# Centripetal and Centrifugal Acceleration and Force

## Forces due to circular motion and centripetal / centrifugal acceleration.

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

=(1)(π n_{rpm }/ 30)^{2}r

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

= mω^{2}r

= m(2 π n_{s})^{2}r

= m(2 π n_{rpm }/ 60)^{2}r

(2)=m(π n_{rpm }/ 30)^{2}r

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) (4.84ft/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.**