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Simple Harmonic Motion

The oscillation which can be expressed in terms of single harmonic function, i.e., sine or cosine function, is called harmonic oscillation.

A harmonic oscillation of constant amplitude and of single frequency under a restoring force whose magnitude is proportional to the displacement and always acts towards mean position is called Simple Harmonic Motion (SHM).

Simple Harmonic MotionSimple Harmonic Motion

A simple harmonic oscillation can be expressed as

y = a sin ωt or y = a cos ωt

Where a = amplitude of oscillation.

Non-harmonic Oscillation

A non-harmonic oscillation is a combination of two or more than two harmonic oscillations.

It can be expressed as y = a sin ωt + b sin 2ωt

(i) Time Period: Time taken by the body to complete one oscillation is known as time period. It is denoted by T.

(ii) Frequency: The number of oscillations completed by the body in one second is called frequency. It is denoted by v.

Its SI unit is ‘hertz’ or ‘second-1.

Frequency = 1 / Time period

(iii) Angular Frequency: The product of frequency with factor 2π is called angular frequency. It is denoted by ω.

Angular frequency (ω) = 2πv

Its SI unit is ‘hertz’ or ‘second-1‘.

(iv) Displacement: A physical quantity which changes uniformly with time in a periodic motion is called displacement. It is denoted by y.

(v) Amplitude: The maximum displacement in any direction from mean position is called amplitude. It is denoted by a.

(vi) Phase: A physical quantity which express the position and direction of motion of an oscillating particle, is called phase. It is denoted by φ.

Simple harmonic motion is defined as the projection of a uniform circular motion on any diameter of a circle of reference.

Question for Simple Harmonic Motion
Try yourself:
What is the time period of a simple harmonic motion?
View Solution

Some Important Formulae of SHM

(i) Displacement in SHM at any instant is given by

y = a sin ωt

or y = a cos ωt

where a = amplitude and

ω = angular frequency.

(ii) Velocity of a particle executing SHM at any instant is given by

v = ω √(a2 – y2)

At mean position y = 0 and v is maximum

vmax = aω

At extreme position y = a and v is zero.

(iii) Acceleration of a particle executing SHM at any instant is given by

A or α = – ω2 y

Negative sign indicates that the direction of acceleration is opposite to the direction in which displacement increases, i.e., towards mean position.

At mean position y = 0 and acceleration is also zero.

At extreme position y = a and acceleration is maximum

Amax = – aω2

(iv) Time period in SHM is given by

T = 2π √Displacement / Acceleration

Graphical Representation:

(i) Displacement – Time Graph:

Simple Harmonic Motion | Physics Class 11 - NEET

(ii) Velocity – Time Graph:

Simple Harmonic Motion | Physics Class 11 - NEET

(iii) Acceleration – Time Graph:

Simple Harmonic Motion | Physics Class 11 - NEET

Note: The acceleration is maximum at a place where the velocity is minimum and vice – versa.

For a particle executing SHM, the phase difference between

(i) Instantaneous displacement and instantaneous velocity

= (π / 2) rad

(ii) Instantaneous velocity and instantaneous acceleration

= (π / 2) rad

(iii) Instantaneous acceleration and instantaneous displacement

= π rad

The graph between velocity and displacement for a particle executing SHM is elliptical.

Force in SHM

We know that, the acceleration of body in SHM is α = -ω2 x

Applying the equation of motion F = ma,

We have, F = – mω2 x = -kx

Where, ω = √k / m and k = mω2 is a constant and sometimes it is called the elastic constant.

In SHM, the force is proportional and opposite to the displacement.

Question for Simple Harmonic Motion
Try yourself:What is the formula for displacement in Simple Harmonic Motion (SHM) at any instant?
View Solution

Energy in SHM

The kinetic energy of the particle is K = 1 / 2 mω2 (A2 – x2)

From this expression we can see that, the kinetic energy is maximum at the centre (x = 0) and zero at the extremes of oscillation (x ± A).

The potential energy of the particle is U = 1 / 2 mω2 x2

From this expression we can see that, the potential energy has a minimum value at the centre (x = 0) and increases as the particle approaches either extreme of the oscillation (x ± A).

Total energy can be obtained by adding potential and kinetic energies. Therefore,

E = K + U

= 1 / 2 mω2 (A2 – x2) + 1 / 2 mω2 x2

= 1 / 2 mω2 A2

where A = amplitude

m = mass of particle executing SHM.

ω = angular frequency and

v = frequency

Changes of Kinetic and Potential Energies during OscillationsChanges of Kinetic and Potential Energies during Oscillations

When a particle of mass m executes SHM with a constant angular frequency (I), then time period of oscillation

T = 2π √Inertia factor / Spring factor

In general, inertia factor = m, (mass of the particle)

Spring factor = k (force constant)

How the different physical quantities (e.g., displacement, velocity, acceleration, kinetic energy etc) vary with time or displacement are listed ahead in tabular form:

S.No.Name of the equationExpression of the equationRemarks

1.

