# Centripetal and Centrifugal Force and Acceleration

## Centripetal and Centrifugal acceleration - force due to circular motion

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Centripetal Force and Centrifugal Force is 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 centripetal acceleration and can be expressed as

a_{c}= v^{2}/ r (1)

where

a_{c}= centripetal acceleration (m/s^{2}, ft/s)^{2}

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

r = circular radius (m, ft)

### Centripetal Force

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

F_{c}= m a_{c}

= m v^{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 an opposite centrifugal force.

### Example - Centripetal Acceleration acting on a Car through a Curve

#### Metric Units

A car of *1000 kg* travels a curve of radius *200 m* at *50 km/h*. The centripetal acceleration can be calculated as

* a _{c} = ((50 km/h)(1000 m/km) / (3600 s/h))^{2} / (200 m)*

* = 0.965 m/s ^{2}*

The centripetal force can bee calculated as

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

= 965 N

#### Imperial Units

A car of *3000 lb* travels a curve of radius *100 ft* at *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)*

* = 0.22 ft/s ^{2}*

The centripetal force can bee calculated as

F_{c}= (94 slugs) (0.22 ft/s)^{2}

= 20.7 lb_{f}

### Online Centripetal (Centrifugal) Calculator - Velocity

### Centripetal (Centrifugal) Force - rpm

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

F_{c}= m v^{2}/ r

= m (n 2 π^{ }r / 60)^{2}/r

= 0.01097 m r n^{2}(3)

where

n = revolution per minute - rpm

### Online Centripetal (Centrifugal) Calculator - rpm

### Centrifugal Force

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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