# Centripetal and Centrifugal Force - Acceleration

## Centripetal and centrifugal acceleration - forces due to circular motion

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 - Centripetal Acceleration 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

The centripetal force can bee calculated as

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

= 965 N

Related to the gravity force - weight:

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

= 9810 N

#### 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

### Online Centripetal (Centrifugal) Calculator - velocity

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)

### Online Centripetal (Centrifugal) Calculator - rpm

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

• Dynamics - Motion - velocity and acceleration, forces and torques
• Mechanics - Forces, acceleration, displacement, vectors, motion, momentum, energy of objects and more

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