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Simple Harmonic Oscillations 2016
Simple Harmonic Oscillations
Q.1
In an engine, a piston oscillates with SHM so that its position varies according to expression x=(5cm) cos
(2t + π/6) where x is in cms and t is in seconds. At t=0, find [a] the position of the particle [b] its velocity
and [c] its acceleration [d] find the period and amplitude of motion.
[Ans. [a] 4.33cm , -5cm/c, -17.3cm/s2, 3.14s, 5cm]
Q.2
A particle moving along the x axis in SHM starts from its equilibrium position, the origin at t=0 and moves
to the right. The amplitude of its motion is 2cm and the frequency is 1.5Hz. [a] Show that the position of
the particle is given by
X= (2cm) sin (3πt)
Determine [b] the maximum speed and the earliest time at which the particle has this speed. [c] the
maximum acceleration and the earliest time at which the particle has this acceleration and [d] the total
distance traveled between t=0 an t=1 seconds.
[Ans.[a] 18.8cm/s, 0.33s [b]178cm/s2, 0.500s [c]12cm]
Q.3
A 7 kg object is hung from the bottom end of a vertical spring fastened to an overhead beam. The object
is set into vertical oscillations having period of 2.6 seconds. Find the force constant of the spring.
[Ans. 40.9N/m]
Q.4
A 0.500 kg object is attached to the spring with force constant 8N/m vibrates in SHM with an amplitude of
10cm. Calculate the [a] maximum value of its speed and acceleration [b] the speed and acceleration when
the object is 6cm from equilibrium position and [c] the time it takes the object to move from x=0 to
x=8cm.
[Ans. [a]40cm/s, 160cm/s2 [b]32cm/s, -96cm/s2 [c] 0.232s]
An automobile having mass of 1000kg is driven into a brick wall in a safety test. The bumper behaves like
a spring constant 5 x 106N/m and compresses 3.16cm as the car is brought to rest. What was the speed of
the car before impact, assuming that the mechanical energy of the car remains constant.
[Ans. 2.23m/s]
Q.6
A 200gm block is attached to a horizontal spring and execute SHM on frictionless surface with period of
0.250s. If the total energy of the system is 2J. Find [a] the force constant of the spring and [b] the
amplitude of the motion.
[Ans. 126N/m, 0.178m]
Q.7
A 50gm block connected to a spring with force constant of 35N/m oscillates on a horizontal frictionless
surface with an amplitude of 4cm. Find [a] the total energy of the system and [b] the speed of the block
when the displacement is 1cm. find [a] the kinetic energy and [d] potential energy when the displacement
is 3cm.
[Ans. [a] 28mJ [b]1.02m/s [c] 12.2mJ [d] 15.8mJ]
Sanjay Chopra 249, Chotti Baradari Part-2[Near Medical College] Jalandhar #98152-15362
Chapter: Simple Harmonic Oscillations
Q.5
1
Simple Harmonic Oscillations 2016
Q.8
A particle executes SHM with an amplitude of 3cm. at what position does its speed equals one half of the
maximum speed?
[Ans. 2.6cm and -2.60cm]
Q.9
A simple pendulum has mass of 0.250kg and length of 1m. if it is displaced through an angle of 15 0 and
then released. What are the [a] the maximum speed [b] the maximum angular acceleration and [c] the
maximum restoring force?
[ans. 0.817m/s , 2.54rad/s2, 0.634N]
Q.10
A particle of mass m slides without friction inside a hemispherical bowl of radius R. Show that if it starts
from small displacement , from equilibrium position, the particle moves in SHM with an angular frequency
equal to that of a simple pendulum of length R.
Q.11
A physical pendulum in the form of a planar body moves in SHM with frequency of 0.450Hz. If the
pendulum has a mass of 2.20kg and the pivot is located 0.350m from the center of mass, determine the
MI of the pendulum about the pivot point.
[Ans.0.944kg m2]
Q.13
A force of 1N is required to stretch a spring by 1.5cm. If the spring is cut into three equal parts, find the
force required to stretch one part by 3cm?
[Ans. 6N]
Q.14
Two particles A and B of equal masses are suspended from massless springs of spring constant 4N/m and
8N/m respectively. If maximum velocities during oscillation are equal, find the ratio of amplitudes of A
and B.
Q.15
Two point masses 3kg and 1kg are connected to opposite ends of horizontal spring of spring constant
300N/m. Find the natural frequency of vibration of the system
Q.16
The time period of simple pendulum of length L is T1 and the time period of uniform rod of length L
pivoted about its one end is T2. Amplitude of oscillation in both the cases is small. Find T1/T2.
