Angular velocity of a particle is the rate of change of angular displacement.
Expression for centripetal force on a particle undergoing uniform circular motion:
i) Suppose a particle is performing U.C.M in anticlockwise direction.
The co-ordinate axes are chosen as shown in the figure.
Let,
A = initial position of the particle which lies on positive X-axis
P = instantaneous position after time t
θ = angle made by radius vector
ω = constant angular speed
\(\vec{r}\)= instantaneous position vector at time t
ii) From the figure,
where, \(\hat{i}\) and \(\hat{j}\) are unit vectors along X-axis and Y-axis respectively.
iv. Velocity of the particle is given as rate of change of position vector.
v. Further, instantaneous linear acceleration of the particle at instant t is given by,
vi. From equation (1) and (2),
Negative sign shows that direction of acceleration is opposite to the direction of position vector. Equation (3) is the centripetal acceleration.
vii) Magnitude of centripetal acceleration is given by a = ω2r
viii) The force providing this acceleration should also be along the same direction, hence centripetal.
This is the expression for the centripetal force on a particle undergoing uniform circular motion.
ix) Magnitude of F = mω2r = mv2/r = mωv
