Unit # 10 Engr Umair Akhtar Physics 10th
Unit No 10
Simple Harmonic Motion and Waves
Extensive Question
Q1: What is simple harmonic motion? What are the necessary conditions for a body to execute simple
harmonic motion?
Ans: Statement:.
Simple harmonic motion occurs when the net force is proportional to the displacement from the mean
position and is always directed toward the mean position and its acceleration is also directly proportional to
this displacement from the mean position, it’s motion is called Simple Harmonic Motion (SHM).
Mathematical Expression:.
a ∝ –x
Fext ∝ x ⟹ Fext = kx
Conditions:.
Following are the necessary conditions to prove the Simple Harmonic Motion.
i. A body executing SHM always vibrates about a fixed position.
ii. Its acceleration is always directed towards the mean position.
iii. The magnitude of acceleration is always directly proportional to it’s displacement from the mean position
i.e. acceleration will be zero at the mean position while it will be maximum at the extreme position.
a ∝ –x
iv. Its velocity is maximum at the mean position and zero on the extreme positions.
Q2: Prove simple harmonic motion with an example or with the help of mass attach to the spring and the
motion of simple pendulum?
Ans: Statement:.
Simple harmonic motion occurs when the net force is proportional to the displacement from the mean
position and is always directed towards the mean position.
Example: Mass attached to a spring
Simplest types of oscillatory motion is that of horizontal mass – spring system.
At Initial Position:. (Point A)
If the spring is stretched or compressed through a small displacement x from its
mean position, it exerted a force F on the mass.
Hooke’s Law:
According to Hook’s law force is directly proportional to the change in length x of the spring i.e.
Fext ∝ x
Fext = –kx
x
Here x is the displacement of the mass from its mean position O,
Spring Constant: K is a constant called spring constant. It defines as
x
The value of K is a measure of the stiffness of the spring. Stiff springs have large K value and soft springs have
small K value.
Negative Sign:
The negative sign shows that the force exerted by the spring is always directed opposite to the displacement
of mass.
Restoring Force:
A restoring force always pushes or pulls the object performing oscillatory motion towards the mean position.
Page 1 of 16
,Unit # 10 Engr Umair Akhtar Physics 10th
Initially the mass m is at rest in mean position O and the resultant force on the mass is zero.
At the point O – A:
Suppose the mass is pulled through a distance x up to extreme position A and then released. The restoring
force exerted by the spring on the mass will pull it towards the mean position O. Due to the restoring force
the mass moves back towards the mean position O. The magnitude of the restoring force decreases with the
distance from the mean position and becomes zero at O. So, the mass gains speed as it moves towards the
mean position and its speed becomes maximum at O.
At the point O – B:
Due to inertia the mass does not stop at the mean position O but continues its motion and reaches the
extreme position B.
As the mass moves from the mean position O to the extreme position B, the restoring force acting on it towards
the mean position steadily increase in strength. Hence the speed of the mass decreases as it moves towards
the extreme position B. Ultimately the mass returns to the mean position due to the restoring force.
Time Period:
The time period of mass attached to spring is
T = 2π √
Conclusion:
Hence the process is repeated, and the mass continues to oscillate back and forth about the mean position O.
Such motion of a mass attached to a spring on horizontal frictionless surface is known as Simple Harmonic
Motion (SHM).
MOTION OF A SIMPLE PENDULUM:
A simple pendulum also exhibits SHM. It consists of a small bob of mass
suspended from a light string of length L fixed at its upper end.
In the equilibrium position O, the net force on the bob is zero and the bob is stationary.
W = mg
At the Point O – A:
If we bring the bob to extreme position A, the net force is not zero. There is no
force acting along the string as the tension in the string cancels the component of
the weight mg cos θ. Hence there is no motion along this direction. The component
of the weight mg sin θ is directed towards the mean position and acts as a restoring
force. Due to this force the bob starts moving towards the mean position O. At O
the bob has got the maximum velocity and due to inertia it does not stop at O rather it continues to move
towards the extreme position B.
At the Point O – B:
During its motion towards point B, the velocity of the bob becomes zero as it reaches the point B. The
restoring force mg sin θ still acts towards the mean position O and due to this force the bob again starts
moving towards the mean position O. In this way, the bob continues its to and fro motion about the mean
position O.
At the Point A – O:
The speed of the bob increases while moving from point A to O due to the restoring force which acts
towards O. Therefore, acceleration of the bob is also directed towards O.
At the Point O – B:
Similarly, when the bob moves from O to B, its speed decreases due to restoring force which again acts
towards O.
