Simple Harmonic Motion, Dynamics, Velocity And Acceleration, Important Topics For JEE 2024

Simple Harmonic Motion : We will learn that When a body repeats its motion after regular time intervals we say that it is in harmonic motion or periodic motion . The time interval after which the motion is repeated is called the time period . If a body moves to and fro on the same path, it is said to perform oscillations .

a = −ω ² x

where ω ² is a positive constant. If x is positive, a is negative and if x is negative, a is positive. This means that the acceleration is always directed towards the centre of oscillation.

SHM is a special type of oscillatory motion in which the restoring force is proportional to the displacement from the mean position (for small displacement from mean position). It is the simplest (easy to analyse) form of oscillatory motion. The amplitude of oscillations of the particle is very small. At any instant the displacement of a particle in oscillatory motion can be expressed in terms of sinusoidal functions (sine and cosine functions).

These functions are called harmonic functions. That is why an oscillatory motion is also called a harmonic motion. A simple harmonic motion can be expressed in terms of one single sine or cosine function or a linear combination of sine and cosine functions. If a particle is moving to and fro about an equilibrium point M (called as the mean position), along a straight line as shown in Figure, then we can call its motion to be SHM, where A and B are extreme positions and

AM = MB = Amplitude ( a )

However, when a body or particle is free to rotate about a given axis executing angular oscillations, then this type of SHM can be regarded as angular SHM.

Dynamics of Simple Harmonic Motion : The restoring force for small displacement x is given by F = – k x , where k is called the force constant of the system. Force constant is defined as the restoring force per unit displacement. Its SI unit is Nm ⁻¹ . If m is mass of the body and a is its acceleration, then for simple harmonic motion.

The general solution to this equation is

Where, a is the amplitude, is the phase (also called as instantaneous phase), is phase constant or initial phase angle (in radian) also called as epoch.

Consider a particle to execute SHM and assume that the path of the particle is on a straight line, then x is the displacement of particle from the mean position at that instant. The displacement of a particle executing SHM at time t is given by

x = a sin (ω t + ϕ)

y = a sin (ω t + ϕ)

Amplitude, a or A , is the maximum value of displacement of the particle from its equilibrium position i.e., mean position. So, amplitude of SHM is half the separation between the extreme points of SHM. It depends upon the energy of the system.

Angular frequency , ω of SHM is and its SI unit is rads ^–1 .

Frequency, f or v is the number of oscillations completed in unit time interval, so we have Its unit is sec ^–1 or Hz.

Time period, T is the smallest time interval after which oscillatory motion gets repeated, so time period T given by

Mathematically at t = T , the displacement x of SHM must satisfy x ( t + T ) = x ( t ). For example,

and

Velocity And Acceleration In Shm : We can write the equation of motion of a particle performing SHM is

y = A sin (ω x + ϕ) …(i)

Differentiating y with respect to ‘ t ’ given velocity of the particle

…(ii)

From Eq. (i), we can write sin (ω t + ϕ) = and we know that

Substituting the value of cos (ω t + ϕ) in Eq. (i), we get

…(iii)

Differentiating Eq. (ii) again w.r.t. time.

Equation (iii) can be rearranged in the form

∙ This shows that velocity-position graph is an ellipse. This is also true for momentum-position graph. The momentum-position graph is known as phase-space graph of the particle.

(a) Displacement-time graphs: The displacement of the particle is always measured from the mean position P wheareas, time t can be calculated from any position of the oscillating particle.

(b) Velocity-time and acceleration-time graphs: If particle starts SHM from mean position, then

Displacement equation, y = A sin ω t …(i)

Differentiation of equation (i) gives velocity of the particle.

…(ii)

Differentiation of Eq. (ii) again gives acceleration of the particle.

Potential Energy Of A Particle In Simple Harmonic Motion : The potential energy U at a displacement x is the work done against the restoring force in moving the body from the mean position to this position.

∙ The maximum value of potential energy is at the extreme position i.e., at x = ± A and minimum value of potential energy is U _min = 0 at the mean position i.e., at x = 0.

Kinetic Energy Of A Particle In Simple Harmonic Motion

Kinetic Energy Of A Particle In Simple Harmonic Motion : If x = A sin (ω t ), then v = A ω cos (ω t ), so kinetic energy K of a particle executing SHM is given by

∙ The maximum value of kinetic energy is at the mean position i.e., at x = 0 and minimum value of kinetic energy is K _min = 0 at the extreme position i.e., at x = ± A .

Mechanical Energy Of A Particle In Simple Harmonic Motion

Mechanical Energy Of A Particle In Simple Harmonic Motion : The mechanical energy E of a particle executing SHM is

So, we observe that total mechanical energy is a constant. The variations of U , K and E with time are shown in Figure.

The variations of U , K and E with displacement are shown in Figure.

At x = 0, U = 0 and the energy is purely kinetic i.e., At extreme points or turning points of the SHM, kinetic energy is zero and the energy is purely potential i.e.,