Oscillations of a Spring (Spring-Mass System)

Oscillation

A repetitive variation, typically in time, of some measure about a central value (often a state of equilibrium).

Simple Harmonic Motion (SHM)

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

Spring Constant (k)

A measure of the stiffness of a spring; it is the ratio of the force applied to the spring to the displacement it produces.

Time Period (T)

The time taken for one complete cycle of oscillation.

Hooke's Law

F is the restoring force, k is the spring constant, x is the displacement from equilibrium.

F = -kx

Angular Frequency (ω)

k is the spring constant, m is the mass.

ω = \sqrt{\frac{k}{m}}

Time Period (T)

Time for one complete oscillation.

T = \frac{2\π}{\ω} = 2\π \sqrt{\frac{m}{k}}

Frequency (f)

Number of oscillations per unit time.

f = \frac{1}{T} = \frac{\ω}{2\π}

Effective Spring Constant (Series)

For springs connected end-to-end.

\frac{1}{k_s} = \frac{1}{k₁} + \frac{1}{k₂} + ...

Effective Spring Constant (Parallel)

For springs connected side-by-side.

k_p = k₁ + k₂ + ...