prove that the centripetal acceleration of a particle moving with a uniform velocity in a circular path is
ar = v²/r = rw² =4π²n²r = 4π²r/ T²
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Class 11
>>Physics
>>Motion in a Plane
>>Uniform Circular Motion
>>Derive the formula for centripetal accel
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Derive the formula for centripetal acceleration of a particle which is moving with a constant speed on a circular path.
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Consider the distance travelled by the body over a tome Δt
Arc length S=VΔt
also S=rΔθ
∴rΔθ=VΔt
Δt
Δθ
=
r
V
⇒
dt
dθ
=
r
V
angular velocity
Now from the second diagram:
sin(Δθ/2)=
2V
ΔV
ΔV=2Vsin(Δθ/2)
Δθ
ΔV
=
Δθ
2Vsin(Δθ/2)
⇒
Δθ
ΔV
=
(Δθ/2)
Vsin(Δθ/2)
⇒
dθ
dV
=lim
Δθ⟶0
(Δθ/2)
Vsin(Δθ/2)
=V
⇒
dθ
dV
=
dθ
dv
dt
dθ
=V×
r
V
=
r
V
2
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