Simple Harmonic Motion (SHM) |Study Notes|

Study Notes of Physics

class - 11ᵗʰ CBSE

Topic - Simple harmonic motion (SHM)
|Part - 1|

PERIODIC MOTION :

• When a body or a moving particle repeats its motion along a definite path after regular intervals of time its motion is said to be Periodic Motion and interval of time is called time period (T). 
• The path of periodic motion may be linear, circular, elliptical or any other curve.
• For example rotation of earth around the sun. 

OSCILLATORY MOTION :

• To and fro type of motion is called Oscillatory Motion. 
• A particle has oscillatory motion when it moves about stable equilibrium position. It need not be periodic and need not have fixed extreme positions. 
• The oscillatory motions in which energy is conserved are also periodic.
• For example motion of pendulum of a wall clock. The force / torque (directed towards equillibrium point) acting in oscillatory motion is called restoring force/torque Damped Oscillations are those in which energy consumed due to some resistive forces and hence total mechanical energy decreases and after some time oscillation will stop. 

Oscillatory Equation : Consider a particle free to move on x-axis is being acted upon by a force given by

F = -kxⁿ

Above equation is called oscillatory equation.
Here

• k is a positive constant
• x is the displacement from mean position

Now following cases are possible depending on the value of n.
Case – I if n is an even integer ( n= 0,2,4........) 

• force is always along negative x-axis whether x is positive or negative 
• Hence, the motion of the particle is not oscillatory. 
• If the particle is released from any position on the x-axis (except x = 0) a force in negative direction of x-axis acts on it and it moves rectilinearly along negative x axis. 

Case -II  If n is an odd integer (1, 3, 5 ........... etc),

• force is along negative  x-axis for x > 0 and along positive  x-axis for x < 0 and zero for x = 0.
• Thus the particle will oscillate about stable equillibrium position x = 0.
• The force in this case is called the restoring force. 

If n = 1 i.e., F = -kx

the motion is said to be SHM (Simple Harmonic Motion)

• If the restoring force / torque acting on the body in oscillatory motion is directly proportional to the displacement of body / particle w.r.t. mean position and is always directed towards equillibrium position then the motion is called Simple Harmonic motion.

TYPES OF SHM :

1. Linear SHM :When a particle moves to and fro about an equilibrium point, along a straight line here A and B are extreme positions and M is mean position so AM = MB = Amplitude. M A B
   
2. Angular SHM :When body/particle is free to rotate about a given axis and executing angular oscillations.
   

ANALYSIS OF MOTION IN LINEAR SHM :

• When the particle is moved away from the mean position or equillibrium position and released, a force (-kx) comes into play to pull it back towards mean position. 
• By the time it gets at mean position it has picked up some kinetic energy and so it overshoots, stopping some where on the other side and it is again pulled back towards the mean position. 
• It is necessary to study the change in speed and acceleration of particle during SHM. Let us
  

NOTE :In the figure shown, path of the particle is a straight line.

(1) Motion of a particle from A to B :

• Initially the particle is at A (mean position) and is moving towards +ve x direction with speed v0 . As the particle is moving towards B, force acting on it towards A is increasing. Consequently its acceleration towards A is increasing in magnitude while its speed decreases and finally it comes to rest momentarily at B.

(2) Motion of a particle from B to A :

• Now the particle starts moving towards A with initial speed v = 0. As the particle is moving towards A, force is acting on it towards A and decreasing as it approaches A. Consequently its acceleration towards A is decreasing in magnitude while its speed increases and finally it comes to A with same speed v = v0

(3) Motion of a particle from A to C :

• The motion of a particle from A to C is qualitatively same as motion of a particle from A to B.

(4) Motion of a particle from C to A :

• It is qualitatively same as motion of a particle from B to A.

CHARACTERISTICS OF SHM :
(1) Mean Position : It is the position where net force on the particle is zero.
(2) Extreme Point : Point where speed of the particle is zero.
(3) Displacement : It is defined as the distance of the particle from the mean position at that instant.
(4) Amplitude : It is the maximum value of displacement of the particle from its mean position.

EP - MP = Amplitude.

depends upon the energy of the system.
(5) Frequency : The frequency of SHM is equal to the number of complete oscillations per unitT ime.

f=1/T   Hz.

(6) Time Period : Smallest time interval after which the oscillatory motion gets repeated is called time period.

T=2π/ω  sec⁻¹

EQUATION OF SIMPLE HARMONIC MOTION :

The necessary and sufficient condition for SHM is

F = -kx

we can write above equation in the following way:

ma = -kx

a = - (k/m)x

d²x/dt² = - (k/m)x

d²x/dt² = - ω²x

d²x/dt² + ω²x = 0 .......(1)

Equation (1) is Double Differential Equation of SHM. Now

d²x/dt² + ω²x = 0

It's solution is x = A sin(ωt+φ)

where
ω= angular frequency = √(k/m)

x = displacement from mean position

k = SHM constant. 
The equality (ωt+φ) is called the phase angle or simply the phase of the SHM 
φ is the initial phase i.e., the phase at t = 0 and depends on initial position and direction of velocity at t = 0. 
To understand the role of φ in SHM, we take two particles performing SHM in the following condition

• As shown in figure above two different waves are shown in figure. One is of red in colour and the other is blue in colour. 
• Red wave is a sin wave that is starts from 0. But the blue wave is starts from 90°.  
• In circle we can see that red and blue dot never meets with each other. This is due to initial phase difference that is 90° in the above case.
•  This initial phase difference  may be 30°, 45°, 60° etc.
• The equation of this waves can be written as- 
• For Red particle :-Y = A sin(ωt)

• For blue particle :- Y = A sin(ωt+90°) or Y = A cos(ωt)

VELOCITY:

• It is the rate of change of particle displacement with respect to time at that instant. Let the displacement from mean position is given by 

x = A sin(ωt+φ) ......(1)

We know that      v = dx/dt 

v = Aω cos(ωt+φ) ......(2)

or

v = Aω sin(ωt+φ + 90°) 

• At mean position (x = 0),  velocity is maximum.       Vmax = Aω

• At extreme position (x = A), velocity is minimum.       Vmin = zero.

• According to equation -1  and equation-2 we can say that x and v has initial phase difference of 90°. 
• That means if  X is min than V should be max. And vice - versa.
• In SHM X and V both are positive that shows that both are in same direction. 
•  Another formula for velocity at any position X. :- 

V = ω√(A² - X²)

Relation between displacement and velocity of SHM

as we know that 

v= ω√(A² - X²)

on squaring both sides, we get 

v² = ω²(A² - X²)

v² = ω²A² - ω²X²

v² + ω²X² = ω²A² 

(v²/ω²A²) + (X²/A²) = 1 ......(3)

It is the equation of parabola. the graph between velocity and displacement is shown in figure below.

ACCELERATION :It is the rate of change of particle's velocity w.r.t. time at that instant.

Acceleration(a) = dv/dt

a = d(Aω cos(ωt+φ))/dt

a = -Aω²sin(ωt+φ)

Note :Negative sign shows that acceleration is always directed towards the mean position. 

At meanposition (x=0), acceleration is minimum.

amin = zero

At extreme position (x = A), acceleration is maximum.

|amax | = Aω²

Graph of Acceleration (A) v/s Displacement (x):

Graph of acceleration, velocity and displacement