Tangential and Centripetal Acceleration

by • 04/07/2012 • GeneralComments (0)334

Tangential Velocity

When a body is moving along a circle or circular path, the velocity of the body along the tangent of the circle is called its tangential velocity.

Vt = r ω

Tangential velocity is not same for every point on the circular path.
Centripetal Acceleration

A body moving along a circular path changes its direction at every instant. Due to this change, the velocity of the body ‘V’ is changing at every instant. Thus body has an acceleration which is called its centripetal acceleration. It is denoted by a(c) or a1 and always directed towards the centre of the circle. The magnitude of the centripetal acceleration a(c) is given as follows

a(c) = V2 / r, ……………………… r = radius of the circular path
Prove That a(c) = V2 / r

Proof

Consider a body moving along a circular path of radius of r with a constant speed V. Suppose the body moves from a point P to a point Q in a small time Δt. Let the velocity of the body at P is V1 and at Q is V2. Let the angular displacement made in this time be ΔO .

Since V1 and V2 are perpendicular to the radial lines at P and Q, therefore, the angle between V1 and V2 is also Δ0, Triangles OPQ and ABC are similar.

Therefore,

|ΔV| / |V1| = Δs / r

Since the body is moving with constant speed

Therefore,

|V1| = |V2| = V

Therefore,

ΔV / V = Vs / r

ΔV = (V / r) Δs

Dividing both sides by Δt

Therefore,

ΔV / Δt = (V/r) (V/r) (Δs / Δt)

taking limit Δt → 0.
Proof That a(c) = 4π2r / T2

Proof

We know that

a(c) = V2 / r

But V = r ω

Therefore,

a(c) = r2 ω2 / r

a(c) = r ω2 …………………. (1)

But ω = Δθ / Δt

For one complete rotation Δθ = 2π, Δt = T (Time Period)

Therefore,

ω = 2π / T

(1) => a(c) = r (2π / T)2

a(c) = 4 π2 r / T2 ……………… Proved

Tangential Acceleration

The acceleration possessed by a body moving along a circular path due to its changing speed during its motion is called tangential acceleration. Its direction is along the tangent of the circular path. It is denoted by a(t). If the speed is uniform (unchanging) the body do not passes tangential acceleration.

Total Or Resultant Acceleration

The resultant of centripetal acceleration a(c) and tangential acceleration a(t) is called total or resultant acceleration denoted by a.
Centripetal Force

If a body is moving along a circular path with a constant speed, a force must be acting upon it. Direction of the force is along the radius towards the centre. This force is called the centripetal force by F(c).

F(c) = m a(c)

F(c) = m v(2) / r ………………… {since a(c) = v2 / r}

F(c) = mr2 ω2 r ………………….. {since v = r ω}

F(c) = mrω2

 

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