DC Pandey Solutions (JEE Main): Simple Harmonic Motion | DC Pandey Solutions for JEE Physics PDF Download

 Page 1


EXERCISES 
For JEE Main 
  Subjective Questions 
  Kinematics of SHM 
Q 1.  A 50 g mass hangs at the end of a massless spring. When 20 g more are added to the 
end of the spring, it stretches 7.0 cm more. 
  (a) Find the spring constant. 
  (b) If the 20 g are now removed, what will be the period of the motion ? 
Q 2.  A body of weight 27 N hangs on a long spring of such stiffness that an extra force of 
9 N stretches the spring by 0.05 m. If the body is pulled downward and released, what 
is the period ? 
Q 3.  A 0.5 kg body performs simple harmonic motion with a frequency of 2 Hz and an 
amplitude of 8 mm. Find the maximum velocity of the body, its maximum 
acceleration and the maximum restoring force to which the body is subjected. 
Q 4.  A body describing SHM has a maximum acceleration of 8 ? m/s 
2
 and a maximum 
speed of 1.6 m/s. Find the period T and the amplitude A. 
Q 5.  Given that the equation of motion of a mass is x=0.20 sin (3.0t) m. Find the velocity 
and acceleration of the mass when the object is 5 cm from its equilibrium position. 
Repeat for x = 0. 
Q 6.  A particle executes simple harmonic motion of amplitude A along the x-axis. At t = 0, 
the position of the particle is x = A/2 and it moves along the positive x-direction. Find 
the phase constant 8, if the equation is written as x = A sin ( ?t + ?). 
Q 7.  A body makes angular simple harmonic motion of amplitude ? / 10 rad and time 
period 0.05 s. If the body is at a displacement 0 = ? /10 rad at t = 0, write the equation 
giving angular displacement as a function of time. 
Q 8.  The equation of motion of a particle started at t = 0 is given by x = 5 sin 20t
3
? ??
?
??
??
, 
where x is in cm and t in sec. When does the particle 
  (a) first come to rest,  (b) first have zero acceleration, (c) first have maximum 
speed. 
Q 9.  A particle executes simple harmonic motion of period 16 s. Two seconds later after it 
passes through the centre of oscillation its velocity is found to be 2 m/s. Find the 
amplitude. 
Q 10.  The period of a particle in SHM is 8 s. At t = 0 it is in its equilibrium position. 
  (a) Compare the distance travelled in the first 4 s and the second 4 s. 
  (b) Compare the distance travelled in the first 2 s and the second 2 s. 
Q 11.  An object of mass 0.8 kg is attached to one end of a spring and the system is set into 
simple harmonic motion. The displacement x of the object as a function of time is 
shown in the figure. With the aid of the data, determine 
Page 2


EXERCISES 
For JEE Main 
  Subjective Questions 
  Kinematics of SHM 
Q 1.  A 50 g mass hangs at the end of a massless spring. When 20 g more are added to the 
end of the spring, it stretches 7.0 cm more. 
  (a) Find the spring constant. 
  (b) If the 20 g are now removed, what will be the period of the motion ? 
Q 2.  A body of weight 27 N hangs on a long spring of such stiffness that an extra force of 
9 N stretches the spring by 0.05 m. If the body is pulled downward and released, what 
is the period ? 
Q 3.  A 0.5 kg body performs simple harmonic motion with a frequency of 2 Hz and an 
amplitude of 8 mm. Find the maximum velocity of the body, its maximum 
acceleration and the maximum restoring force to which the body is subjected. 
Q 4.  A body describing SHM has a maximum acceleration of 8 ? m/s 
2
 and a maximum 
speed of 1.6 m/s. Find the period T and the amplitude A. 
Q 5.  Given that the equation of motion of a mass is x=0.20 sin (3.0t) m. Find the velocity 
and acceleration of the mass when the object is 5 cm from its equilibrium position. 
Repeat for x = 0. 
Q 6.  A particle executes simple harmonic motion of amplitude A along the x-axis. At t = 0, 
the position of the particle is x = A/2 and it moves along the positive x-direction. Find 
the phase constant 8, if the equation is written as x = A sin ( ?t + ?). 
Q 7.  A body makes angular simple harmonic motion of amplitude ? / 10 rad and time 
period 0.05 s. If the body is at a displacement 0 = ? /10 rad at t = 0, write the equation 
giving angular displacement as a function of time. 
Q 8.  The equation of motion of a particle started at t = 0 is given by x = 5 sin 20t
3
? ??
?
??
??
, 
where x is in cm and t in sec. When does the particle 
  (a) first come to rest,  (b) first have zero acceleration, (c) first have maximum 
speed. 
Q 9.  A particle executes simple harmonic motion of period 16 s. Two seconds later after it 
passes through the centre of oscillation its velocity is found to be 2 m/s. Find the 
amplitude. 
Q 10.  The period of a particle in SHM is 8 s. At t = 0 it is in its equilibrium position. 
  (a) Compare the distance travelled in the first 4 s and the second 4 s. 
  (b) Compare the distance travelled in the first 2 s and the second 2 s. 
Q 11.  An object of mass 0.8 kg is attached to one end of a spring and the system is set into 
simple harmonic motion. The displacement x of the object as a function of time is 
shown in the figure. With the aid of the data, determine 
 
