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.
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