Angular Frequency and Period

Understanding Oscillation Timing

When an object undergoes simple harmonic motion (SHM), it repeats its motion in a regular cycle. We quantify this timing using three related values:

Angular Frequency \( \omega \): How fast the object oscillates in radians per second.
Frequency \( f \): How many full cycles it completes per second (measured in hertz, Hz).
Period \( T \): How long it takes to complete one full cycle (measured in seconds).

\[ \omega = 2\pi f \quad \text{and} \quad T = \frac{1}{f} = \frac{2\pi}{\omega} \]

Mass-Spring Systems

For both horizontal and vertical spring systems (assuming no damping and friction), the motion follows Hooke’s Law: \( F = -kx \). This produces SHM when the mass is displaced from equilibrium.

Formulas for a Mass-Spring System

\[ \omega = \sqrt{\frac{k}{m}}, \quad f = \frac{1}{2\pi} \sqrt{\frac{k}{m}}, \quad T = 2\pi \sqrt{\frac{m}{k}} \]

- \( k \): Spring constant (N/m) - \( m \): Mass (kg) - These equations apply to **horizontal** spring systems directly. - For **vertical springs**, the equilibrium position shifts due to gravity, but the formulas remain the same for small oscillations.

Simple Pendulum

A simple pendulum consists of a mass suspended from a light, inextensible string. When displaced slightly from equilibrium, it undergoes SHM due to the restoring force of gravity.

Formulas for a Simple Pendulum

\[ \omega = \sqrt{\frac{g}{L}}, \quad f = \frac{1}{2\pi} \sqrt{\frac{g}{L}}, \quad T = 2\pi \sqrt{\frac{L}{g}} \]

- \( g \): Acceleration due to gravity (≈ 9.8 m/s²) - \( L \): Length of the pendulum (from pivot to center of mass) - These equations are valid only for **small angles** (typically \( \theta < 15^\circ \)) where the restoring force is proportional to displacement.

Comparing oscillatory systems: Mass-Spring vs. Pendulum

Which Variables Affect the Period?

Increasing mass in a spring system slows down the oscillation (larger \( T \)).
Increasing spring stiffness (larger \( k \)) makes the motion faster (smaller \( T \)).
For pendulums, longer strings increase the period (slower swing).
The mass of the pendulum bob does not affect the period at all!

Key Takeaway

All of these systems obey the same basic differential equation:

\[ \frac{d^2x}{dt^2} = -\omega^2 x \]

This equation guarantees that the system’s motion is sinusoidal, with angular frequency \( \omega \) determined by system parameters like mass, spring constant, or pendulum length.