Angular Velocity

How far will the body travel in one revolution?

How many radians will the body travel through in one revolution?

C = rC/t = (t)r

v = r = v/r

C = 2r metres

=2radians

= v/r

This implies:

• That larger velocities will produce larger angular velocities

• And that larger radii will produce smaller angular velocities– Eg: a runner on the inside will have an

advantage over a runner on the outside

The chain ruley = f(x)y’ = f’(x) = dy/dx

Sometimes we don’t have a relationship between y and x. However, we have relationship between y and u, and a relationship between u and x.

ie: y = f(u) and u = f(x)

We can can use these to find a relationship between y and x

Using the ‘chain rule’

If: y = f(u) and u = f(x)

Then:f’(y) = dy/du and f’(u) = du/dx

And:dy/dx = …

dy x dudu dx

Calculating Centripetal Acceleration

p(t) =v(t) =

dp/dt =

=

==

r

r.cos

r.sin

(r.cos + r.sindp/dt(-r.sinddt +

r.cosddt)-r(sinddt -

cosddt)-r(sin - cos

-r(sin - cos

Calculating Centripetal Acceleration

v =a =

dv/dt =

====

r

r.cos

r.sin

-r(sin - cosdv/dt-r(cosddt +

sinddt)-r(cos+sin)-r(cos+ sin)-(r.cos+ r.sin)-p

Calculating Centripetal Acceleration

a =IaI =

==

r

r.cos

r.sin

-pI-I.r(v/r).rv/ r

a = v2/r

This makes sense …The body will change direction more quickly if:

The velocity is higherThe radius is smaller

For higher velocity, this is easy to imagineAs the radius increases, the circumference

becomes straighter (eg: the horizon) and the body’s change in direction is very gradual, even at high speeds.

Calculating Centripetal Force

Since: F = maAnd : a = v2/r

Centripetal force is simply:

F = mv2/r

Angular velocity