  (a) the amplitude A of the motion,  (b) the angular frequency ?,  
  (c) the spring constant k,   (d) the speed of the object at t = 1.0 s and 
  (e) the magnitude of the object's acceleration at t = 1.0 s. 
Q 12.  (a) The motion of the particle in simple harmonic motion is given by x = a sin ?t. 
If its speed is u, when the displacement is x1 and speed is v, when the displacement is 
x2, show that the amplitude of the motion is 
   
1/ 2
2 2 2 2
12
22
v x u x
a
vu
?? ?
?
??
?
??
 
(b) A particle is moving with simple harmonic motion in a straight line. When the 
distance of the particle from the equilibrium position has the values x1 and x2, the 
corresponding values of velocity are u1 and u2, show that the period is 
   
1/ 2
22
21
22
12
xx
T2
uu
?? ?
??
??
?
??
 
  Energy in SHM 
Q 13.  A spring-mass oscillator has a total energy E0 and an amplitude x0. 
  (a) How large will K and U be for it when x = 
0
1
x
2
?
 
 
  (b) For what value of x will K =U ? 
Q 14.  Show that the combined spring energy and gravitational energy for a mass m hanging 
from a light spring of force constant k can be expressed as 
2
1
ky
2
, where y is the 
distance above or below the equilibrium position. 
Q 15.  The masses in figure slide on a frictionless table m 1 but not m 2, is fastened to the 
spring. If now m 1 and m 2 are pushed to the left, so that the spring is compressed a 
distance d, what will be the amplitude of the oscillation of m 1 after the spring system 
is released ? 
 
Q 16.  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 
Page 3


EXERCISES 
For JEE Main 
  Subjective Questions 
  Kinematics of SHM 
Q 1.  A 50 g mass hangs at the end of a massless spring. When 20 g more are added to the 
end of the spring, it stretches 7.0 cm more. 
  (a) Find the spring constant. 
  (b) If the 20 g are now removed, what will be the period of the motion ? 
Q 2.  A body of weight 27 N hangs on a long spring of such stiffness that an extra force of 
9 N stretches the spring by 0.05 m. If the body is pulled downward and released, what 
is the period ? 
Q 3.  A 0.5 kg body performs simple harmonic motion with a frequency of 2 Hz and an 
amplitude of 8 mm. Find the maximum velocity of the body, its maximum 
acceleration and the maximum restoring force to which the body is subjected. 
Q 4.  A body describing SHM has a maximum acceleration of 8 ? m/s 
2
 and a maximum 
speed of 1.6 m/s. Find the period T and the amplitude A. 
Q 5.  Given that the equation of motion of a mass is x=0.20 sin (3.0t) m. Find the velocity 
and acceleration of the mass when the object is 5 cm from its equilibrium position. 
Repeat for x = 0. 
Q 6.  A particle executes simple harmonic motion of amplitude A along the x-axis. At t = 0, 
the position of the particle is x = A/2 and it moves along the positive x-direction. Find 
the phase constant 8, if the equation is written as x = A sin ( ?t + ?). 
Q 7.  A body makes angular simple harmonic motion of amplitude ? / 10 rad and time 
period 0.05 s. If the body is at a displacement 0 = ? /10 rad at t = 0, write the equation 
giving angular displacement as a function of time. 
Q 8.  The equation of motion of a particle started at t = 0 is given by x = 5 sin 20t
3
? ??
?
??
??
, 
where x is in cm and t in sec. When does the particle 
  (a) first come to rest,  (b) first have zero acceleration, (c) first have maximum 
speed. 
Q 9.  A particle executes simple harmonic motion of period 16 s. Two seconds later after it 
passes through the centre of oscillation its velocity is found to be 2 m/s. Find the 
amplitude. 
Q 10.  The period of a particle in SHM is 8 s. At t = 0 it is in its equilibrium position. 
  (a) Compare the distance travelled in the first 4 s and the second 4 s. 
  (b) Compare the distance travelled in the first 2 s and the second 2 s. 
Q 11.  An object of mass 0.8 kg is attached to one end of a spring and the system is set into 
simple harmonic motion. The displacement x of the object as a function of time is 
shown in the figure. With the aid of the data, determine 
 
