CIRCULAR MOTION

Ex.4 A particle lis constrained to move in a circular path of radius r = 6m. Its velocity varies with time accordingto the relation v = 2t (m/s). Determine its (i) centripetal acceleration, (ii) tangential acceleration,(iii) instantaneous acceleration at (a) t = 0 sec. and (b) t = 3 sec.

Sol. (a) At = 0, v = 0, Thus ar = 0

but dvdt =2 thus at = 2 m/s

2 and a = 2 2t ra a+ = 2 m/s2

(b) At t = 3 sec. v = 6 m/s so ar =

2 2v (6) 6r 6

= = m/s2

and tdv

a 2dt

= = m/s2 Therefore, a = 2 2a 2 6= + = 40 m/s2

Ex.5 The kinetic energy of a particle moving along a circle of radius r depends on distance covered s asK = As2 where A is a const. Find the force acting on the particle as a function of s.

Sol. According to given problem

12 mv

2 = As2 or v = s

2Am

...........(1)

So 2 2

r

v 2Asa

r mr= = ...........(2)

Further more as at = dv dv ds

.

dt ds dt=

= v dvds ...........(3)

from eqn. (1), v s (2A /m)= dv 2Ads m= ...........(4)

Substitute values from eqn. (1) & eqn. (4) in eqn. (3)

t2A 2A 2As

a sm m m

= =

so 2 2r ta a a= + =

2 222As 2Asmr m

+

i.e. 22As

a 1 [s / r]m

= +

so F = ma = 2As 21 [s / r]+

Ex.6 A particle of mass m is moving in a circular path of constant radius r such that its centripetal accelerationas is varying with time t as a

c = k2 rt2, where k is a constant . Determine the power delivered to particle

by the forces acting on it.