Unit 6: Oscillations and Dynamics of Periodic Motion

Simple Harmonic Motion (SHM)

A type of periodic motion where the restoring force is proportional to the displacement and acts in the opposite direction.

Equilibrium Position

The position where the net force on a system is zero.

Restoring Force

A force that acts to return a displaced object to its equilibrium position.

Hooke's Law

States that the restoring force is proportional to the displacement: F = -kx.

Differential Equation of SHM

The equation describing the motion of SHM: d²x/dt² + (k/m)x = 0.

Angular Frequency (ω)

A measure of how quickly an object is oscillating, defined as ω = √(k/m) for a mass-spring system.

Period (T)

The time taken to complete one full cycle of motion.

Frequency (f)

The number of cycles per second, measured in Hertz (Hz).

Amplitude (A)

The maximum displacement from the equilibrium position.

Phase Constant (φ)

A constant that determines the position of the waveform at t = 0 in the SHM equation.

Potential Energy (U)

The energy stored due to the position of a mass in a force field, in SHM: U = (1/2)kx².

Kinetic Energy (K)

The energy of an object due to its motion, in SHM: K = (1/2)mv².

Total Mechanical Energy (E)

The sum of potential and kinetic energy in an ideal SHM system, E = K + U.

Max Potential Energy

Occurs at maximum displacement, U_max = (1/2)kA².

Max Kinetic Energy

Occurs at the equilibrium position, Kmax = (1/2)m(vmax)².

Mass-Spring System Period (T_s)

The period of a mass-spring system given by T_s = 2π√(m/k).

Physical Pendulum

A rigid body swinging about an axis outside its center of mass.

Restoring Torque (τ)

The torque that acts to return the system to equilibrium, proportional to angular displacement.

Simple Pendulum

A weight suspended from a pivot that swings in a circular arc.

Small Angle Approximation

Assumes sin(θ) is approximately equal to θ for small values of θ, simplifying calculations in pendulum motion.

Period of a Simple Pendulum (T_p)

The period given by T_p = 2π√(L/g), independent of mass.

Velocity Function (v(t))

The derivative of position in SHM, v(t) = -Aωsin(ωt + φ).

Acceleration Function (a(t))

The derivative of velocity in SHM, a(t) = -Aω²cos(ωt + φ).

Mistake: Calculator Mode

Always use RADIANS mode for SHM calculations involving trigonometric functions.

Mistake: Confusing Frequency and Angular Frequency

Frequency (f) is in Hz (cycles/s), while angular frequency (ω) is in rad/s.

Mistake: Pendulum Mass in Period Formula

The period of a simple pendulum does not depend on mass; T = 2π√(L/g).