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Simple Harmonic Motion Formula: Types, Solved Examples

Simple Harmonic Motion (SHM) refers to a type of periodic motion in which an object oscillates back and forth around an equilibrium position
authorImageMurtaza Mushtaq20 Sept, 2023
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Simple Harmonic Motion Formula

What Is Simple Harmonic Motion?

In Simple Harmonic Motion Formula , there are two key factors at play: a restoring force and a mass (or equivalent inertial element). A restoring force proportional to the displacement is generated, directing the object back towards its equilibrium. As the object moves closer to the equilibrium, this force decreases, and it increases as the object moves farther away. The acceleration of a particle executing simple harmonic motion is given by a(t) = -ω 2 x(t). The example of simple harmonic motion includes a mass attached to a spring and a pendulum. In the case of a mass-spring system, when the spring is compressed or stretched, the resulting oscillations exhibit simple harmonic motion. Similarly, a pendulum's swinging motion back and forth is also a form of simple harmonic motion when the angles are small.

Simple Harmonic, Periodic And Oscillation Motion And Its Difference

Simple Harmonic Motion (SHM): Simple Harmonic Motion (SHM) refers to a type of periodic motion in which an object oscillates back and forth around an equilibrium position under the influence of a restoring force that is proportional to its displacement from the equilibrium position. Functions like sine or cosine. SHM is characterised by its amplitude (maximum displacement from the equilibrium), frequency (number of oscillations per unit time).

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Periodic Motion: In a regular pattern over time. It doesn't necessarily have to be a simple harmonic motion. A periodic motion can exhibit various patterns. The defining feature of periodic motion is its repetition at regular intervals. The period of a motion is the length of time it takes for one full cycle to occur, and the frequency is the reciprocal of the period.

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Oscillation: Oscillation is a general term that describes any repetitive back-and-forth movement around a central point or position. It can include both simple harmonic motion and other types of periodic motion. Oscillation can occur in various systems, including mechanical systems like pendulums, electronic systems like alternating current (AC) circuits, and even in waves like sound and light.

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Differences: The key differences between these terms are as follows: Type of Motion:
  • A type of periodic motion where the restoring force is directly proportionate to the displacement is known as simple harmonic motion (SHM).
  • Periodic motion is a broader term that encompasses any motion.
  • Oscillation is a general term for repetitive back-and-forth motion, which can include both periodic and non-periodic motions.

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Restoring Force:
  • A restoring force that is proportionate to the displacement from equilibrium is a feature of SHM.
  • Periodic motion may or may not involve a linear restoring force.
  • Oscillation does not specify the nature of the restoring force.
Mathematical Representation:
  • SHM is often mathematically described using trigonometric functions like sine or cosine.
  • Periodic motion can be described using various mathematical functions, not necessarily limited to trigonometric functions.
  • Oscillation is a broader concept and does not have a specific mathematical representation.

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Types Of Simple Harmonic Motion

SHM can be categorized into two main types: linear (translational) SHM and angular SHM. 1. Linear (Translational) Simple Harmonic Motion: In linear SHM, the motion occurs along a straight line. A spring that is stretched or compressed and then released. Mathematical Condition: The equation of motion for linear SHM can be expressed as follows: x ( t )= A ⋅cos( ωt + ϕ ) Where:
  • x ( t ) is the displacement of the object from its equilibrium position at time t.
  • The oscillation's largest departure from equilibrium is represented by A, the oscillation's amplitude.
  • ω is the angular frequency of the oscillation, related to the frequency f by ω =2 πf
  • ϕ is the phase angle, which determines the initial phase of the motion.
2. Angular Simple Harmonic Motion: In angular SHM, the motion occurs in a circular path. Mathematical Condition: The equation of motion for angular SHM can be expressed as follows: θ ( t )= θ 0 ⋅cos( ωt + ϕ ) Where:
  • θ ( t ) is the angular displacement of the object from its equilibrium position at time t .
  • θ 0​   is the amplitude of angular displacement (maximum angle from the equilibrium position).
  • ω and ϕ have the same meanings as in linear SHM.

