Radial Acceleration: Formula, Derivation, Units

Radial acceleration happens when a body is moving in a circular motion. The rate of change of angular velocity is directed towards the radius of the body. It is symbolized as ar and the main cause behind this acceleration is the centripetal force. 

Sir Issac Newton, a great English mathematician, and physicist developed this theory when he was discussing the nature of the gravitational force of the earth.

We know that acceleration is the rate at which velocity changes equally, at equal intervals of time. This happens in terms of both speed and magnitude. It is a vector quantity and is denoted by a meter per second square. 

When an object is moving in a curved linear path, it will undergo two kinds of motion. 

1. Tangential acceleration, or
2. Radial acceleration

Keyterms: Acceleration, Tangential acceleration, Radial acceleration, Motion, Linear path, Centripetal force, Circular Motion, Velocity, Angular velocity, Distance

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Tangential Acceleration

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This can be defined as the rate of change of tangential velocity with respect to time. This concept is related to circular motion and can be called the same as linear acceleration. If a body undergoes tangential acceleration, it means its movement is more likely to be inclined and directed towards the tangential direction. In a uniform circular motion of a body, the tangential acceleration becomes zero. This is because the angular velocity is constant due to the centripetal force hitting perpendicular to the velocity. 

A car accelerating in a circular path can be called concerning an example of this acceleration. The SI unit for tangential acceleration is radian per second square. It goes by the formula, 

at = Δv/Δt,

Whereat implies tangential acceleration

Δv denotes the angular velocity, and 

Δt denotes the change in the amount of time taken

It’s distance formula, at=v.dv/ds

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Radial Acceleration

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When a body moves in a circular path, it is said to be in radial acceleration. It can be said that when the angular velocity of a moving body changes in a unit of time, it is a radial acceleration. An acceleration, where the movement of the body is concerned along the radius and is directed towards the center is radial acceleration. This kind of movement/ acceleration takes place in a uniform circular motion. It should be remembered that the centripetal acceleration is always acting towards the center, whereas, in radial acceleration, is either going towards the center or away from it. The force acts in the radial direction here. 

Radial Acceleration

For example, if you are sitting in a merry-go-round, it is because of radial acceleration and the centripetal force acting, causing the swing to move in a round motion. The radial aspect is an important factor behind the circular movement of the body along the radius.

Read more: Centripetal and Centrifugal force

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Formula

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Circle                                                                                Triangle

We know that,

By applying the property of similar triangles

QR/PQ = i/r

Considering QR as the arc, 

AB = v × dt

v + dv ≈ dv × QR/PQ

dv/v * vxdt/r

dv/v

dv/v = v²/r

Ar = v²/r

Read More to Know About: Centripetal Acceleration

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Derivation

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Radial acceleration is symbolized as ‘ar‘ because it is directed towards the center. The factor behind this radial movement is the action of the centripetal force acting upon it.

The force on a moving body can be denoted as, 

F = ma, (where m is the mass of the body, and a is the acceleration)

It originates from Newton’s second law of motion. It explains that the force produced on a body is equal to the mass of the body multiplied by its acceleration. 

The equation for the centripetal force acting on a body is, mv²/r

Equating both the equations, we get,

mar = mv²/r, which ultimately gives us,

ar = mv²/r (centripetal acceleration/radial acceleration)

Read more: Types of Friction

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Units

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The units of measurement of radial acceleration are radians per second squared, and meters per second squared. 

These two units can be written like ωs^-2 or ms^-2.

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Things to Remember

• Radial acceleration is always directed towards the center of the circle, it is for this reason that it is also called centripetal acceleration. This acceleration is normal to instant velocity. Suppose a motion is not linear, for example, a rectilinear motion, the radial acceleration will be zero in this case, irrespective of the fact whether it is uniform or not. 
• Angular acceleration can be divided into two aspects, Radial, and Tangential acceleration. Radial acceleration is measured in terms of Radians per the second square which is represented as ωs-2. The tangential acceleration can be defined as an existing element of angular acceleration that is tangential to the circular path. 
• The tangential acceleration of a body is said to be 0 when it is in a uniform circular motion. The reason behind this is that speed remains constant, which leads to such a situation. The magnitude of the tangential acceleration is equal to the rate of change of speed of the particle in relation to time and it always remains tangential to the path it is on.
• At any given instance, the magnitude of the radial acceleration is v2/r where v is the speed and r is the radius of curvature acting along the body. The direction of radial acceleration happens along the radius. 
• Planets revolving around the sun, children playing in swings and merry-go-round, driving in a circular path, whirling a stone tied to a string, and the dryer doing its job in the washing machine can all be called examples of radial acceleration. 

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