Time Period:
The time taken by a vibrating body in complete one vibration is called time period.
Time period of simple pendulum is
T = 2π √
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, Unit # 10 Engr Umair Akhtar Physics 10th
Conclusion:
When an object oscillates about either side of a fixed position (mean position) such that its acceleration is
directly proportional to its displacement from the mean position, and is always directed towards the mean
position, this motion is called SHM.
Q3: Think of several examples in everyday life of motion that are simple harmonic?
Ans: Following are the everyday life examples of motion that are simple harmonic.
i. Up and down motion of leaf in water pond.
ii. Motion of pendulum.
iii. Motion of hands of clock.
iv. Motion of a plucked string fixed at both its end.
Q4: What are damped oscillation? How damping progressively reduces the amplitude of oscillation?
Ans: Definition:.
The oscillations of a system in the presence of some resistive force are damped oscillation.
Explanation:
Vibratory motion of ideal systems in the absence of any friction or residence
continues indefinitely under the action of a restoring force. Practically, in all
systems the force of friction retards the motion, so the systems do not oscillate
indefinitely. The friction reduces the mechanical energy of the system passes,
and the motion is said to be damped. This damping progressively reduces the
amplitude of the motion.
Example:
Shock absorbers in automobiles are one practical application of damped
motion.
Construction and working:
A shock absorber consists of a piston moving through a liquid such as oil. The
upper part of the shock absorber is firmly attached to the body of the car. When
the car travels over a bump on the road, the car may vibrate violently. The
shock absorbers damp these vibrations and convert their energy into heat
energy of the oil.
Conclusion:
Hence, the oscillations of a system in the presence of some resistive force are damped oscillations.
Q5: How can you define the term wave? Elaborate the difference between mechanical and electromagnetic
waves? Give examples of each?
Ans: Wave
A wave is a disturbance in the medium which causes the particles of the medium to undergo vibratory
motion about their mean position in equal intervals of time.
Types of waves:
There are two categories of waves:
i. Mechanical waves
ii. Electromagnetic waves
Mechanical Waves:
Waves which require any medium for their propagation are called mechanical waves.
Example:
Water waves, sound waves and waves produced in the strings and springs.
Electromagnetic waves:
Waves which do not require any medium for their propagation are called electromagnetic waves.
Page 3 of 16
Unit No 10
Simple Harmonic Motion and Waves
Extensive Question
Q1: What is simple harmonic motion? What are the necessary conditions for a body to execute simple
harmonic motion?
Ans: Statement:.
Simple harmonic motion occurs when the net force is proportional to the displacement from the mean
position and is always directed toward the mean position and its acceleration is also directly proportional to
this displacement from the mean position, it’s motion is called Simple Harmonic Motion (SHM).
Mathematical Expression:.
a ∝ –x
Fext ∝ x ⟹ Fext = kx
Conditions:.
Following are the necessary conditions to prove the Simple Harmonic Motion.
i. A body executing SHM always vibrates about a fixed position.
ii. Its acceleration is always directed towards the mean position.
iii. The magnitude of acceleration is always directly proportional to it’s displacement from the mean position
i.e. acceleration will be zero at the mean position while it will be maximum at the extreme position.
a ∝ –x
iv. Its velocity is maximum at the mean position and zero on the extreme positions.
Q2: Prove simple harmonic motion with an example or with the help of mass attach to the spring and the
motion of simple pendulum?
Ans: Statement:.
Simple harmonic motion occurs when the net force is proportional to the displacement from the mean
position and is always directed towards the mean position.
Example: Mass attached to a spring
Simplest types of oscillatory motion is that of horizontal mass – spring system.
At Initial Position:. (Point A)
If the spring is stretched or compressed through a small displacement x from its
mean position, it exerted a force F on the mass.
Hooke’s Law:
According to Hook’s law force is directly proportional to the change in length x of the spring i.e.
Fext ∝ x
Fext = –kx
x
Here x is the displacement of the mass from its mean position O,
Spring Constant: K is a constant called spring constant. It defines as
x
The value of K is a measure of the stiffness of the spring. Stiff springs have large K value and soft springs have
small K value.
Negative Sign:
The negative sign shows that the force exerted by the spring is always directed opposite to the displacement
of mass.
Restoring Force:
A restoring force always pushes or pulls the object performing oscillatory motion towards the mean position.
Page 1 of 16
,Unit # 10 Engr Umair Akhtar Physics 10th
Initially the mass m is at rest in mean position O and the resultant force on the mass is zero.