Displacement-time

Simple Harmonic Motion | Physics Class 11 - NEET

x varies between and - A

2.

Velocity-time Simple Harmonic Motion | Physics Class 11 - NEET

Simple Harmonic Motion | Physics Class 11 - NEET

v varies between + Aω and - A ω

3.

Acceleration-time Simple Harmonic Motion | Physics Class 11 - NEET

Simple Harmonic Motion | Physics Class 11 - NEET

a varies between + Aω2 and - Aω2

4.

Kinetic energy-time Simple Harmonic Motion | Physics Class 11 - NEET

Simple Harmonic Motion | Physics Class 11 - NEET

K varies between and Simple Harmonic Motion | Physics Class 11 - NEET

5.

Potential energy-time
Simple Harmonic Motion | Physics Class 11 - NEET

Simple Harmonic Motion | Physics Class 11 - NEET

U varies between and

Simple Harmonic Motion | Physics Class 11 - NEET

6.

Total energy-time
(E = K + U)

Simple Harmonic Motion | Physics Class 11 - NEET

E is constant

7.

Velocity-displacement

Simple Harmonic Motion | Physics Class 11 - NEET

v = 0 at x = ±4 and at x = 0 , v = ± Aω

8.

Acceleration-displacement

a = - ω2X

Simple Harmonic Motion | Physics Class 11 - NEET

9.

Kinetic energy-displacement

Simple Harmonic Motion | Physics Class 11 - NEET

Simple Harmonic Motion | Physics Class 11 - NEET

10.

Potential energy-d displacement

Simple Harmonic Motion | Physics Class 11 - NEET

Simple Harmonic Motion | Physics Class 11 - NEET

11.

Total energy-d displacement

Simple Harmonic Motion | Physics Class 11 - NEET

x = ± A
 E is constant

 Oscillations of Liquid in a U – tube

If a liquid is filled up to height h in both limbs of a U-tube and now liquid is depressed upto a small distance y in one limb and then released, then liquid column in U-tube start executing SHM.

The time period of oscillation is given by

T = 2π √h / g

Oscillations of a floating cylinder in liquid is given by

T = 2π √l / g

where I = length of the cylinder submerged in liquid in equilibrium.

Oscillations of Liquid In U-TubeOscillations of Liquid In U-Tube

Vibrations of a Loaded Spring

When a spring is compressed or stretched through a small distance y from mean position, a restoring force acts on it.

Restoring force (F) = – ky

where k = force constant of spring.

If a mass m is suspended from a spring then in equilibrium,

mg = kl

This is also called Hooke’s law.

Time period of a loaded spring is given by:

T = 2π √m / k

Simple Harmonic Motion | Physics Class 11 - NEET

When two springs of force constants k1 and k2 are connected in parallel to mass m as shown in figure, then

(i) Effective force constant of the spring combination

k = k1 + k2

(ii) Time period T = 2π √m / (k1 + k2)

When two springs of force constant k1 and k2 are connected in series to mass m as shown in figure, then

(i) Effective force constant of the spring combination

1 / k = 1 / k1 + 1 / k2

Simple Harmonic Motion | Physics Class 11 - NEET

(ii) Time period T = 2π √m(k1 + k2) / k1k2

The document Simple Harmonic Motion | Physics Class 11 - NEET is a part of the NEET Course Physics Class 11.
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FAQs on Simple Harmonic Motion - Physics Class 11 - NEET

1. What is Simple Harmonic Motion (SHM)?
Ans. Simple Harmonic Motion is a type of periodic motion where an object oscillates back and forth around an equilibrium position, with the acceleration of the object proportional and opposite to its displacement from the equilibrium position.
2. What are some terms related to SHM?
Ans. Some terms related to Simple Harmonic Motion include amplitude, frequency, period, equilibrium position, restoring force, and displacement.
3. What are some important formulae of SHM?
Ans. Some important formulae of Simple Harmonic Motion include the formula for the period of oscillation T = 2π√(m/k), the formula for the angular frequency ω = √(k/m), and the formula for the velocity of the object v = Aωcos(ωt).
4. How is force related to SHM?
Ans. In Simple Harmonic Motion, the force acting on the object is directly proportional and opposite to the displacement of the object from the equilibrium position, as described by Hooke's Law (F = -kx), where k is the spring constant and x is the displacement.
5. What is the role of energy in SHM?
Ans. In Simple Harmonic Motion, the total mechanical energy (kinetic energy + potential energy) of the object remains constant throughout the motion, with the kinetic energy being maximum at the equilibrium position and the potential energy being maximum at the extreme positions.
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