Solutions Of Differential Equations Of SHM

The following are the solutions to the differential equation for the Simple Harmonic Motion:
  • x= A sin t
(This solution when the particle is in its mean position point (O) in figure (a)
  • x 0 = A sin
(When the particle is at the position & (not at mean position) in figure (b)
  • x = A sin ( t+ )
(When the particle at Q at in figure (b) (any time t).

Time Period And Frequency Of SHM

The time period and frequency of SHM are important characteristics that describe the motion.
  • Time Period (T): The time required for one full cycle of motion is the SHM time period. In other words, it is the amount of time required for the object to oscillate fully before returning to its initial position. The time interval is often represented by the letter "T" and is expressed in seconds.
  • Frequency (f): The number of full cycles or oscillations that take place in a given period of time is the frequency of SHM. It is the time period's inverse, and it is frequently denoted by the letter "f." Hertz (Hz) is a unit of frequency measurement, where 1 Hz equals one cycle per second.
The relationship between time period (T) and frequency (f) in SHM is given by the equation: f = 1 / T Conversely: T = 1 / f

Phase In SHM

The condition of a vibrating or oscillating particle with respect to its displacement and vibration direction at any given instant is known as the phase of the particle. A particle's expression and location in relation to time. x = A sin (ωt + Φ) Where (ωt + Φ) is the phase of the particle

Energy In Simple Harmonic Motion (SHM)

There are two main forms of energy involved in SHM: potential energy and kinetic energy.
  • Potential Energy (PE): Potential energy in SHM is related to how far an object is moved from its equilibrium position. As the displacement changes, the potential energy changes as well. A mass-spring system, in which a mass is connected to a spring and is capable of oscillating around its equilibrium position, is the most typical example of SHM. This system's potential energy is determined by:
  • PE = 0.5  kx 2
where:
  • PE is the potential energy.
  • k is the spring constant, a measure of the stiffness of the spring.
  • x represents the mass's deviation from its equilibrium position.
Potential energy is at its maximum (and kinetic energy is at its minimum) when the object is at the extreme points of its motion (maximum displacement), and potential energy is at its minimum (kinetic energy is at its maximum) when the object is at the equilibrium position (zero displacement).
  • Kinetic Energy (KE): Kinetic energy in SHM is associated with the object's velocity as it moves back and forth. The kinetic energy varies as the velocity changes. An oscillating object's kinetic energy is given by:
KE = 0.5 mv 2 where:
  • KE is the kinetic energy.
  • m is the mass of the object.
  • v is the velocity of the object.
Kinetic energy is at its maximum (and potential energy is at its minimum) when the object passes through the equilibrium position, and kinetic energy is at its minimum (potential energy is at its maximum) when the object is at the extreme points of its motion. Total Mechanical Energy (E): In the absence of non-conservative forces like friction or air resistance, the total mechanical energy of an object undergoing SHM remains constant. This is a consequence of the conservation of mechanical energy. Potential energy and kinetic energy are added to create total mechanical energy (E): E = PE + KE

Simple Harmonic Motion Formula FAQs

What is Simple Harmonic Motion (SHM)?

An item oscillates back and forth around an equilibrium position in simple harmonic motion (SHM), a form of periodic motion, under the influence of a restoring force that is proportional to the object's displacement from the equilibrium position.

How does the mass affect SHM?

In a simple spring-mass system, the mass does not affect the frequency of SHM, but it does affect the amplitude. A larger mass will result in a smaller amplitude for the same amount of energy.

How does the spring constant affect SHM?

The spring constant affects the frequency of SHM. A higher spring constant will result in a higher frequency of oscillation.

What is the relationship between SHM and circular motion?

SHM can be thought of as a projection of circular motion onto a single dimension. The projection of uniform circular motion results in simple harmonic motion.
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