Ch.5 Uniform Circular Motion – Handouts
Uniform Circular Motion: is the motion of an object traveling at a constant
speed on a circular path.
Rotation is motion around an internal axis and Revolution is motion around an
external axis.
Time period: is the time in seconds taken to make one complete rotation /
revolution around a circle.
Linear / tangential speed or velocity: is the circumference of the circle divided
by the time period and measured in m/sec.
v
2 R

Rotational / angular speed or velocity ‘’ is measured in radians / sec.

2

An object in uniform circular motion is changing its direction every instant, so its
velocity is changing even if the speed is constant. Therefore, it is said to be
accelerating. This is called centripetal acceleration, which for an object
v2
traveling with a velocity v is given by ac 
. The centripetal force acting
r
mv 2
on an object of mass m and velocity v is Fc 
. For a can tied to a
r
string and rotated in a circle the centripetal force is the tension in the string. For
planets going around the sun the centripetal force is the gravitational field of the
sun, for electrons in orbit it is the electric field of the nucleus, though, the exact
nature of electron orbits are determined by quantum mechanics.
For objects moving in a circular plane (for example a merry go round), all the
objects irrespective of the radial distance have the same rotational speed , but
different linear speeds. The greater the distance from the center, the greater the
linear speed. V = r
When a passenger in a car turns around a sharp curve, he continues to travel in
his original velocity due to inertia, until he meets the car door which provides the
centripetal force to keep him in. The greater the speed of the car and the sharper
the curve, the greater is the centripetal force. Static friction provides the
centripetal force for vehicles turning around a sharp curve. Fc = fs OR
mv 2
 s mg . Solving we get, v   s.r.g
r
. This gives the maximum velocity
the car can travel and still manage the curve. Since mass does not appear in the
equation, the safe speed is the same whether it is a bicycle or a truck. The
reliance on static friction can be eliminated by banking the curves. For a curve
banked at an inclination of  with the ground, the x-component of the normal
force provides the centripetal force. Fc = FN sin  = mv2 / r. The y-component
of the normal force is balanced by the weight or FN cos  = mg. Dividing the two
we get tan  = v2 / rg or  = tan-1 (v2 / rg)
For a satellite in orbit around the earth, gravitational force provides the centripetal
force. Fc = mv2/ r = GmME / r2. Solving we get v 
GME
. Consequently, for a
r
given orbit, all satellites irrespective of the mass will have exactly the same
orbital speed. The same equation holds for the planets orbiting the sun, except
that the ME will be replaced by the mass of the sun or any other celestial object
as the case may be.
For geosynchronous satellite where the time period is fixed as 24 hr = 86,400
sec, the orbital radius is also fixed. v 
GME 2 r
=
.To avoid the physiological ill

r
effects of weightlessness, artificial gravity can be provided by rotating the space
station at a speed such that the centripetal force can provide a force equal to that
v2
g
of earth’s gravity.
r
or
v
rg