Question

Derivation of centripetal acceleration in uniform circular motion

Answer 1

Centripetal acceleration:

If an object moves in a circle it changes its velocity i.e. it is accelerating. This acceleration is known as the centripetal acceleration.

or we can say that the centripetal acceleration is defined as the rate of change of tangential velocity.

Now we know that, the centripetal force required to keep a particle of mass ‘m’ revolving in a circular path of radius ‘r’ with a speed ‘v’ is,

F = mv 2 /r

Now, centripetal acceleration (a c ) is the radial acceleration due to the centripetal force.

Using,

a c = F/m

=> a c = [mv 2 /r]/m

=> a c = v 2 /r

The centripetal acceleration is directed along the radius in the circular motion.

Answer 2

$\huge \mathfrak{answer}$

Centripetal Acceleration. Centripetal acceleration is the rate of change of tangential velocity: Consider an object moving in a circle of radius r with constant angular velocity. The tangential speed is constant, but the direction of the tangential velocity vector changes as the object rotates.
A body that moves in a circular motion (of radius r) at constant speed (v) is always being accelerated. The acceleration is at right angles to the direction of motion (towards the center of the circle) and of magnitude v2 / r.

The direction of acceleration is deduced by symmetry arguments. If the acceleration pointed out of the plane of the circle, then the body would leave the plane of the circle; it doesn't, so it isn't. If the acceleration pointed in any direction other than perpendicular (left or right) then the body would speed up or slow down. It doesn't.Now for the magnitude. Consider the distance traveled by the body over a small time increment Δt:
We can calculate the arc length s as both the distance traveled
The angular velocity of the object is thus v / r (in radians per unit of time.)

The right half of the diagram is formed by putting the tails of the two v vectors together. Note that Δθ is the same in both diagrams.
(distance = rate ×time = v Δt) and using the definition of
a radian (arc = radius ×angle in radians = r Δθ:)