  (a) the amplitude A of the motion,  (b) the angular frequency ?,  
  (c) the spring constant k,   (d) the speed of the object at t = 1.0 s and 
  (e) the magnitude of the object's acceleration at t = 1.0 s. 
Q 12.  (a) The motion of the particle in simple harmonic motion is given by x = a sin ?t. 
If its speed is u, when the displacement is x1 and speed is v, when the displacement is 
x2, show that the amplitude of the motion is 
   
1/ 2
2 2 2 2
12
22
v x u x
a
vu
?? ?
?
??
?
??
 
(b) A particle is moving with simple harmonic motion in a straight line. When the 
distance of the particle from the equilibrium position has the values x1 and x2, the 
corresponding values of velocity are u1 and u2, show that the period is 
   
1/ 2
22
21
22
12
xx
T2
uu
?? ?
??
??
?
??
 
  Energy in SHM 
Q 13.  A spring-mass oscillator has a total energy E0 and an amplitude x0. 
  (a) How large will K and U be for it when x = 
0
1
x
2
?
 
 
  (b) For what value of x will K =U ? 
Q 14.  Show that the combined spring energy and gravitational energy for a mass m hanging 
from a light spring of force constant k can be expressed as 
2
1
ky
2
, where y is the 
distance above or below the equilibrium position. 
Q 15.  The masses in figure slide on a frictionless table m 1 but not m 2, is fastened to the 
spring. If now m 1 and m 2 are pushed to the left, so that the spring is compressed a 
distance d, what will be the amplitude of the oscillation of m 1 after the spring system 
is released ? 
 
Q 16.  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 the time period of the motion of the block, 
  (b) the energy stored in the spring when the block passes through the equilibrium 
position and 
  (c) the kinetic energy of the block at this position. 
Q 17.  In figure, k = 100 N/m, M = 1 kg and F = 10 N. 
 
  (a) Find the compression of the spring in the equilibrium position. 
(b) A sharp blow by some external agent imparts a speed of 2 m/s to the block 
towards left. Find the sum of the potential energy of the spring and the kinetic energy 
of the block at this
 
instant. 
  (c) Find the time period of the resulting simple harmonic motion. 
  (d) Find the amplitude. 
  (e) Write the potential energy of the spring when the block is at the left extreme. 
  (f) Write the potential energy of the spring when the block is at the right extreme. 
The answers of (b), (e) and (f) are different. Explain why this does not violate the 
principle of conservation of energy? 
Q 18.  A point particle of mass 0.1 kg is executing SHM of amplitude 0.1 m. When the 
particle passes through the mean position, its kinetic energy is 8 × 10
-3
 J. Write down 
the equation of motion of this particle when the initial phase of oscillation is 45°. 
Q 19.  Potential energy of a particle in SHM along x-axis is given by : 
     U =10 + (x - 2)
2
 
Here, U is in joule and x in metre. Total mechanical energy of the particle is 26 J. 
Mass of the particle is 2 kg. Find : 
  (a) angular frequency of SHM, 
  (b) potential energy and kinetic energy at mean position and extreme position, 
  (c) amplitude of oscillation, 
  (d) x-coordinates between which particle oscillates. 
  Time period of SHM 
Q 20.  A pendulum has a period T for small oscillations. An obstacle is placed directly 
beneath the pivot, so that only the lowest one-quarter of the string can follow the 
pendulum bob when it swings to the left of its resting position. The pendulum is 
released from rest at a certain point. How long will it take to return to that point ? In 
Page 4


