Uniform Circular Motion
A particle undergoes UNIFORM CIRCULAR v
MOTION is it travels around in a circular arc at a
CONSTANT SPEED. Note that although the
speed does not change, the particle is in fact
ACCELERATING since the DIRECTION OF THE
VELOCITY IS CHANGING with time. a
The velocity vector is tangential to the instantaneous
direction of motion of the particle.
The (centripetal) acceleration is directed towards the centre of the circle
Radial vector (r) and the velocity vector (v) are always perpendicular
The PERIOD OF REVOLUTION time taken for theparticle to go
around the circle. If the speed (i.e., the magnitude of the velocity for
UCM) of the particle v, the time taken is, by definition
• Acceleration by definition
| v | | r |
Circular motion with not constant speed
• Velocity change in general case can be
replace by two orthogonal components-
tangential and radial.
tangential - change in speed magnitude
radial –change in direction
d |v | v
a ar at r
Fig. 4.19, p.95
Forces in Uniform Circular Motion
Recalling that for a body moving in a circular arc or radius, r, with
constant speed, v, the MAGNITUDE of the ACCELERATION, a, is
given by a = v2/r, where a is called the centripetal acceleration.
We can say that a centripetal force accelerates a body by changing the
direction of that body’s velocity without changing its speed.
Note that this centripetal force is not a ‘new’ force, but rather a
consequence of another external force, such as friction, gravity or
tension in a string.
Examples of circular motion are
(1) Sliding across your seat when your car rounds a bend:
The centripetal force (which here is the frictional force between
the car wheels and the road) is enough to cause the car to accelerate
inwards in the arc. However, often the frictional force between you
and your seat is not strong enough to make the passenger go in this
arc too. Thus, the passenger slides to the edge of the car, when its push
(or normal force) is strong enough to make you go around the arc.
(2) the (apparent) weightlessness of astronauts on the space shuttle.
Here the centripetal force which causes the space shuttle to orbit the
earth in a circular orbits is caused by the gravitational force of the
earth on all parts of the space shuttle (including the astronauts).The
centripetal force is equal on all areas of the astronauts body to he/she
feels no relative extra pull etc. on any specific area, giving rise to a
sensation of weightlessness.
Note that the magnitude of the centripetal FORCE is given, from
Newton’s second law by : F = ma = m v2/r
Note that since the speed, radius and mass are all CONSTANTS so
is the MAGNTIUDE OF THE CENTRIPETAL FORCE. However,
DIRECTION IS NOT CONSTANT, varying continuously so as to
point towards the centre of a circle.
How fast a constant speed does the roller r
coasters have to go to ‘loop the loop’
of radius r ?
At the top of the loop, the free body
forces on the roller coaster are gravity Fg N
(downwards) and the normal
force (also inwards). The total acceleration is
also inwards (i.e., in the downwards direction).
Fy ,net N Fg m a y , limit at N 0 (no contact!
thus, Fg m a y m. m g
g v gr
i.e., independent of mass!