274
6 Oscillations
Since the spring and the wire are in series, the effective spring constant k eff is
given by
k eff =
k k
k + k
(2)
The time period of oscillations is given by
T = 2π
m
k eff
(3)
Combining (1), (2) and (3)
T = 2π
m(Y A + k L)
Y Ak
6.44 In Fig. 6.15, C is the point of contact around which the masses M and m rotate.
As it is the instantaneous centre of zero velocity, the equation of motion is of
the form τ c = I c ¨
θ , where I c is the moment of inertia of masses M and m
with respect to point C. Now
I c =
1
2
M R
2
+ M R
2
+ md
2
(1)
where d
2
= L
2
+ R
2
− 2RL cos θ.
(2)
For small oscillations, sin θ θ , cos θ 1 and
I c =
3M R 2
2
+ m(L − d)
2
(3)
Therefore the equation of motion become
3M R 2
2
+ m(L − d)
2
¨
θ = −mgL sin θ = −mgLθ
or ¨
θ +
mgL
3M R 2 /2 + m(L − d) 2 θ = 0
∴ ω =
mgL
3M R 2 /2 + m(L − d) 2 rad/s
6.45 Figure 6.21 shows the semicircular disc tilted through an angle θ compared to
the equilibrium position (b). G is the centre of mass such that a = OG =
4r
3π
,
where r is the radius.