EXERCISES 
For JEE Main 
  Subjective Questions 
  Kinematics of SHM 
Q 1.  A 50 g mass hangs at the end of a massless spring. When 20 g more are added to the 
end of the spring, it stretches 7.0 cm more. 
  (a) Find the spring constant. 
  (b) If the 20 g are now removed, what will be the period of the motion ? 
Q 2.  A body of weight 27 N hangs on a long spring of such stiffness that an extra force of 
9 N stretches the spring by 0.05 m. If the body is pulled downward and released, what 
is the period ? 
Q 3.  A 0.5 kg body performs simple harmonic motion with a frequency of 2 Hz and an 
amplitude of 8 mm. Find the maximum velocity of the body, its maximum 
acceleration and the maximum restoring force to which the body is subjected. 
Q 4.  A body describing SHM has a maximum acceleration of 8 ? m/s 
2
 and a maximum 
speed of 1.6 m/s. Find the period T and the amplitude A. 
Q 5.  Given that the equation of motion of a mass is x=0.20 sin (3.0t) m. Find the velocity 
and acceleration of the mass when the object is 5 cm from its equilibrium position. 
Repeat for x = 0. 
Q 6.  A particle executes simple harmonic motion of amplitude A along the x-axis. At t = 0, 
the position of the particle is x = A/2 and it moves along the positive x-direction. Find 
the phase constant 8, if the equation is written as x = A sin ( ?t + ?). 
Q 7.  A body makes angular simple harmonic motion of amplitude ? / 10 rad and time 
period 0.05 s. If the body is at a displacement 0 = ? /10 rad at t = 0, write the equation 
giving angular displacement as a function of time. 
Q 8.  The equation of motion of a particle started at t = 0 is given by x = 5 sin 20t
3
? ??
?
??
??
, 
where x is in cm and t in sec. When does the particle 
  (a) first come to rest,  (b) first have zero acceleration, (c) first have maximum 
speed. 
Q 9.  A particle executes simple harmonic motion of period 16 s. Two seconds later after it 
passes through the centre of oscillation its velocity is found to be 2 m/s. Find the 
amplitude. 
Q 10.  The period of a particle in SHM is 8 s. At t = 0 it is in its equilibrium position. 
  (a) Compare the distance travelled in the first 4 s and the second 4 s. 
  (b) Compare the distance travelled in the first 2 s and the second 2 s. 
Q 11.  An object of mass 0.8 kg is attached to one end of a spring and the system is set into 
simple harmonic motion. The displacement x of the object as a function of time is 
shown in the figure. With the aid of the data, determine 
 
  (a) the amplitude A of the motion,  (b) the angular frequency ?,  
  (c) the spring constant k,   (d) the speed of the object at t = 1.0 s and 
  (e) the magnitude of the object's acceleration at t = 1.0 s. 
Q 12.  (a) The motion of the particle in simple harmonic motion is given by x = a sin ?t. 
If its speed is u, when the displacement is x1 and speed is v, when the displacement is 
x2, show that the amplitude of the motion is 
   
1/ 2
2 2 2 2
12
22
v x u x
a
vu
?? ?
?
??
?
??
 
(b) A particle is moving with simple harmonic motion in a straight line. When the 
distance of the particle from the equilibrium position has the values x1 and x2, the 
corresponding values of velocity are u1 and u2, show that the period is 
   
1/ 2
22
21
22
12
xx
T2
uu
?? ?
??
??
?
??
 
  Energy in SHM 
Q 13.  A spring-mass oscillator has a total energy E0 and an amplitude x0. 
  (a) How large will K and U be for it when x = 
0
1
x
2
?
 
 
  (b) For what value of x will K =U ? 
Q 14.  Show that the combined spring energy and gravitational energy for a mass m hanging 
from a light spring of force constant k can be expressed as 
2
1
ky
2
, where y is the 
distance above or below the equilibrium position. 
Q 15.  The masses in figure slide on a frictionless table m 1 but not m 2, is fastened to the 
spring. If now m 1 and m 2 are pushed to the left, so that the spring is compressed a 
distance d, what will be the amplitude of the oscillation of m 1 after the spring system 
is released ? 
 
