248
6 Oscillations
6.27 A SHM is given by y = 8 sin
2π t
τ
+ ϕ
, the time period being 24 s. At
t = 0, the displacement is 4 cm. Find the displacement at t = 6 s.
6.28 In a vertical spring-mass system, the period of oscillation is 0.89 s when the
mass is 1.5 kg and the period becomes 1.13 s when a mass of 1.0 kg is added.
Calculate the mass of the spring.
6.29 Consider two springs A and B with spring constants k A and k B , respectively,
A being stiffer than B, that is, k A > k B . Show that
(a) when two springs are stretched by the same amount, more work will be
done on the stiffer spring.
(b) when two springs are stretched by the same force, less work will be done
on the stiffer spring.
6.30 A solid uniform cylinder of radius r rolls without sliding along the inside
surface of a hollow cylinder of radius R, performing small oscillations. Determine the time period.
6.2.2 Physical Pendulums
6.31 Consider the rigid plane object of weight Mg shown in Fig. 6.7, pivoted about
a point at a distance D from its centre of mass and displaced from equilibrium
by a small angle ϕ. Such a system is called a physical pendulum. Show that
the oscillatory motion of the object is simple harmonic with a period given by
T = 2π
I
Mg D
where I is the moment of inertia about the pivot point.
Fig. 6.7
6.32 A thin, uniform rod of mass M and length L swings from one of its ends
as a physical pendulum (see Fig. 6.8). Given that the moment of inertia of a
6 Oscillations
6.27 A SHM is given by y = 8 sin
2π t
τ
+ ϕ
, the time period being 24 s. At
t = 0, the displacement is 4 cm. Find the displacement at t = 6 s.
6.28 In a vertical spring-mass system, the period of oscillation is 0.89 s when the
mass is 1.5 kg and the period becomes 1.13 s when a mass of 1.0 kg is added.
Calculate the mass of the spring.
6.29 Consider two springs A and B with spring constants k A and k B , respectively,
A being stiffer than B, that is, k A > k B . Show that
(a) when two springs are stretched by the same amount, more work will be
done on the stiffer spring.
(b) when two springs are stretched by the same force, less work will be done
on the stiffer spring.
6.30 A solid uniform cylinder of radius r rolls without sliding along the inside
surface of a hollow cylinder of radius R, performing small oscillations. Determine the time period.
6.2.2 Physical Pendulums
6.31 Consider the rigid plane object of weight Mg shown in Fig. 6.7, pivoted about
a point at a distance D from its centre of mass and displaced from equilibrium
by a small angle ϕ. Such a system is called a physical pendulum. Show that
the oscillatory motion of the object is simple harmonic with a period given by
T = 2π
I
Mg D
where I is the moment of inertia about the pivot point.
Fig. 6.7
6.32 A thin, uniform rod of mass M and length L swings from one of its ends
as a physical pendulum (see Fig. 6.8). Given that the moment of inertia of a