Simple Harmonic Motion: A Comprehensive Guide
1) Definition of Linear Simple Harmonic Motion
Linear Simple Harmonic Motion (SHM) is a periodic motion where the restoring force (or acceleration) is always directed towards the mean position and its magnitude is directly proportional to the displacement from the mean position.
2) Differential Equation of Linear SHM
The differential equation of SHM is:
(a) Expression for Acceleration in SHM
But a =d2x is the acceleration of the particle performing SHM.
dt2
a= -ω2.x
This is the expression for acceleration.
(b) Expression for Velocity in SHM
Integrating both sides, we get
Where C is the constant of integration.
Let A be the maximum displacement (amplitude) of the particle in SHM.
Thus, atx A, v = 0,
Substituting v = 0 and x = ±A in equation (II), we get
Using the value of C in equation (2), we get
v2 = ω2(A2 -x2)
v = + ω√A2-x2)
This is the expression for velocity
3) Free Oscillation
When the oscillator is allowed to oscillate by displacing its position from the equilibrium position, it oscillates with a frequency which is equal to the natural frequency of the oscillator. Such an oscillation or vibration is known as free oscillation or free vibration.
4) Period of a Simple Pendulum Performing SHM
Let m is Mass of the bob.
Lis the Length of mass-less string A free-body diagram to the forces acting on the bob, θ-angle made by the string with the vertical, T is tension along the string g is the acceleration due to gravity Radial acceleration = ω2L Net radial force-T-mg cosθ Tangential acceleration is provided by mg sinθ Torques, τ=L (mg sinθ)
According to Newton’s law of rotational motion, τ=la
Where, I is the moment of inertia
a- Angular acceleration
la= -mg sinθ L
If θ is very small, then sinθ≈θ so, la= -mgeθL= -(mgL)θ
We know, it will follow simple harmonic motion when a= -ω2θ
So, I (-ω2θ)= – (mgL)θ Moment of inertia, I-mL2
So, mL2w² = mgl
ω√g/L
T=2π =2π√L/g
ω
Here it is clear that the time period of a simple pendulum is directly proportional to the square root of the length of the pendulum and inversely proportional to the square root of acceleration due to gravity
5) Laws of Simple Pendulum
At a given place, the period of a simple pendulum is
where,
L-length of the simple pendulum,
gis the acceleration due to gravity at that place.
From the above expression, the laws of a simple pendulum are as follows:
Law of length: The period of a simple pendulum at a given place. (g constant) is directly proportional to the square root of its length.
• Law of acceleration due to gravity: The period of a simple pendulum of a given length (L. constant) is inversely proportional to the square root of the acceleration due to gravity.
• Law of mass: The period of a simple pendulum does not depend on the mass.
Law of isochronism: The period of a simple pendulum does not depend on its amplitude (for small amplitude).
6) Forced Oscillations
The body executing vibration initially vibrates with its natural frequency and due to the presence of external periodic force, the body later vibrates with the frequency of the applied periodic force. Such vibrations are known as forced vibrations.
Example: Sound boards of stringed instruments
7) Second’s Pendulum
A pendulum with the time period of oscillation equal to two seconds is known as a seconds pendulum
8) Projection of UCM along Diameter
Linear SHM is defined as the linear periodic motion of a body, in which the restoring force (or acceleration) is always directed towards the mean position and its magnitude is directly proportional to the displacement from the mean position
There is basic relation between SH.M. and U.C.M. that is very useful in understanding. S.H.M. For an object performing UCM the projection of its motion along any diameter of
its path executes S.H.M Consider particle ‘P’ is moving along the circumference of circle of radius ‘a’ with constant angular speed of win anticlockwise direction as shown in figure.
Particle Palong circumference of circle has its projection particle on diameter AB at point. M. Particle P is called reference particle and the circle on which it moves, its projection moves back and forth along the horizontal diameter, AB
10)Derive an expression for Differential Equation of S.H.M
i) In linear S.H.M the force is directed towards the mean position and its magnitude is directly proportional to the body from the mean position
f f= -kx ,….(1)
where k is force constant and x is the displacement from the mean position
ii)According to newton’s second law of motion,
f=ma….(2)
from equations (1)and (2)
ma= – kx….(3)
iii)The velocity of the particle is given by,v=dx
Acceleration,a=dv = d2x (4) dt
dt dt2
Substituting equation (4) in equation (3),
md2x= -kx iv)Substituting k =ω2, where ω is the angular frequency,
dt2 m
d2x = kx=0 d2x +ω2x=0 This is the differential equation
dt2 m dt2 of linear S.H.M