Determine the correct algebraic manipulation to convert between period T and angular frequency ω for a simple harmonic oscillator | Step-by-Step Solution

Mastering Angular Frequency 🌊

What We're Solving:

We need to derive the relationship between angular frequency ω and period T for simple harmonic motion, then determine the correct formula for angular frequency. This is a fundamental relationship that connects the timing of oscillations to the physical properties of the system!

The Approach:

We'll work step-by-step using two methods: first connecting ω and T through basic definitions, then deriving ω from the physics of simple harmonic motion. This dual approach will help you see both the mathematical and physical foundations of this important relationship.

Step-by-Step Solution:

Step 1: Connect Angular Frequency to Period Angular frequency ω tells us how many radians per second the oscillator completes. Since one complete oscillation covers 2π radians and takes time T (the period):

ω = 2π/T

This means: T = 2π/ω

Step 2: Set Up the Physics For a mass-spring system in simple harmonic motion, we use Hooke's Law and Newton's second law:

• Restoring force: F = -kx (negative because it opposes displacement)
• Newton's second law: F = ma = m(d²x/dt²)

Step 3: Create the Differential Equation Combining these: m(d²x/dt²) = -kx

Rearranging: d²x/dt² = -(k/m)x

Step 4: Recognize the Standard Form This is the standard differential equation for simple harmonic motion: d²x/dt² = -ω²x

Comparing our equation with the standard form: ω² = k/m

Step 5: Solve for Angular Frequency Taking the square root: ω = √(k/m)

Notice how this makes physical sense! Stiffer springs (larger k) make faster oscillations, while heavier masses (larger m) make slower oscillations.

The Answer:

The correct relationship is ω = √(k/m), NOT √(m/k).

The complete set of relationships for simple harmonic motion:

• ω = √(k/m)
• T = 2π/ω = 2π√(m/k)
• f = 1/T = (1/2π)√(k/m)

Memory Tip:

Think "Kick Mass" - the k goes on top when you want to kick (increase) the frequency! A stiffer spring (bigger k) on top makes ω bigger, while more mass (bigger m) on bottom makes ω smaller. This matches our intuition: heavy things oscillate slowly, springy things oscillate quickly! 🏃‍♂️💨

Great job working through this fundamental relationship - you've just connected the mathematical description of oscillations to their physical causes!