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# 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

ac = v2 / r

= ω2

= (2 π nrps)2 r

= (2 π nrpm / 60)2 r

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

where

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

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

r = circular radius (m, ft)

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

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

### Centripetal Force

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

Fc = m ac

= m v2 / r

= m ω2 r

= m (2 π ns)2

= m (2 π nrpm / 60)2

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

where

Fc = centripetal force (N, lbf)

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

#### 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

ac = ((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

Fc = (1000 kg) (0.965 m/s2)

= 965 N

= 0.97 kN

Related to the gravity force - weight:

Fg = (1000 kg) (9.81 m/s2)

= 9810 N

= 9.8 kN

#### 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

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

= 4.84 ft/s2

The centripetal force can bee calculated as

Fc = (94 slugs) (4.84 ft/s2)

= 455 lbf

### 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

Fc = 0.01097 m r nrpm2                              (3)

where

nrpm = 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)

nrpm - revolutions per minute (rpm)

### 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.

## Related Topics

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Motion - velocity and acceleration, forces and torque.

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