Uniform circular motion: motion in a circular path at constant speed.
Is there an acceleration involved here?
Yes - the velocity changes because its direction changes.
A ball is being whirled in a circle. If the string is released when the ball is at the position shown, which path will the ball follow?
If the string is released there is no force to deflect the path of the ball, so it will continue in a straight line, following path 2.
r = the radius of the circular path
T = the period, the time to go around once
v = 2πr/T
As in straight-line motion, the relationship between a and v is the same as that between v and r:
a = 2πv/T
Combining these two equations gives us:
centripetal acceleration: ac = v2/r
For motion on circular paths it can be useful to describe motion using angular variables. Instead of asking how much distance has been covered, we sometimes ask how much of an angle something has spun through. There are equivalent questions for velocity and acceleration.
Distance: s = rθ
Velocity: v = rω
Acceleration: at = rα
This acceleration involves a speeding up or slowing down of an object as it moves along a circular path, and is equal to zero for uniform circular motion. The a is in a direction tangent to the circle, so its the tangential acceleration. This is very different from the centripetal acceleration, which acts in the radial direction.
Keeping in mind that a free-body diagram shows all the forces acting on an object, and that those forces come from interactions between that object and other objects, what does the free-body diagram look like for the Earth in its roughly circular orbit around the Sun?
The only interaction we've got to worry about is the force of gravity. The Sun exerts a gravitational force on the Earth that points toward the Sun.
Knowing the distance to the Sun and how long it takes the Earth to orbit the Sun, we can calculate the acceleration the Earth is experiencing.
r = 150 million km = 1.5 x 1011 m
T = 1 year = π x 107 s
v = 2π r / t = 3 x 104 m/s = 30000 m/s = 30 km/s.
ac = v2/r = 9 x 108 / 1.5 x 1011 = 6 x 10-3 m/s2
Pretty small, but just right to keep us in orbit.
The centripetal acceleration is the special form the acceleration has when an object is experiencing uniform circular motion. It is:
ac = v2 / r
and is directed toward the center of the circle.
Newton's second law can then be written as:
ΣF = ma = mv2/r
I prefer NOT to use the phrase "centripetal force" because it makes you think there is a magical force that appears when an object is experiencing uniform circular motion. There is no such thing, and, in my opinion, you should never put a centripetal force on a free-body diagram.
When an object is experiencing uniform circular motion there is definitely a net force directed toward the center of the circle, but this force comes from one or more of the standard forces we've discussed already. Depending on the situation, it could be the force of gravity, the normal force, tension, friction, some combination of these, or even a combination of components of these.