# 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 }= v^{2}/ r

=ω^{2}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/s^{2}, ft/s)^{2}

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 v^{2}/ r

= m ω^{2}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

#### 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/s ^{2}*

* = 0.1 g *

* where *

* 1 g = acceleration of gravity (9.81 m/s ^{2}) *

The centripetal force can bee calculated as

F_{ c }= (1000 kg) ( 0.965 m/s^{2})

= 965 N

= 0.97 kN

Related to the gravity force - weight :

* F _{ g } = (1000 kg) (9.81 m/s^{2}) *

* = 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/s ^{2}) *

* = 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/s ^{2}*

The centripetal force can bee calculated as

F_{ c }= (94 slugs) (4.84 ft/s^{2})

= 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.

### 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_{ rpm }^{2}(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.

### 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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