Q 16.  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 the time period of the motion of the block, 
  (b) the energy stored in the spring when the block passes through the equilibrium 
position and 
  (c) the kinetic energy of the block at this position. 
Q 17.  In figure, k = 100 N/m, M = 1 kg and F = 10 N. 
 
  (a) Find the compression of the spring in the equilibrium position. 
(b) A sharp blow by some external agent imparts a speed of 2 m/s to the block 
towards left. Find the sum of the potential energy of the spring and the kinetic energy 
of the block at this
 
instant. 
  (c) Find the time period of the resulting simple harmonic motion. 
  (d) Find the amplitude. 
  (e) Write the potential energy of the spring when the block is at the left extreme. 
  (f) Write the potential energy of the spring when the block is at the right extreme. 
The answers of (b), (e) and (f) are different. Explain why this does not violate the 
principle of conservation of energy? 
Q 18.  A point particle of mass 0.1 kg is executing SHM of amplitude 0.1 m. When the 
particle passes through the mean position, its kinetic energy is 8 × 10
-3
 J. Write down 
the equation of motion of this particle when the initial phase of oscillation is 45°. 
Q 19.  Potential energy of a particle in SHM along x-axis is given by : 
     U =10 + (x - 2)
2
 
Here, U is in joule and x in metre. Total mechanical energy of the particle is 26 J. 
Mass of the particle is 2 kg. Find : 
  (a) angular frequency of SHM, 
  (b) potential energy and kinetic energy at mean position and extreme position, 
  (c) amplitude of oscillation, 
  (d) x-coordinates between which particle oscillates. 
  Time period of SHM 
Q 20.  A pendulum has a period T for small oscillations. An obstacle is placed directly 
beneath the pivot, so that only the lowest one-quarter of the string can follow the 
pendulum bob when it swings to the left of its resting position. The pendulum is 
released from rest at a certain point. How long will it take to return to that point ? In 
answering this question, you may assume that the angle between the moving string 
and the vertical stays small throughout the motion. 
 
Q 21.  An object suspended from a spring exhibits oscillations of period T. Now, the spring 
is cut in half and the two halves are used to support the same object, as shown in 
figure. Show that the new period of oscillation is 772. 
 
Q 22.  Assume that a narrow tunnel is dug between two diametrically opposite points of the 
earth. Treat the earth as a solid sphere of uniform density. Show that if a particle is 
released in this tunnel, it will execute a simple harmonic motion. Calculate the time 
period of this motion. 
Q 23.  A simple pendulum is taken at a place where its separation from the earth's surface is 
equal to the radius of the earth. Calculate the time period of small oscillations if the 
length of the string is 1.0 m. Take g = ?
2
 m/s
2
 at the surface of the earth. 
Q 24.  The string, the spring and the pulley shown in figure are light. Find the time period of 
the mass m. 
 
Q 25.  A solid cylinder of mass M = 10kg and cross-sectional area A =20cm
2
 is suspended 
by a spring of force constant k =100 N/m and hangs partially immersed in water. 
Calculate the period of small oscillations of the cylinder. 
Q 26.  Pendulum A is a physical pendulum made from a thin, rigid and uniform rod whose 
length is d. One end of this rod is attached to the ceiling by a frictionless hinge, so that 
the rod is free to swing back and forth. Pendulum B is a simple pendulum whose 
length is also d. Obtain the ratio 
A
B
T
T
of their periods for small angle oscillations. 
Page 5


