Find the time period of small oscillations of the following systems.
(p) A metre stick suspended through the 20 cm mark.
(q) A ring of mass m and radius r suspended through a point on its periphery.
(i)2π√2√2a3g (ii) 1.51 sec (iii) 2π√3r3g (iv)T=2π√2Rg
p - (ii) ; q - (iv)
The ruler is suspended at 20 cm mark.C.O.M is at 50cm so the C.O.M is going in a circle of 30 cm radius.
Circle of 30 cm radius so if ruler is displaced by an angleθ and left,then torque of mg will try to restore it.
Torque of mg=0.3 mg sin θ = -l α . . . (1)
Iruler=mL212+md2 (L=1m;d=30 cm)
Iruler=m[112+(30100)2]
= m (0.173)
eqn (1) becomes 0.3 mg θ=−Irα
(∵θissmall,sinθ≃θ)
α=−0.3mgθIr(α=−ω2θ)
T=2πω
=2π√Ir0.3mg
= 1.51 sec
(q)
If displaced by angle θ (small) restoring torque will be - mgR sin θ = lα
α=mgRθI(α=−ω2θ)
ω2=mgRI
T=2π√2Rg