Spring Period — Oscillating Scale Pan (Worked Example)

Given

• m = 2.0 kg

• k = 250 N/m

Find

The period T.

Principle

Vertical or horizontal, a mass-spring system obeys T = 2π√(m/k) for small oscillations.

Formula

Solution

1. Write the formula: $T = 2\pi\sqrt{m/k}$

2. Substitute: $T = 2\pi\sqrt{2.0/250}$

3. Inside: $2/250 = 0.008$; $\sqrt{0.008} = 0.0894$

4. Multiply: $T = 2\pi \times 0.0894 = 0.562\ \mathrm{s}$

Unit check

√(kg ÷ kg/s²) = s — correct.

What this means physically

0.562 s — about 1.78 oscillations per second. Heavier masses swing slower; this square-root dependence is why adding mass is an easy way to slow a vibration down.