EXERCISES 
For JEE Main 
  Subjective Questions 
  Kinematics of SHM 
Q 1.  A 50 g mass hangs at the end of a massless spring. When 20 g more are added to the 
end of the spring, it stretches 7.0 cm more. 
  (a) Find the spring constant. 
  (b) If the 20 g are now removed, what will be the period of the motion ? 
Q 2.  A body of weight 27 N hangs on a long spring of such stiffness that an extra force of 
9 N stretches the spring by 0.05 m. If the body is pulled downward and released, what 
is the period ? 
Q 3.  A 0.5 kg body performs simple harmonic motion with a frequency of 2 Hz and an 
amplitude of 8 mm. Find the maximum velocity of the body, its maximum 
acceleration and the maximum restoring force to which the body is subjected. 
Q 4.  A body describing SHM has a maximum acceleration of 8 ? m/s 
2
 and a maximum 
speed of 1.6 m/s. Find the period T and the amplitude A. 
Q 5.  Given that the equation of motion of a mass is x=0.20 sin (3.0t) m. Find the velocity 
and acceleration of the mass when the object is 5 cm from its equilibrium position. 
Repeat for x = 0. 
Q 6.  A particle executes simple harmonic motion of amplitude A along the x-axis. At t = 0, 
the position of the particle is x = A/2 and it moves along the positive x-direction. Find 
the phase constant 8, if the equation is written as x = A sin ( ?t + ?). 
Q 7.  A body makes angular simple harmonic motion of amplitude ? / 10 rad and time 
period 0.05 s. If the body is at a displacement 0 = ? /10 rad at t = 0, write the equation 
giving angular displacement as a function of time. 
Q 8.  The equation of motion of a particle started at t = 0 is given by x = 5 sin 20t
3
? ??
?
??
??
, 
where x is in cm and t in sec. When does the particle 
  (a) first come to rest,  (b) first have zero acceleration, (c) first have maximum 
speed. 
Q 9.  A particle executes simple harmonic motion of period 16 s. Two seconds later after it 
passes through the centre of oscillation its velocity is found to be 2 m/s. Find the 
amplitude. 
Q 10.  The period of a particle in SHM is 8 s. At t = 0 it is in its equilibrium position. 
  (a) Compare the distance travelled in the first 4 s and the second 4 s. 
  (b) Compare the distance travelled in the first 2 s and the second 2 s. 
Q 11.  An object of mass 0.8 kg is attached to one end of a spring and the system is set into 
simple harmonic motion. The displacement x of the object as a function of time is 
shown in the figure. With the aid of the data, determine 
 
  (a) the amplitude A of the motion,  (b) the angular frequency ?,  
  (c) the spring constant k,   (d) the speed of the object at t = 1.0 s and 
  (e) the magnitude of the object's acceleration at t = 1.0 s. 
Q 12.  (a) The motion of the particle in simple harmonic motion is given by x = a sin ?t. 
If its speed is u, when the displacement is x1 and speed is v, when the displacement is 
x2, show that the amplitude of the motion is 
   
1/ 2
2 2 2 2
12
22
v x u x
a
vu
?? ?
?
??
?
??
 
(b) A particle is moving with simple harmonic motion in a straight line. When the 
distance of the particle from the equilibrium position has the values x1 and x2, the 
corresponding values of velocity are u1 and u2, show that the period is 
   
1/ 2
22
21
22
12
xx
T2
uu
?? ?
??
??
?
??
 
  Energy in SHM 
Q 13.  A spring-mass oscillator has a total energy E0 and an amplitude x0. 
  (a) How large will K and U be for it when x = 
0
1
x
2
?
 
 
  (b) For what value of x will K =U ? 
Q 14.  Show that the combined spring energy and gravitational energy for a mass m hanging 
from a light spring of force constant k can be expressed as 
2
1
ky
2
, where y is the 
distance above or below the equilibrium position. 
Q 15.  The masses in figure slide on a frictionless table m 1 but not m 2, is fastened to the 
spring. If now m 1 and m 2 are pushed to the left, so that the spring is compressed a 
distance d, what will be the amplitude of the oscillation of m 1 after the spring system 
is released ? 
 
Q 16.  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 the time period of the motion of the block, 
  (b) the energy stored in the spring when the block passes through the equilibrium 
position and 
  (c) the kinetic energy of the block at this position. 
Q 17.  In figure, k = 100 N/m, M = 1 kg and F = 10 N. 
 
  (a) Find the compression of the spring in the equilibrium position. 
(b) A sharp blow by some external agent imparts a speed of 2 m/s to the block 
towards left. Find the sum of the potential energy of the spring and the kinetic energy 
of the block at this
 
instant. 
  (c) Find the time period of the resulting simple harmonic motion. 
  (d) Find the amplitude. 
  (e) Write the potential energy of the spring when the block is at the left extreme. 
  (f) Write the potential energy of the spring when the block is at the right extreme. 
The answers of (b), (e) and (f) are different. Explain why this does not violate the 
principle of conservation of energy? 
Q 18.  A point particle of mass 0.1 kg is executing SHM of amplitude 0.1 m. When the 
particle passes through the mean position, its kinetic energy is 8 × 10
-3
 J. Write down 
the equation of motion of this particle when the initial phase of oscillation is 45°. 
Q 19.  Potential energy of a particle in SHM along x-axis is given by : 
     U =10 + (x - 2)
2
 
