Simple Harmonic Motion Questions

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Very Short Answers Questions [1 Mark Questions]

Ques. What are the main types of simple harmonic motion?

Ans. There are mainly two types of simple harmonic motion (SHM)

• Linear SHM
• Angular SHM

Ques. What is meant by phase difference?

Ans. It is a physical quantity on which the position of the particle or body executing SHM depends. The difference in the phase angle of the two bodies executing SHM is called phase difference.

Ques. What is meant by linear simple harmonic motion (linear SHM)?

Ans. A particle executing simple harmonic motion is said to be linear simple harmonic motion if the oscillations of the particle about the mean position are in a straight line.

Ques. What is meant by angular simple harmonic motion?

Ans. Angular SHM is defined as the oscillatory motion in which the torque for angular acceleration of a body is directly proportional to the angular displacement and its direction is opposite to that of the angular displacement.

Ques. The formula for the acceleration of a body executing SHM is

1. ωx
2. - ωx
3. - ω²x
4. ω²x

Ans. The correct answer is c. - ω²x

Explanation: The formula of acceleration in simple harmonic motion is given by

a = - ω²x

Where x is the displacement of the particle from the mean position.

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Short Answers Questions [2 Marks Questions]

Ques. Define simple harmonic motion.

Ans. A particle is said to execute simple harmonic motion if it moves to and fro about a mean position under the action of restoring force which is directly proportional to its displacement from the mean position and is always directed toward the mean position.

It is the simplest form of oscillatory motion.

Ques. Write the formula of simple harmonic motion.

Ans. The formula of simple harmonic motion is given by

x = A sin ωt

or, x = A cos ωt

Where

• x = displacement of the body from the mean position at any instant of time t.
• A = maximum displacement, also known as amplitude of the displacement of the body
• ω = angular frequency

Ques. What is the main connection between simple harmonic motion and harmonic motion?

Ans. The oscillation that can be expressed in terms of a single harmonic function is called harmonic oscillation. In this type of oscillation, a restoring force acts on the particle is proportional to the displacement but in the opposite direction.

A harmonic oscillation of constant amplitude and of single frequency is called Simple harmonic motion.

Ques. What is meant by the time period of SHM?

Ans. For a particle executing simple harmonic motion, the time period is defined as the time taken by the particle to complete one oscillation.

Also Read:

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Long Answers Questions [3 Marks Questions]

Ques. A simple harmonic oscillator has a displacement equation x = 5 cosωt. Show that whatever may be the value of ω, the simple harmonic oscillator is at its extreme position at t = 0.

Ans. The extreme position is marked by the maximum displacement (amplitude) from the mean position.

For displacement equation, x = 5 cosωt

The extreme position represents, x = 5

Now, at t = 0, we have

x = 5 cos(ω x 0) = 5 cos 0 = 5

Therefore, at t = 0, the particle is at its extreme position.

Ques. A particle executes SHM with amplitude A and the time period T. When the displacement from the equilibrium position is half the amplitude, what fraction of the total energy is kinetic and potential?

Ans. The total energy in simple harmonic motion is given by

E = 1/2 KA²

Where

• K is a constant
• A is the amplitude of the oscillation

Let the displacement of the particle be x when the i.e half the amplitude of oscillation

x = A/2

The potential energy, U = 1/2 Kx²

⇒ U = 1/2 K (A/2)² = 1/4 x (1/2kA²) = 1/4 E = 25% of E

Also kinetic energy, K.E. = Total energy E - potential energy U

⇒ K.E. = E - 1/4 E = 3/4 E = 75% of E

Ques. A particle of mass 0.2 kg is executing SHM of amplitude 0.2 m. When it passes through the mean position, its kinetic energy is 64 x 10^-3 J. Obtain the equation of motion of this particle if the initial phase of oscillation is π/4.

Ans. Given

• The amplitude of the oscillation, A = 0.2 m
• Phase difference, Φ = π/4
• Mass of the particle, m = 0.2 kg

Kinetic energy at distance x from the mean position is given by

K.E. = 1/2 mω² (A² - x²)

At mean position x = 0, kinetic energy is 64 x 10^-3 J (Given)

⇒ 64 x 10^-3 = 1/2 x 0.2 x ω² (0.22 - 0)

⇒ ω² = 16

⇒ ω = 4 rad/s

Hence the equation of motion at the mean position is given by

x = A sin (ωt + Φ)

⇒ x = 0.2 sin (4t + π/4)

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Very Long Answers Questions [5 Marks Questions]

Ques. What are the characteristics of simple harmonic motion?

Ans. The followings are the characteristics of simple harmonic motion

• It is a type of oscillatory motion and a particular case of periodic motion.
• A restoring force must act on the body.
• The body must have acceleration in the opposite direction of displacement and the acceleration is directly proportional to the displacement.
• The system must have inertia.
• The energy of the system oscillates between kinetic energy and potential energy but the total energy remains constant provided there is no loss of energy due to external or internal frictional force.

Ques. Show that the displacement equation x = 5 + 4cosωt can represent a simple harmonic motion. Represent the mean and extreme position of the oscillator on the X-axis.

Ans. The given displacement equation can be written as

x - 5 = 4cosωt

Let x - 5 = X, then X = 4cosωt

The above equation represents an SHM with a mean position marked by X = 0 and extreme positions marked by X = ± 4.

Mean position:

It is given by X = 0

⇒ x - 5 = 0

⇒ x = 5

Thus the mean position is located at x = 5.

Extreme position:

It is given by X = ± 4

⇒ x - 5 = ± 4

⇒ x = 5 ± 4

⇒ x = 9 or 1

Thus two extreme positions are x = 1 and x = 9.

The representation of the mean and extreme position of the oscillator on the X-axis is shown below

Ques. Two particles oscillate simple harmonically with the same amplitude and same frequency along the same straight line. They cross each other when moving in opposite directions, each time their displacement is 1/√2 times their amplitude. What is the phase difference between them?

Ans. Let's assume the equations of motions of two particles are 

y₁ = A sinωt and 

y₂ = A sin (ωt + Φ)

Where

• A is the maximum amplitude
• Φ is the phase difference

For y₁ = A/√2 and y₂ = A/√2, we have

A/√2 = A sinωt and A/√2 = A sin(ωt + Φ)

⇒ sin ωt = 1/√2 and sin(ωt + Φ) = 1/√2

⇒ ωt = π/4 and ωt + Φ = π/4 or 3π/4

As the two particles are not in the same phase

⇒ ωt + Φ = 3π/4

⇒ Φ = 3π/4 - π/4 = π/2

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