[Ans. √[3/2]
Q.17
A body is dropped along the hole drilled across diameter of earth. Show that it executes SHM. Assume
earth to be a homogeneous sphere, find the time period of its motion.
[Ans. 5065 second]
Q.18
In damped oscillations, the amplitude of oscillations is reduced to one fourth of its initial value 10cm at
the end of 50 oscillations. What will be its amplitude at the end of 150 oscillations?
[Ans. 1.56mm]
Sanjay Chopra 249, Chotti Baradari Part-2[Near Medical College] Jalandhar #98152-15362
Chapter: Simple Harmonic Oscillations
[Ans. √2]
2
Simple Harmonic Oscillations 2016
Q.19
A simple pendulum with a brass bob has time period T. the bob is now immersed in a non viscous liquid
and oscillated. If the density if the liquid is 1/9 that of brass find the time period of the same pendulum.
[Ans. 3T/√8]
Q.20
A light point fixed to one prong of tuning fork touches a vertical plate. The fork is set vibrating and the
plate is allowed to fall freely. Eight complete oscillations are counted when the plate falls through 10cm.
what is the frequency of the tuning fork?
[Ans. 56 Hz]]
Q.21
The block of mass m1 is fastened as shown in the figure to the
spring and the block of mass m2 is placed against it. [a] find the
compression of the spring in the equilibrium position [b] The
blocks are pushed further distance
2
m1  m2  gsinθ against the
k
spring and released. Find the position where the two blocks
separate. [c] what is the common speed of the blocks at the time of separation?
[Ans. [a]
k
[b] when spring acquires natural length [c]
The spring shown in figure is unstretched when a man starts pulling on
the cord. The mass of the block is M. If the man exerts a constant force
F, find [a] the amplitude and time period of the motion of the block [b]
the energy stored in the spring when the block passes through
equilibrium position and [c] the kinetic energy of the block at this
position.
[Ans. [a]
Q.23
M
F2
F2
F
, 2π
, [b]
[c]
]
k
2k
2k
k
A particle of mass m is attached to three springs A , B and C as shown in the
figure with each having spring constant k. if the particle is pushed slightly
against the spring C and released, find the time period of the oscillation.
[Ans. 2π
Q.24
3
m1  m2 g sin  ]
k
m
]
2k
The springs shown in the figure are all unstretched n the beginning when
a man starts to pulling the block, the man exerts a constant force F on
the block. Find the amplitude and frequency of the motion of the block.
Sanjay Chopra 249, Chotti Baradari Part-2[Near Medical College] Jalandhar #98152-15362
Chapter: Simple Harmonic Oscillations
Q.22
m1  m2 g sin 
3
Simple Harmonic Oscillations 2016
A rectangular plate of side a and b suspended from the ceiling by two parallel
strings of length L each. The separation between the strings is d. the plate is
displaced slightly n its plane keeping strings tight. Show that it will execute SHM
and find its time period.
[Ans. 2π
Q.26
l
]
g
A spherical ball of mass m and radius r rolls without slipping on a rough concave surface of radius R. it
takes small oscillations about the lowest point. Find its time period.
[Ans. 2
7 R  r 
]
5g
Q.27
A simple pendulum of length 40cm is taken inside a deep mine. Assume for the time being that the mine
is 1600km deep. Calculate the time period of the pendulum there. Radius of earth is 6400km.[Ans. 1.47 s]
Q.28
Assume that a tunnel is dug along a chord of the earth at a perpendicular distance R/2 from the earth’s
center. The wall of the tunnel is frictionless. [a] find the gravitational force exerted by the earth on a
particle of mass m placed in the tunnel at a distance of x from the center of the tunnel [b] Find the
component of this force along the tunnel and perpendicular to the tunnel. [c] find the normal force
exerted by wall on the particle.[d] find the resultant force on the particle[e] Show that the particle in
tunnel executes SHM and find its time period.
GMm 2 R 2
R3
GMm GMm
GMm GMm
x 
x,
x [d]2π
[Ans. [a]
[b]
[c]
,
4
GM
R3
2R 2
2R 2
R3
R3
Q.29
A simple pendulum of length l is suspended through the ceiling of the elevator [a] is going up with
acceleration a0 [b] is going down with an acceleration a0 and [c] moving with uniform velocity.
[Ans. [a] 2π
Q.30
]
l
l
[b]2π
]
g  a0
g  a0
A simple pendulum is fixed in a car has a time period of 4 seconds when the car is moving uniformly on a
horizontal road. When the accelerator is pressed, the time period changes to 3.99seconds. Making an
approximate analysis, find the acceleration of the car.
