** Tangential Acceleration vs Centripetal Acceleration
**

Acceleration is the rate of change of velocity, and when expressed using calculus, it is the time derivative of the velocity. Tangential acceleration and centripetal acceleration are components of the acceleration for a particle or a rigid body in a circular motion.

**Tangential Acceleration**

Consider a particle moving along a path as shown in the diagram. At the instance considered, the particle is in angular motion, and the velocity of the particle is tangential to the path.

The rate of change of tangential velocity is defined as the tangential acceleration, and it is denoted by ** a_{t}**.

**a _{t }= dv_{t}/dt**

However, this does not account for the total acceleration of the particle. According to Newton’s first law, for a particle to deviate from the rectilinear path and turn, there must be another force; hence we can deduce that there must be an acceleration component directed perpendicular to the tangential acceleration component, i.e. towards the point O at the instance shown. This component of acceleration is known as the * normal acceleration*, and it is denoted by

*.*

**a**_{n}**a _{n }= v_{t}^{2}/r**

If * u_{t }*and

*are the unit vectors in the tangential and normal directions, the resultant acceleration can be given by the following expression.*

**u**_{n}**a = a _{t}u_{t }+ a_{n}u_{n }= (dv_{t}/dt) u_{t }+ (v_{t}^{2}/r) u_{n}**

**Centripetal Acceleration**

Now consider that the force inducing the normal acceleration is constant. In this case, the particle enters a circular path with a radius r. This is a special case in angular motion, and the normal acceleration is given the term centripetal acceleration. The force driving the circular motion is known as the * centripetal force*.

The centripetal acceleration is also given by the above expression, but angular relations in velocity and acceleration can be used to give it in terms of the angular velocity.

Therefore,

**a _{c }= v_{t}^{2}/r = -rω^{2}**

(Negative sign indicate that the acceleration pointed in the opposite direction of the radius vector)

The net acceleration can be obtained by the resultant of the two components a_{c} and a_{t}.

**What is the difference between Tangential Acceleration and Centripetal Acceleration?**

• Tangential and centripetal accelerations are two components of the acceleration of a particle/body in a circular motion.

• The tangential acceleration is the rate of change of tangential velocity, and it is always tangential to the circular path, and normal to the radius vector.

• Centripetal acceleration is pointed towards the center of the circle, and this acceleration component is the major factor that keeps the particle in the circular path.

• For a particle in a circular motion, the acceleration vector always lies within the circular path.

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