Definition: Simple Harmonic Motion (SHM) is a specific type of periodic motion where a particle moves to and fro about a fixed equilibrium position. In this motion, the restoring force acting on the particle is always proportional to its displacement from the equilibrium position and is directed towards it.
Kinematics of Simple Harmonic Motion
To understand SHM, we must first look at the displacement function. Any particle executing SHM follows a sinusoidal path, represented by the equation x(t) = A sin(ωt + φ), where A is the amplitude, ω is the angular frequency, and φ is the initial phase constant. The motion is defined by its periodicity, meaning the particle repeats its path after a fixed interval known as the Time Period (T), where T = 2π/ω.
Velocity and acceleration are derived from the displacement using calculus. By differentiating the displacement with respect to time, we find the velocity v(t) = Aω cos(ωt + φ). Differentiating once more yields the acceleration a(t) = -Aω² sin(ωt + φ). A critical observation here is that a = -ω²x. This relationship is the hallmark of SHM; the acceleration is directly proportional to the negative of the displacement, confirming that the force is always pulling the particle back toward the center.
Dynamics and the Restoring Force
From Newton’s second law, F = ma, we can substitute the SHM acceleration to find the restoring force: F = -mω²x. If we define a constant k = mω², we obtain the familiar Hooke’s Law form, F = -kx. This confirms that any system obeying a linear restoring force will naturally oscillate in SHM.
The energy in an SHM system is conserved and oscillates between potential and kinetic forms. The potential energy at any displacement x is U = ½kx², while the kinetic energy is K = ½mv². The total mechanical energy E = U + K = ½kA² remains constant throughout the motion. At the equilibrium position (x=0), the energy is entirely kinetic, whereas at the extreme positions (x=±A), the energy is entirely potential.
SHM and Circular Motion
There is a profound geometric relationship between SHM and Uniform Circular Motion (UCM). If a particle moves in a circle of radius A with a constant angular velocity ω, its projection on any diameter of the circle executes SHM. This projection method is a powerful tool for solving complex phase problems in competitive exams.
The projection of a uniform circular motion onto a diameter is mathematically identical to the displacement, velocity, and acceleration vectors of a simple harmonic oscillator.
When analyzing these systems, we often use the Phasor Diagram. A phasor is a rotating vector that represents the state of the oscillator. The length of the phasor corresponds to the amplitude, and the angle it makes with the reference axis represents the phase (ωt + φ). This visual representation simplifies the addition of multiple SHM waves.
Important Facts and Formulas
| Parameter | Formula |
|---|---|
| Restoring Force | F = -kx |
| Angular Frequency | ω = √(k/m) |
| Time Period | T = 2π√(m/k) |
| Total Energy | E = ½kA² |
| Velocity at displacement x | v = ω√(A² – x²) |
Key Points to Remember
- Equilibrium Position: The point where the net force is zero and potential energy is minimum.
- Phase Angle: Determines the state of oscillation at t=0; it shifts the sine wave along the time axis.
- Frequency (f): The number of oscillations per unit time, given by f = 1/T = ω/2π.
- Extreme Points: Velocity is zero, acceleration is maximum (a_max = ω²A), and kinetic energy is zero.
- Mean Position: Velocity is maximum (v_max = ωA), acceleration is zero, and potential energy is zero.
- Restoring Force: Always acts towards the mean position, regardless of which side of the equilibrium the particle is on.
Previous Year Question Hints
- Energy Distribution: Expect questions asking for the displacement x where kinetic energy equals potential energy. (Hint: Set ½kx² = ½m(ω²(A²-x²)), solve for x).
- Superposition: Be prepared to find the resultant amplitude when two SHM equations of the same frequency but different phases are added. Use the phasor addition method.
Quick Revision Summary
- SHM is defined by the differential equation d²x/dt² + ω²x = 0.
- The restoring force is linear: F = -kx.
- Total mechanical energy is proportional to the square of the amplitude (E ∝ A²).
- The time period is independent of the amplitude for a simple harmonic oscillator.
- Velocity leads displacement by a phase of π/2 radians.
- Acceleration leads velocity by a phase of π/2 radians.
- The projection of uniform circular motion is a perfect model for SHM.
- Always check units when calculating k or ω to avoid dimensional errors in exam problems.
