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6 Oscillations
6.4 A particle performs SHM with a period of 16 s. At time t = 2 s, the particle
passes through the origin while at t = 4 s, its velocity is 4 m/s. Show that the
amplitude of the motion is 32
√
2/π .
[University of Dublin]
6.5 Show that given a small vertical displacement from its equilibrium position
a floating body subsequently performs simple harmonic motion of period
2π
√
V /Ag where V is the volume of displaced liquid and A is the area of
the plane of floatation. Ignore the viscous forces.
6.6 Imagine a tunnel bored along the diameter of the earth assumed to have constant
density. A box is thrown into the tunnel (chute). (a) Show that the box executes
SHM inside the tunnel about the centre of the earth. (b) Find the time period of
oscillations.
6.7 A particle which executes SHM along a straight line has its motion represented
by x = 4 sin(πt/3 + π/6). Find (a) the amplitude; (b) time period; (c) frequency; (d) phase difference; (e) velocity; (f) acceleration, at t = 1 s, x being
in cm.
6.8 (a) At what distance from the equilibrium position is the kinetic energy equal
to the potential energy for a SHM?
(b) In SHM if the displacement is one-half of the amplitude show that the
kinetic energy and potential energy are in the ratio 3:1.
6.9 A mass M attached to a spring oscillates with a period 2 s. If the mass is
increased by 2 kg, the period increases by 1 s. Assuming that Hooke’s law is
obeyed, find the initial mass M.
6.10 A particle vibrates with SHM along a straight line, its greatest acceleration is
5π 2 cm/s 2 , and when its distance from the equilibrium is 4 cm the velocity of
the particle is 3π cm/s. Find the amplitude and the period of oscillation of the
particle.
6.11 If the maximum acceleration of a SHM is α and the maximum velocity is
β, show that the amplitude of vibration is given by β 2 /α and the period of
oscillation by 2πβ/α.
6.12 If the tension along the string of a simple pendulum at the lowest position is
1% higher than the weight of the bob, show that the angular amplitude of the
pendulum is 0.1 rad.
6.13 A particle executes SHM and is located at x = a, b and c at time t 0 , 2t 0 and
3t 0 , respectively. Show that the frequency of oscillation is
1
2π t 0
cos −1 a + c
2b
.
6.14 A 4 kg mass at the end of a spring moves with SHM on a horizontal frictionless
table with period 2 s and amplitude 2 m. Determine (a) the spring constant;
(b) maximum force exerted on the spring.
6 Oscillations
6.4 A particle performs SHM with a period of 16 s. At time t = 2 s, the particle
passes through the origin while at t = 4 s, its velocity is 4 m/s. Show that the
amplitude of the motion is 32
√
2/π .
[University of Dublin]
6.5 Show that given a small vertical displacement from its equilibrium position
a floating body subsequently performs simple harmonic motion of period
2π
√
V /Ag where V is the volume of displaced liquid and A is the area of
the plane of floatation. Ignore the viscous forces.
6.6 Imagine a tunnel bored along the diameter of the earth assumed to have constant
density. A box is thrown into the tunnel (chute). (a) Show that the box executes
SHM inside the tunnel about the centre of the earth. (b) Find the time period of
oscillations.
6.7 A particle which executes SHM along a straight line has its motion represented
by x = 4 sin(πt/3 + π/6). Find (a) the amplitude; (b) time period; (c) frequency; (d) phase difference; (e) velocity; (f) acceleration, at t = 1 s, x being
in cm.
6.8 (a) At what distance from the equilibrium position is the kinetic energy equal
to the potential energy for a SHM?
(b) In SHM if the displacement is one-half of the amplitude show that the
kinetic energy and potential energy are in the ratio 3:1.
6.9 A mass M attached to a spring oscillates with a period 2 s. If the mass is
increased by 2 kg, the period increases by 1 s. Assuming that Hooke’s law is
obeyed, find the initial mass M.
6.10 A particle vibrates with SHM along a straight line, its greatest acceleration is
5π 2 cm/s 2 , and when its distance from the equilibrium is 4 cm the velocity of
the particle is 3π cm/s. Find the amplitude and the period of oscillation of the
particle.
6.11 If the maximum acceleration of a SHM is α and the maximum velocity is
β, show that the amplitude of vibration is given by β 2 /α and the period of
oscillation by 2πβ/α.
6.12 If the tension along the string of a simple pendulum at the lowest position is
1% higher than the weight of the bob, show that the angular amplitude of the
pendulum is 0.1 rad.
6.13 A particle executes SHM and is located at x = a, b and c at time t 0 , 2t 0 and
3t 0 , respectively. Show that the frequency of oscillation is
1
2π t 0
cos −1 a + c
2b
.
6.14 A 4 kg mass at the end of a spring moves with SHM on a horizontal frictionless
table with period 2 s and amplitude 2 m. Determine (a) the spring constant;
(b) maximum force exerted on the spring.