Sample Questions on Simple Harmonic Motion

Question 1: Why is Harmonic Motion Periodic?

Solution: 

The sine wave can represent a harmonic motion. When a spring is stretched from its mean position, it oscillates to and fro about the mean position under the influence of a restoring force that is always directed towards the mean position and whose magnitude at any instant is proportional to the body’s displacement from the mean position at that instant. When there is no friction, the motion tends to be periodic. The harmonic motion is periodic in this case.

Question 2: What are Periodic and Non-Periodic Changes?

Solution: 

Periodic changes are those that occur at regular intervals of time, such as the occurrence of day and night, or the change of periods in your school. Non-periodic changes are those that do not occur on a regular basis, such as the freezing of ice to water.

Question 3: What is the period of the Earth’s revolution around the sun and around its polar axis? what is the motion Earth performs explain?

Solution: 

The earth’s revolution around the sun takes one year, and its revolution around its polar axis takes one day. The motion of earth is periodic because after some interval of time it repeats its path.

Question 4: What is the frequency of SHM? How time periods and frequency are related?

Solution: 

The frequency of SHM is the number of oscillations performed by a particle per unit of time. Hertz, or r.p.s. (rotations per second), is the SI unit of frequency. Frequency and time period are related as:

Frequency, (f) = 1/ Time period (T)    

Question 5: A spring with a spring constant of 1200 N m–1 is mounted on a horizontal table. A 3 kg mass is attached to the free end of the spring. The mass is then pulled sideways to a distance of 2.0 m before being released. Determine the following:

1. The frequency of oscillations,
2. Maximum acceleration of the mass, and
3. The maximum speed of the mass.

Solution:

Given:

• Spring Constant, k = 1200 N/m.
• Mass of Object, m = 3 kg.
• Displacement, x = 2 m.

(1) Frequency of Oscillation:

We know that frequency (f) = 1/Time period (T)          T = 2π/ω and ω = √k/m]

Therefore,

f = (1/2π)√k/m

  = (1/2 × 3,14) √1200/3 = 3.18 Hz.

(2) Maximum Acceleration:

Maximum Acceleration (a) = ω²x

where, ω = Angular frequency = √k/m

Therefore, a = x(k/m)

a = 2 × (1200/3) 

a= 800 m/s².

(3) Maximum Speed:

Maximum Speed (V) = ωx 

Put, ω = √k/m.

Therefore, V = x(√k/m)

V = 2 × (√1200/3) 

V = 40 m/s.

Question 6: A mass of 2 kg is attached to the end of the spring with a spring constant of 50 N/m. What is the period of the resulting simple harmonic motion? (π = 3.14)

Solution:

Formula for time period is

T = 2π√(k/m)

where,

• m is the mass
• k is the spring constant

Thus, T = 2π√(50/2) 

⇒ T = 2π√(25) 

⇒ T = 2π/5 

⇒ T ≈ 1.26 s

So, the time period of the SHM is approximately 1.26 s.

Question 7: A block of mass 0.5 kg is attached to the end of the spring (spring constant =100 N/m). If The block is displaced 0.1 m from its equilibrium position then what is the maximum speed of the block during its motion?

Solution:

The maximum speed of the block is given by:

v_max = Aω

where, 

• A is Amplitude of Motion
• ω is Angular Frequency

Also, angular frequency ω is given by:

ω = √(k/m)

where, 

• m is the mass
• k is the spring constant

Given: 

• Amplitude(A) = 0.1 m
• k = 100 N/m
• m = 0.5 Kg

⇒ v_max  = 0.1 × √(100/0.5) 

⇒ v_max  = 0.1 × √(1000/5) 

⇒ v_max  = 0.1 × √(200) 

⇒ v_max  = √2 

So, the maximum speed of the block during its motion is √2  m/s.

Simple Harmonic Motion is a fundament concept in the study of motion, especially oscillatory motion; which helps us understand many physical phenomena around like how strings produce pleasing sounds in a musical instrument such as the sitar, guitar, violin, etc., and also, how vibrations in the membrane in drums and diaphragms in telephone and speaker system creates the precise sound. Understanding Simple Harmonic Motion is key to understanding these phenomena. 

In this article, we will grasp the concept of Simple Harmonic Motion (SHM), its examples in real life, the equation, and how it is different from periodic motion.