Module 7.1: Mechanisms of Oscillatory Motion

Simple Harmonic Motion (SHM)

A specific type of periodic motion defined by a restoring force proportional to displacement.

Restoring Force

A net force that acts to bring an object back to its equilibrium position, opposite to displacement.

Hooke's Law

The relationship defined by F_{restoring} = -kx, where F is the restoring force, k is the stiffness constant, and x is the displacement.

Amplitude (A)

The maximum displacement from the equilibrium position, representing magnitude and always positive.

Period (T)

The time it takes to complete one full cycle of motion.

Frequency (f)

The number of cycles completed per unit of time, measured in Hertz (Hz).

Relationship between Period and Frequency

T = 1/f; they are inversely related.

Period of a Spring-Mass System

T_s = 2π√(m/k), where m is the mass and k is the spring stiffness.

Effect of mass on Spring-Mass Period

More mass leads to more inertia, resulting in a larger period (T).

Effect of spring constant on Spring-Mass Period

A stiffer spring (higher k) results in a smaller period (T).

Period of a Simple Pendulum

T_p = 2π√(ℓ/g), where ℓ is the length of the string and g is the gravitational field strength.

Effect of length on Pendulum Period

Longer string results in a larger period (T).

Effect of gravity on Pendulum Period

Stronger gravity leads to a smaller period (T).

Mass Independence for Pendulum

The mass of the bob does not affect the period of a simple pendulum.

Equilibrium Condition in SHM

At equilibrium position (x=0), velocity is maximum, and acceleration and net force are zero.

Turning Points in SHM

At maximum amplitude (±A), velocity is zero, and acceleration and net force are at maximum magnitude.

Friction and Damping in SHM

In ideal SHM, without friction or damping, amplitude remains constant over time.

Units of Period

The SI unit of Period (T) is seconds (s).

Units of Frequency

The SI unit of Frequency (f) is Hertz (Hz), where 1 Hz = 1 cycle/s.

Calculating Spring Constant (k)

k can be derived using k = (4π²m)/(T²).

Mistake: Acceleration vs Velocity

Acceleration is maximum when velocity is zero at the turning points, not the other way around.

Common Mistake: Mass on a Pendulum

Adding mass to a pendulum does not affect its period.

Amplitude and Period Relation

In ideal SHM, amplitude does not affect the period; period is independent of amplitude.

Total Energy in SHM

The total mechanical energy in simple harmonic motion remains constant if no non-conservative forces act on the system.

Kinematics of SHM

SHM involves a constant exchange of potential energy and kinetic energy during motion.

Phase of SHM

The phase describes the position and direction of motion at any point in time.

Energy Conservation in SHM

At maximum displacement, energy is all potential; at equilibrium, energy is all kinetic.