[Ans. g/10]
Q.31
A simple pendulum of length l is suspended from the ceiling of a car moving with speed v on a circular
road of radius r. [a] find the tension in the string when it at rest with respect to the car. [b] Find the time
period of the small oscillations.
[Ans. [a] ma [b] 2π
Sanjay Chopra 249, Chotti Baradari Part-2[Near Medical College] Jalandhar #98152-15362
l
]
a
Chapter: Simple Harmonic Oscillations
Q.25
4
Simple Harmonic Oscillations 2016
Q.32
A uniform rod of length l is suspended by an end and is made to undergo small oscillations. Find the
length of the simple pendulum having time period equal to that of the rod.
[Ans. 2l/3]
Q.33
A uniform disc of radius r is suspended through a small hole made in the disc. Find the minimum possible
time period of the disc for small oscillations. What should be the distance of the hole from the center for
it to have minimum time period?
[Ans. 2π
Q.34
r 2
, r/√2]
g
A hollow sphere of radius 2cm is attached to 18cm long thread to make a pendulum. Find the time period
of the oscillation of this pendulum. How does it differ from the time period calculated using the formula
for simple pendulum.
[Ans. 0.89 s, it is about 0.3% larger than calculated value]
A uniform disc of mass m and radius r is suspended through a wire attached to its center. If the time
period of the torsional oscillations be T, what is torsional constant of the wire.
[Ans.
Q.36
A simple pendulum of length L and mass m has a spring of force constant k connected to it at a distance h
below its point of suspension. Find the frequency of vibrations of the system for small values of
amplitude.
[Ans.
Q.37
1
2
mgL  kh2
]
mL2
A light rod of length L2 has a small ball of mass m2 fixed at one end and another ball of mass m1 fixed on it
at a distance L1 from free end. The rod is supported at end O and is free to rotate about horizontal axis at
O. The rod is slightly displaced from its equilibrium position which is vertical and released. Find the time
period of oscillation and the length of equivalent pendulum?
[Ans. 2π
Q.38
2 2 mr 2
]
T2
m L2  m2 L22
L
,L= 1 1
]
g
m1 L1  m2 L2
With the assumption of no slipping, determine the mass m of the block which must be placed on the top
of 6kg cart in order that the system has time period of 0.75s. What is the minimum coefficient of static
friction for block not to slip relative to the cart if the cart is displaced 500 from equilibrium position and
released.
[Ans. 2.55kg, 0.358]
Q.39
A ring of radius r is suspended from a point on its circumference. Determine the angular frequency of
small oscillations.
Sanjay Chopra 249, Chotti Baradari Part-2[Near Medical College] Jalandhar #98152-15362
Chapter: Simple Harmonic Oscillations
Q.35
5
Simple Harmonic Oscillations 2016
[Ans. ω =
Q.40
A pendulum clock is mounted in an elevator which starts going up with constant acceleration a. at height
h the acceleration of the car reverses, its magnitude remaining the same. How soon after the start of the
motion will the clock show the right time again?
[Ans. t =
Q.41
3m
, 47.9m/s2]
8k
A particle is constrained to move on a smooth circular wire of radius r which rotates uniformly about
diameter which is vertical. If in the position of relative rest the radius drawn to the particle makes an
angle α with the vertical, find the period of small oscillations about this position.
[Ans. 2π
Q.43
2h  g  a  g  a 

]
a 
g  g  a 
A 14 kg cylinder can roll without slipping on a 300 incline. A belt is
attached to the rim of the cylinder and a spring hold the , cylinder at rest
in the position shown. If the cylinder moves down 50mm and released
determine [a] the period of vibration [b] the maximum acceleration of the
center of the cylinder.
[Ans. 2π
Q.42
g
]
2r

r cos 
g 1  cos 2 

1/ 2
]
A 7kg disk is free to rotate about a horizontal axis passing through its center C.
determine the period of oscillations of the disk if the springs have sufficient tension in
them to prevent the string from slipping on the disk as it oscillates. The radius of the disc
is 10cm and the spring constant of both the springs is 600N/m.
Q.44
A thin rod of length L and uniform cross section is pivoted at its lowest point P
inside a stationary homogeneous and non viscous liquid. The rod is free to rotate
in vertical plane about horizontal axis passing through P. the density d1 of the
material of the rod is smaller than density d2 of the liquid. The rod is displaced by
small angle θ from its equilibrium position and released. Show that the motion of
the rod is SHM and find the angular frequency of oscillations.
[Ans. ω =
3g d 2  d1 
]
2d 1 L
Sanjay Chopra 249, Chotti Baradari Part-2[Near Medical College] Jalandhar #98152-15362
Chapter: Simple Harmonic Oscillations
[Ans. 0.34seconds]
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