At the point O – A:
Suppose the mass is pulled through a distance x up to extreme position A and then released. The restoring
force exerted by the spring on the mass will pull it towards the mean position O. Due to the restoring force
the mass moves back towards the mean position O. The magnitude of the restoring force decreases with the
distance from the mean position and becomes zero at O. So, the mass gains speed as it moves towards the
mean position and its speed becomes maximum at O.
At the point O – B:
Due to inertia the mass does not stop at the mean position O but continues its motion and reaches the
extreme position B.
As the mass moves from the mean position O to the extreme position B, the restoring force acting on it towards
the mean position steadily increase in strength. Hence the speed of the mass decreases as it moves towards
the extreme position B. Ultimately the mass returns to the mean position due to the restoring force.
Time Period:
The time period of mass attached to spring is
T = 2π √
Conclusion:
Hence the process is repeated, and the mass continues to oscillate back and forth about the mean position O.
Such motion of a mass attached to a spring on horizontal frictionless surface is known as Simple Harmonic
Motion (SHM).
MOTION OF A SIMPLE PENDULUM:
A simple pendulum also exhibits SHM. It consists of a small bob of mass
suspended from a light string of length L fixed at its upper end.
In the equilibrium position O, the net force on the bob is zero and the bob is stationary.
W = mg
At the Point O – A:
If we bring the bob to extreme position A, the net force is not zero. There is no
force acting along the string as the tension in the string cancels the component of
the weight mg cos θ. Hence there is no motion along this direction. The component
of the weight mg sin θ is directed towards the mean position and acts as a restoring
force. Due to this force the bob starts moving towards the mean position O. At O
the bob has got the maximum velocity and due to inertia it does not stop at O rather it continues to move
towards the extreme position B.
At the Point O – B:
During its motion towards point B, the velocity of the bob becomes zero as it reaches the point B. The
restoring force mg sin θ still acts towards the mean position O and due to this force the bob again starts
moving towards the mean position O. In this way, the bob continues its to and fro motion about the mean
position O.
At the Point A – O:
The speed of the bob increases while moving from point A to O due to the restoring force which acts
towards O. Therefore, acceleration of the bob is also directed towards O.
At the Point O – B:
Similarly, when the bob moves from O to B, its speed decreases due to restoring force which again acts
towards O.
Time Period:
The time taken by a vibrating body in complete one vibration is called time period.
Time period of simple pendulum is
T = 2π √
Page 2 of 16
, Unit # 10 Engr Umair Akhtar Physics 10th
Conclusion:
When an object oscillates about either side of a fixed position (mean position) such that its acceleration is
directly proportional to its displacement from the mean position, and is always directed towards the mean
position, this motion is called SHM.
Q3: Think of several examples in everyday life of motion that are simple harmonic?
Ans: Following are the everyday life examples of motion that are simple harmonic.
i. Up and down motion of leaf in water pond.
ii. Motion of pendulum.
iii. Motion of hands of clock.
iv. Motion of a plucked string fixed at both its end.
Q4: What are damped oscillation? How damping progressively reduces the amplitude of oscillation?
Ans: Definition:.
The oscillations of a system in the presence of some resistive force are damped oscillation.
Explanation:
Vibratory motion of ideal systems in the absence of any friction or residence
continues indefinitely under the action of a restoring force. Practically, in all
systems the force of friction retards the motion, so the systems do not oscillate
indefinitely. The friction reduces the mechanical energy of the system passes,
and the motion is said to be damped. This damping progressively reduces the
amplitude of the motion.
Example:
Shock absorbers in automobiles are one practical application of damped
motion.
Construction and working:
A shock absorber consists of a piston moving through a liquid such as oil. The
upper part of the shock absorber is firmly attached to the body of the car. When
the car travels over a bump on the road, the car may vibrate violently. The
shock absorbers damp these vibrations and convert their energy into heat
energy of the oil.
Conclusion:
Hence, the oscillations of a system in the presence of some resistive force are damped oscillations.
Q5: How can you define the term wave? Elaborate the difference between mechanical and electromagnetic
waves? Give examples of each?
Ans: Wave
A wave is a disturbance in the medium which causes the particles of the medium to undergo vibratory
motion about their mean position in equal intervals of time.
Types of waves:
There are two categories of waves:
i. Mechanical waves
ii. Electromagnetic waves
Mechanical Waves:
Waves which require any medium for their propagation are called mechanical waves.
Example:
Water waves, sound waves and waves produced in the strings and springs.
Electromagnetic waves:
Waves which do not require any medium for their propagation are called electromagnetic waves.
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