Motion in a Vertical Circle
Consider a ball moving
in a circular path as
shown.
What happens to the speed of the ball?
What happens to the tension in the string?
Vertical circle
Or here
V min
Fw
F s1
Because of gravity, the
speed of the ball is not
uniform.
The ball accelerates on the
downward path and
decelerates on the upward
path.
Speed is at a minimum
at the top and a
maximum at the bottom.
F s2
Fw
V max
At the top:
Fc 1  Fs 1  Fw
2
Fc 1 
mv min
r
 Fs 1  Fw
Since vmax > vmin then
At the bottom:
Fs2 > Fs1
Fc 2  Fs 2  Fw
i.e. the string
tension is greater
at the bottom!
2
Fc 2 
mv max
r
 Fs 2  Fw
Recall
This implies that there is some velocity below which the ball
will not move in a circular path. The string will slacken at the
ball’s highest point.
This could also happen
if a car did not have
enough speed to make
a loop de loop.
The minimum value is when Fs1 = 0. At this time , the ball just
makes it around the circle.
At the top:
Fc 1  Fs 1  Fw
2
mv min
 0  mg
r
2
v min
 g
r
v
rg
This value of vmin is called the critical velocity. It
depends only on the radius of the circle and the
acceleration due to gravity. It does NOT depend on
the mass of the object in motion.
A satellite in orbit around a planet moves with the
critical velocity for it’s radius of orbit (altitude)
See animation
Note that the radius of orbit for a satellite is equal to the
radius of the planet (Earth) plus it’s altitude.
alt
Re
R o rbit
Sample
problem #3