What is Simple Harmonic Motion?
Simple Harmonic Motion (SHM) is a type of periodic oscillatory motion about a fixed point(equilibrium position) in which the restoring force acting on the body is directly proportional to its displacement from the mean position and acts always towards mean position or equilibrium position.
Mathematically,
Restoring Force F
F = –
F = Restoring force (N)
k = Spring constant (N/m)
x = Displacement from mean position (m)
The negative sign indicates that the force always acts towards the equilibrium position.
Examples of Simple Harmonic Motion
- Small oscillations of a simple pendulum.
- Vibrations of tuning forks.
- Oscillation of a floating body
- Oscillation of a spring-mass system.
- vibration of a loaded cantilever beam.
Characteristics of SHM
- The motion is periodic and oscillatory.
- The restoring force is directly proportional to the displacement from the equilibrium position and is always directed towards the equilibrium.
- The acceleration is directly proportional to the displacement from the equilibrium position and is always directed towards the equilibrium.
- The frequency or time period is independent of amplitude.
- velocity of the particle is maximum at mean position and zero at extreme position (where x= A)
Important Terms in SHM
Mean position or equilibrium position is the point about which the body or particle oscillates periodically.
Amplitude: It is the maximum displacement of the particle from mean position.
Time period: It is defined as the time taken by the particle to complete one full oscillation.
It is denoted by T.
Unit: Second.
Frequency: It is the number of complete oscillation made by the particle per second.
Unit: Hz (hertz)
Differential Equations of SHM:
For a particle moving in SHM,
The restoring force is directly proportional to the displacement from the equilibrium position and is always directed towards the equilibrium.
F ∝ -x
F =-kx (k= forcr constant , x = displacement )
From newton’s second law we can write force,
F = ma (m= mass, a= acceleration )
Therefore, restoring force
$$ \frac{d^2x}{dt^2}+\omega^2x=0 $$
This is the differential equation of SHM
General equation of the above equation,
x= Asin (ωt+φ)
= Amplitude
= Time (s)
= Angular frequency (rad/s)
This is the displacement equation of SHM
Velocity in SHM
The displacement equation of a particle performing SHM is given by
x= Asin (ωt+φ)
therefore velocity(v) = $$ v= \frac{dx}{dt} = A ω cos (ωt+φ) $$
$$ or, v= A ω \sqrt{1- sin^2 (ωt+φ)} $$
$$ or, v= A ω \sqrt{1-\left(\frac{x}{A}\right)^2} $$
$$ or, v= A ω \sqrt{\frac{A^2-x^2}{A^2}} $$
therefore, $$ v= ± ω \sqrt{(A)^2-(x)^2} $$
When x=0, at mean position :
$$ v= ω \sqrt{(A)^2-(0)^2} $$
$$ v_{max}= A ω $$
When x=A, at extreme position :
$$ v= ω \sqrt{(A)^2-(A)^2} $$
$$ v_{min}= 0 $$
Acceleration in SHM
The displacement equation of a particle performing SHM is given by
x= Asin (ωt+φ)
therefore acceleration(a) = $$ a= \frac{dv}{dt} = -A ω^2 sin (ωt+φ) $$
$$ or, a= – A ω^2 \frac{x}{A} $$
$$ or, a= – ω^2 x $$
When x=0, at mean position :
$$ or, a= -ω^2 x $$
$$ a_{min}= 0 $$
When x=A, at extreme position :
$$ or, a=- ω^2 A $$
$$ a_{max}=- ω^2 A $$
Kinetic and potential energy in SHM
Potential energy:
Potential energy in SHM $$ P.E = \frac{1}{2} k x^2 $$
$$ P.E = \frac{1}{2} m ω^2 x^2 $$
Kinetic energy:
We know that velocity in SHM $$ v= ± ω \sqrt{(A)^2-(x)^2} $$
$$ v^2= ω^2 (A^2-x^2) $$
Therefore Kinetic energy $$ K.E= \frac{1}{2} m v^2 $$
$$ K.E= \frac{1}{2} m ω^2 (A^2-x^2) $$
Therefore total energy(T.E) = P.E + K.E
$$ T.E= \frac{1}{2} m ω^2 x^2 + \frac{1}{2} m ω^2 (A^2-x^2) $$
$$ or, T.E= \frac{1}{2} m ω^2 x^2 + \frac{1}{2} m ω^2 A^2 -\frac{1}{2} m ω^2 x^2) $$
$$ therefore T.E= \frac{1}{2} m ω^2 A^2 $$
What is Simple Harmonic Motion (SHM)?
Simple Harmonic Motion (SHM) is a type of periodic oscillatory motion about a fixed point(equilibrium position) in which the restoring force acting on the body is directly proportional to its displacement from the mean position and acts always towards mean position or equilibrium position.