Here, U is in joule and x in metre. Total mechanical energy of the particle is 26 J. 
Mass of the particle is 2 kg. Find : 
  (a) angular frequency of SHM, 
  (b) potential energy and kinetic energy at mean position and extreme position, 
  (c) amplitude of oscillation, 
  (d) x-coordinates between which particle oscillates. 
  Time period of SHM 
Q 20.  A pendulum has a period T for small oscillations. An obstacle is placed directly 
beneath the pivot, so that only the lowest one-quarter of the string can follow the 
pendulum bob when it swings to the left of its resting position. The pendulum is 
released from rest at a certain point. How long will it take to return to that point ? In 
answering this question, you may assume that the angle between the moving string 
and the vertical stays small throughout the motion. 
 
Q 21.  An object suspended from a spring exhibits oscillations of period T. Now, the spring 
is cut in half and the two halves are used to support the same object, as shown in 
figure. Show that the new period of oscillation is 772. 
 
Q 22.  Assume that a narrow tunnel is dug between two diametrically opposite points of the 
earth. Treat the earth as a solid sphere of uniform density. Show that if a particle is 
released in this tunnel, it will execute a simple harmonic motion. Calculate the time 
period of this motion. 
Q 23.  A simple pendulum is taken at a place where its separation from the earth's surface is 
equal to the radius of the earth. Calculate the time period of small oscillations if the 
length of the string is 1.0 m. Take g = ?
2
 m/s
2
 at the surface of the earth. 
Q 24.  The string, the spring and the pulley shown in figure are light. Find the time period of 
the mass m. 
 
Q 25.  A solid cylinder of mass M = 10kg and cross-sectional area A =20cm
2
 is suspended 
by a spring of force constant k =100 N/m and hangs partially immersed in water. 
Calculate the period of small oscillations of the cylinder. 
Q 26.  Pendulum A is a physical pendulum made from a thin, rigid and uniform rod whose 
length is d. One end of this rod is attached to the ceiling by a frictionless hinge, so that 
the rod is free to swing back and forth. Pendulum B is a simple pendulum whose 
length is also d. Obtain the ratio 
A
B
T
T
of their periods for small angle oscillations. 
Q 27.  A solid cylinder of mass m is attached to a horizontal spring with force constant k. 
The cylinder can roll without slipping along the horizontal plane. (See the 
accompanying figure.) Show that the centre of mass of the cylinder executes simple 
harmonic motion with a period 
3m
T2
2k
?? , if displaced from mean position. 
 
Q 28.  A cord is attached between a 0.50 kg block and a spring with force constant k 
=20N/m. The other end of the spring is attached to the wall and the cord is placed 
over a pulley (I = 0.60 MR
2
) of mass 5.0 kg and radius 0.50 m. (See the 
accompanying figure). Assuming no slipping occurs, what is the frequency of the 
oscillations when the body is set into motion ? 
 
  Combination of two or more simple harmonic motions 
Q 29.  Two linear SHM of equal amplitudes A and frequencies ? and 2 ?
 
are impressed on a 
particle along x and y-axes respectively. If the initial phase difference between them 
is ?/2.
 
Find the resultant path followed by the particle. 
Q 30.  Three simple harmonic motions of equal amplitudes A and equal time periods in the 
same direction combine. The phase of the second motion is 60° ahead of the first and 
the phase of the third motion is 60° ahead of the second. Find the amplitude of the 
resultant motion. 
Q 31.  A particle is subjected to two simple harmonic motions in the same direction having 
equal amplitudes and equal frequency. If the resultant amplitude is equal to the 
amplitude of the individual motions. Find the phase difference between the individual 
motions. 
Q 32.  A particle is subjected to two simple harmonic motions given by 
   x 1 = 2.0 sin(100 ?t) and x 2 = 2.0 sin(120 ?t + ?/3) 
where, x is in cm and t in second. Find the displacement of the particle at (a) t = 
0.0125, (b) t = 0.025. 
Solutions 
1.  (a) 20 × 10
-3 
g = k × 7 × 10
-2
 
   = 2.8 N/m as ?mg = k ?x 
  (b)