Centripetal Acceleration: Formula, Derivation, and Applications

Centripetal acceleration is the rate at which the tangential velocity of a body in circular motion changes.

• The net force that accelerates an object in a circular motion is known as Centripetal force.
• The centripetal acceleration is directed toward the center, which is perpendicular to the motion of the body.
• In a curved path, the centripetal force maintains the constant velocity of a body.
• Christian Huygens provided the first mathematical explanation of centripetal acceleration in 1659.

Key Terms: Centripetal acceleration, Force, Centripetal force, Vectors, Circular motion, Tangential velocity, Dimensional formula, SI unit, Newton’s second law of motion

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What Is Centripetal Acceleration?

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The property of the motion of an object in a circular path is called centripetal acceleration.

• Centripetal acceleration is the acceleration of an object moving in a circle with its acceleration vector pointing in the direction of the center of the circle.
• In your daily life, you probably have observed many examples of centripetal acceleration.
• Centripetal acceleration occurs when you drive a car in a circle; a satellite revolving around the Earth also experiences this acceleration.
• The meaning of Centripetal is "towards the center."
• The SI unit of centripetal acceleration is m/s².
• It is a vector quantity, and its direction is always towards the center of the circular path.
• The dimensional formula of centripetal acceleration is M⁰L¹T^-2.

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What Is Centripetal Force?

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A force acting on a body moving around the center of a circular path is known as a centripetal force.

• This force keeps the body moving in a circular path.
• Whenever a body moves in a circular path, the centripetal force acts on it.
• It is directed toward the center of the circular path.
• An object experiences an increasing centripetal force toward the center of a circle when it travels around it at a constant speed.
• Centripetal refers to the tendency to move in a direction toward the center.
• In the case of a satellite, the centripetal force is always due to the force of gravity.
• For a swing ball, this force is provided by the tension in the string.
• It is the frictional force between the car and the road that provides the centripetal force.

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Centripetal Acceleration Formula

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The formula for centripetal acceleration for a body moving in a circular path is given by

\(a_c=\frac {v^2}{r}\)

Where

• a_c is the centripetal acceleration
• v is the linear velocity of the body
• r is the radius of the circular path

The formula for centripetal force is given by

\(F=ma_c=\frac {mv^2}{r}\)

Where m is the mass of the body.

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Derivation of Centripetal Acceleration

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Consider the body of mass m moving in a circular path of radius r. Then according to Newton’s second law of motion, the force acting on the body is given by

\(F=ma\)

Here, a is the acceleration, the rate of change of velocity with respect to time.

From the above diagram, we have

\(\vec {PQ}+ \vec {QS} = \vec {PS}\)

\(\Rightarrow - \vec {v_1}+ \vec {v_2}=\Delta \vec { v}\)

\(\Rightarrow \Delta v= v_2 -v_1\)

Now the triangle AOB and PQS are similar, therefore

\(\frac {\Delta v}{AB}=\frac {v}{r}\)

Also, we have

 \(AB = v \Delta t\)

On substituting, we get

\(\frac {\Delta v}{v \Delta t}=\frac {v}{r}\)

\(\Rightarrow \frac {\Delta v}{\Delta t}=\frac {v^2}{r}\)

But, \(\frac {\Delta v}{\Delta t}=a\) i.e. the centripetal acceleration

Therefore, Centripetal acceleration, \(a=\frac {v^2}{r}\)

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Applications of Centripetal Acceleration

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The following are the applications of centripetal acceleration

• When a car turns, the friction force acting on the car's spun wheels generates the centripetal force required for circular motion.
• The centripetal force required for circular motion is provided by the force of gravity acting on the moon as it orbits the Earth.
• The centripetal force required for circular motion is provided by the tension force acting on a bucket of water linked to a thread and spun in a circle.

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Things to Remember 

• Centripetal acceleration is the rate at which the tangential velocity of a body in circular motion changes.
• It is a vector quantity.
• The direction of centripetal acceleration is towards the center of the circle.
• Its SI unit is m/s².
• The dimensional formula of centripetal acceleration is M⁰L¹T^-2.
• The formula of centripetal acceleration is, a = v²/r.
• A force acting on a body moving around the center of a circular path is known as a centripetal force.
• The centripetal force maintains the constant velocity of a body.