One of the most significant and basic forms of periodic motion that is studied in physics is simple harmonic motion (SHM). It serves as the foundation for understanding a variety of wave and oscillatory phenomena in both classical and contemporary physics. When an item oscillates back and forth around a fixed position while being influenced by a restoring force proportional to its displacement, this motion is called as SHM.
Definition:
A form of periodic motion known as simple harmonic motion occurs when the restoring force acting on a body is always oriented toward the mean position and is directly proportional to the displacement of the body from its mean position. In terms of mathematics, it is stated as:
F = −kx
where:
F is the restoring force,
x is the displacement from the equilibrium (mean) position,
k is the force constant (or spring constant),
The negative sign indicates that the force is directed opposite to the displacement.
A simple harmonic motion is the simplest form of oscillation. A particular type of periodic motion in which a particle moves to and fro repeatedly about a mean position under the influence of a restoring force is termed as simple harmonic motion (SHM).
A body is undergoing simple harmonic motion if it has an acceleration which is
Directed towards a fixed point, and
Proportional to the displacement of the body from that point.
Acceleration a α – x – a = – kx or d2x / dt2 = – kx, where x = displacement at any instant t.
The time after which motion repeats itself is T, it will remain fixed, no matter what location we choose as the initial (t = 0) location. The speed is maximum for zero displacement (at x = 0) and zero at the extremes of motion.
Angular frequency of SHM ω = 2π / T
Characteristics of SHM:
1. Oscillatory and Periodic: The motion repeats itself at regular intervals of time.
2. Restoring Force: Proportional to displacement and directed toward the equilibrium position.
3. Mean Position: The central point where the net force on the body is zero.
4.Amplitude (A): Maximum displacement from the mean position.
5. Time Period (T): Time taken for one complete oscillation.
6. Frequency (f): Number of oscillations per second;
f = 1 / T.
7. Phase: Describes the state of motion (position and direction) at a given time.
Equation of SHM:
The general equation for SHM in terms of displacement as a function of time is:
x(t) = A sin(ωt + ϕ)
or
x(t) = A cos(ωt + ϕ)
where:
x(t) is the displacement at time t,
A is the amplitude,
ω is the angular frequency (ω = 2πf),
ϕ is the phase constant, which depends on initial conditions.
Velocity and Acceleration in SHM:
Velocity:
v(t) = dx / dt = Aω cos(ωt + ϕ)
Acceleration:
a(t) = d2x / dt2 = −Aω2 sin(ωt + ϕ) = −ω2x
Acceleration in SHM is also directly proportional to displacement but opposite in direction.
Energy in SHM:
There are two main types of energy involved in SHM:
1.Kinetic Energy (K.E.): Energy due to motion.
K.E.= 1 / 2mv2 =1 / 2mω2 (A2 − x2)
2. Potential Energy (P.E.): Stored energy due to displacement.
P.E.= 1 / 2kx2 = 1 / 2mω2x2
Total Energy (E) remains constant:
E = K.E.+ P.E.= 1 / 2mω2A2
This conservation of energy is a hallmark of SHM in ideal (non-damped) systems.
Examples of SHM:
Oscillations of a mass-spring system.
Motion of a simple pendulum (for small angles).
Vibrating tuning forks.
Oscillations of atoms in a crystal network.
Motion of a swing.
Applications of SHM:
SHM plays a vital role in various fields such as:
Designing clocks and watches (pendulum clocks, quartz watches),
Understanding molecular vibrations in chemistry,
Designing suspension systems in vehicles,
Engineering structures to withstand vibrational stress.
