The Physics
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# Centripetal Force

## Summary

Ways to accelerate
A force that is… will make a moving object… or…
parallel to velocity change speed (speed up) accelerate (in the usual sense of the word)
antiparallel to velocity change speed (slow down) decelerate (which is just another way to accelerate)
perpendicular to velocity change direction (turn) accelerate centripetally (which is also just another way to accelerate)
• A centripetal acceleration (ac)…
• occurs whenever a moving object changes direction
• does not change the speed of an object
• acts at right angles to the velocity at any instant
• is directed toward the center of a circle (center seeking)
• A centripetal force (Fc)…
• is the force that makes a moving object change direction
• is not a particular force, but the name given to whatever force or combination of forces is responsible for a centripetal acceleration
• acts at right angles to the velocity at any instant
• is directed toward the center of a circle (center seeking)
• Directions in circular motion:
• Velocity (v) is tangential (lies on a tangent to the path).
• Centripetal acceleration and centripetal force are centripetal (point toward the center of a circle).
• Centripetal acceleration is perpendicular to velocity.
• Centripetal force is parallel to centripetal acceleration.
• Equations [with SI units]
• speed and velocity
linear quantities
 v = 2πr T
where v = speed [m/s] π = the mathematical constant r = radius of circular path [m] T = period of revolution [s]
angular quantities
 v = ω × r
where v = translational velocity vector [m/s] r = radius vector [m] ω = angular velocity vector [rad/s]
• frequency and period
linear quantities
 ƒ = 1 T
where  f = frequency of revolution [Hz = 1/s = s−1] T = period of revolution [s]
angular quantities
 ω = 2π = 2πƒ T
where  ω = magnitude of angular velocity or angular frequency [rad/s] π = the mathematical constant T = period of revolution [s] f = frequency of revolution [Hz = 1/s = s−1]
• centripetal acceleration
linear quantities
 ac = v2 = 4π2r r T2
where  ac = centripetal acceleration [m/s2] v = speed [m/s] r = radius of circular path [m] π = the mathematical constant T = period of revolution [s]
angular quantities
 ac = −ω2r = ω × v
where  ac = centripetal acceleration vector [m/s2] ω = magnitude of angular velocity or angular frequency [rad/s] r = radius vector [m] ω = angular velocity vector [rad/s] v = translational velocity vector [m/s]
• A centrifugal force
• is a fictitious force (a.k.a. an apparent force or a pseudo-force)
• is not the result of an iteraction between objects
• ceases to exist when an object stops moving in a circle
• is an illusion experienced in a rotating reference frame
• appears to be directed away from the center of a circle (center fleeing)
• Circular motion in which the speed is constant is called uniform circular motion.