This document discusses circular motion and centripetal acceleration. It defines centripetal acceleration as the acceleration an object experiences when moving in a circular path, which causes a change in the direction of motion towards the center of the circle. The document provides equations for centripetal acceleration, relating it to the object's velocity, radius of the circular path, and angular speed. Examples are given of forces that provide centripetal acceleration, such as gravity keeping the moon in orbit or friction keeping cars from slipping outward while turning.

1. 2.4 – Circular Motion
Topic 2 - Mechanics

2. Acceleration
● Recall that when a nett force acts on an object
it causes an acceleration.
● This is the basis of Newton's second law
● This acceleration will cause a change in
velocity of the object
● It will change speed,
● It will change direction
● It will change speed and direction

3. Centripetal Acceleration
● Consider an object that is forced to go in a
circle by a string tied at the centre of that circle.
● IF the string is cut, the object will move off with
a linear velocity.
● IF the object is moving at constant speed then;
● There can be no component of acceleration in the
direction of the velocity.
● There must still be an acceleration towards the
centre of the circle because the string is there.
● The direction of the velocity must change by this
radial acceleration

4. Centripetal Acceleration
● Acceleration is the rate
of change of velocity.
● Assume that the object
moves from A to B in
time t.
● The change in velocity
can be found
graphically by v – u
u
v
r
rθ
u
v
s
θ

5. Centripetal Acceleration
● Consider that the
displacement triangle
and the velocity triangle
are mathematically
similar if the speed and
radius of the circle are
constant.
u
v
r
rθ
u
v
 v
v
=
s
r
s
θ

6. Centripetal Acceleration
● The average velocity is defined
as:
Substituting this in the previous
equation gives:
● Which rearranges to:
u
v
r
rθ
u
v
v=
s
 t
s
θ
 v
v
=
v  t
r
 v
 t
=
v2
r
=ac

7. Centripetal Acceleration
● The object travels through a total angle of
2π radians in one orbit in a time Period of
T
● The angular speed is therefore
● The angle turned in time t is therefore
(distance = speed x time):
● Degrees can be converted into radians by:
u
v
r
rθ
u
v
s
θ
=
2
T
rad =
deg
360
×2
= t

8. Centripetal Acceleration
● Using the angular
speed the orbital speed
is thus:
● Which makes the
acceleration equation:
u
v
r
rθ
u
v
s
θ
v=
2 r
T
=r 
ac=r 2

9. Centripetal Acceleration
● Mathematically, the position vector of a particle at
time t is given by:
● Differentiating for v gives:
● Differentiating again gives the acceleration:
r=rcos  xr sin  y
r=rcos t xr sin t y
v=
d
dt
r =−r sin t xr cos t y
a=
d
dt
v=
d2
dt
2
r=−r 
2
cos t x−r 
2
sin  t y
a=−2
r cost xr sin t y
a=−
2
r
x
y
r
rsinθ
r cos θ

10. What causes this Acceleration?
● As has been shown, the uniform circular
acceleration occurs along a radius (radially) as
shown by the r in the equation and towards the
centre of the circle, as shown by the -.
● Any force that acts in this way will cause a
centripetal acceleration.
● e.g. the gravitational pull of the Earth on the Moon
causes the acceleration that keeps the Moon in
orbit.
● The sideways friction force (towards the centre) that
keeps a car's tyres from slipping outwards causes
the acceleration to make the car turn the corner.

11. Centrifugal Acceleration
● A centrifugal (centre-fleeing) acceleration does
not really exist.
● It is a consequence of Newton's first law that
states that objects with inertia will continue in a
straight line unless an external force acts.
● When we go around a corner in a car, we feel that
we are being thrown outwards.
● In reality our inertia causes us to try to go straight.
● The forces acting on the car tyres makes the car
accelerate (turn) across us.
● We collide with the side of the car, which exerts a
sideways force on us (Newton's 3rd
), that causes us

12. Questions
● The Moon orbits the Earth at a radius of
3.2x106
m in a time period of 28.5 days. What is
the centripetal acceleration of the Moon?
● The Earth orbits the Sun at a radius of 150
million km in a year. What is the centripetal
acceleration of the Earth?
● A formula one car travels around a bend of
radius 25m at 110kmh-1
. What is its centripetal
acceleration?

13. Questions
● The tyres of a car of mass 1500kg can exert a
maximum sideways force of 500N when
cornering. What is the maximum speed that
this car can travel around a bend of radius 8m
and radius 24m?
● A formula 1 car has a mass of 625kg. The
driver wants to take a corner at no less than
90kmh-1
using tyres that can exert a maximum
sideways force of 35kN. What is the minimum
radius corner that the car can cope with before
slipping?