Simple Harmonic Motion (SHM)

• Simple Harmonic Motion (SHM) is characterized by a restoring force on a moving object.
• The force is directly proportional to the object's displacement magnitude.
• The force acts towards the equilibrium position.

Key Concepts of SHM

• Periodic motion repeats itself after a fixed time interval.
• Oscillation involves repetitive variation around a central value or between states.
• The equilibrium position is where the net force on the object equals zero.
• Restoring force returns an object to its equilibrium position.
• Displacement refers to the object's distance from its equilibrium position.
• Amplitude (A) is the maximum displacement from equilibrium.
• Period (T) is the time for one complete oscillation.
• Frequency (f) is the number of oscillations per unit time, measured in Hertz (Hz).
• 1 Hz equals 1 oscillation per second.
• Frequency is the reciprocal of the period (f = 1/T).
• Angular frequency (ω) measures the oscillation rate in radians per second (rad/s).
• Angular frequency relates to frequency and period as ω = 2πf = 2π/T.

Mathematical Description of SHM

• Displacement (x) in SHM is described by sinusoidal functions of time (t).
• Functions include: x(t) = A cos(ωt + φ) or x(t) = A sin(ωt + φ).
• A represents amplitude.
• ω represents angular frequency.
• φ represents the phase constant, determining the initial position at t=0.

Velocity in SHM

• Velocity (v) is the time derivative of displacement: v(t) = dx/dt.
• Differentiating x(t) = A cos(ωt + φ) gives v(t) = -Aω sin(ωt + φ).
• Differentiating x(t) = A sin(ωt + φ) gives v(t) = Aω cos(ωt + φ).
• Maximum speed (v_max) occurs at the equilibrium position.
• v_max equals Aω.

Acceleration in SHM

• Acceleration (a) is the time derivative of velocity: a(t) = dv/dt.
• Differentiating v(t) = -Aω sin(ωt + φ) yields a(t) = -Aω^2 cos(ωt + φ).
• Acceleration can be written as a(t) = -ω^2 x(t) since x(t) = A cos(ωt + φ).
• The negative sign indicates acceleration is opposite to displacement, directed towards equilibrium.
• Maximum acceleration (a_max) occurs at maximum displacement (x = ±A).
• a_max equals Aω^2.

Force in SHM

• Force is defined as F = ma, or F = m(-ω^2 x) = -mω^2 x in SHM, according to Newton's second law.
• It acts as a restoring force, directed towards the equilibrium position.
• The effective spring constant, k = mω^2, relates to system stiffness.
• Therefore, F = -kx

Energy in SHM

• Total mechanical energy (E) is constant, comprising kinetic (KE) and potential energy (PE).
• Kinetic energy is KE = (1/2)mv^2 = (1/2)mA^2ω^2 sin^2(ωt + φ).
• Potential energy is PE = (1/2)kx^2 = (1/2)mω^2A^2 cos^2(ωt + φ).
• Total mechanical energy E = KE + PE = (1/2)mω^2A^2, remains constant.

Examples of SHM

• Simple Pendulum: A mass (m) suspended by a string of length (L) oscillates with SHM at small angles (θ).
• Angular frequency is ω = √(g/L), with g as gravitational acceleration.
• Period is T = 2π√(L/g).
• Mass-Spring System: A mass (m) on a spring (k) oscillates in SHM when displaced.
• Angular frequency is ω = √(k/m).
• Period is T = 2π√(m/k).

Damped Oscillations

• Dissipative forces, such as friction or air resistance, cause damped oscillations.
• Oscillation amplitude decreases over time, eventually stopping.
• Underdamped systems oscillate with decreasing amplitude.
• Critically damped systems return to equilibrium fastest without oscillation.
• Overdamped systems return to equilibrium slowly without oscillating.

Forced Oscillations and Resonance

• Forced oscillations arise from an external periodic force on an oscillating system.
• The system oscillates at the driving frequency of the external force.
• Resonance occurs when the driving frequency nears the system’s natural frequency.
• Resonance leads to a large oscillation amplitude.
• Energy transfers efficiently from the driving force to the system at resonance.

Understand simple harmonic motion, a periodic motion where the restoring force is proportional to displacement. Key concepts: periodic motion, oscillation, equilibrium, restoring force, displacement, amplitude and period.