Centripetal and Centrifugal Acceleration Force

Centripetal and Centrifugal Force are the action-reaction force pair associated with circular motion.

Centripetal Acceleration

Velocity is a vector - specifying how fast (or slow) a distance is covered and the direction of the movement. Since the velocity vector (the direction) of a body changes when moved in a circle - there is an acceleration.

This  acceleration is named the centripetal acceleration - and can be expressed as

a_c = v² / r  

    = ω² r 

    = (2 π n_rps)² r   

    = (2 π n_rpm / 60)² r   

    =  (π n_rpm / 30)² r                      (1)

where

a_c = centripetal acceleration (m/s², ft/s²)

v = tangential velocity (m/s, ft/s)

r = circular radius (m, ft)

ω = angular velocity (rad/s)

n_rps = revolutions per second (rev/s, 1/s)

n_rpm = revolutions per min (rev/min, 1/min)

Centripetal Force

According Newton's second law the centripetal force can be expressed as

F_c = m a_c

    = m v² / r  

    = m ω² r        

    = m (2 π n_s)² r 

    = m (2 π n_rpm / 60)² r 

    =  m (π n_rpm / 30)² r                             (2)

where

F_c = centripetal force (N, lb_f)

m = mass (kg, slugs)

According to Newton's Third Law the centripetal force acting on the object has a centrifugal force of the same magnitude acting in the opposite direction.

Example - the Centripetal Acceleration and Force acting on a Car through a Curve

• Banked Turn

Metric Units

A car with mass 1000 kg drives through a curve with radius 200 m at speed 50 km/h. The centripetal acceleration can be calculated as

a_c = ((50 km/h) (1000 m/km) (1/3600 h/s))² / (200 m)

   = 0.965 m/s²

   = 0.1 g

where

1 g = acceleration of gravity (9.81 m/s²)

The centripetal force can bee calculated as

F_c = (1000 kg) (0.965 m/s²)

    = 965 N

    = 0.97 kN 

Related to the gravity force - weight:

F_g = (1000 kg) (9.81 m/s²)

    = 9810 N

    = 9.8 kN

• car traction force

Imperial Units

A car with weight (gravity force) 3000 lb travels through a curve with radius 100 ft with speed 15 miles/h.

The mass of the car can be calculated as

m = (3000 lb) / (32 ft/s²)

   = 94 slugs

The centripetal acceleration can be calculated as

a_c = ((15 miles/h)(5280 ft/mile) / (3600 s/h))² / (100 ft)

   = 4.84 ft/s²

The centripetal force can bee calculated as

F_c = (94 slugs) (4.84 ft/s²)

    = 455 lb_f

Centripetal (Centrifugal) Calculator - velocity

This calculator can be used if the velocity of an object is known - like a car in a turning curve.

m - mass of object (kg)

v - velocity of object (m/s)

r - radius of curve (m)

Centripetal (Centrifugal) Force - rpm

Equation (2) can be modified to express centripetal or centrifugal force as a function of revolution per minute - rpm - as

F_c = 0.01097 m r n_rpm²                              (3)

where

n_rpm = revolution per minute (rpm)

Centripetal (Centrifugal) Calculator - rpm

This calculator can be used if revolution speed of an object is known - like a turning bowl in a lathe.

m - mass (kg)

n_rpm - revolutions per minute (rpm)

r - radius (m)

• Acceleration Unit Converter

Centrifugal Force

Force is an abstraction representing the push and pull interaction between objects.  Newton's third law states that 

• for every acting force there is an equal and opposite reaction force

Therefore there must be an equal and opposite reaction force to the Centripetal Force - the Centrifugal Force.

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