Simple Harmonic Motion: Tricky Questions Solved (3)

hard • 2 min read
Tags Simple Harmonic Motion

Question

Two springs of stiffness k1=200k_1 = 200 N/m and k2=300k_2 = 300 N/m are connected to a block of mass m=2m = 2 kg. Find the period of oscillation if the springs are connected (a) in parallel, (b) in series.

Solution — Step by Step

When springs are in parallel, both stretch by the same amount but each provides its own restoring force. The forces add:

keff=k1+k2=200+300=500 N/mk_{\text{eff}} = k_1 + k_2 = 200 + 300 = 500 \text{ N/m} Tp=2πmkeff=2π2500=2π0.004T_p = 2\pi\sqrt{\tfrac{m}{k_{\text{eff}}}} = 2\pi\sqrt{\tfrac{2}{500}} = 2\pi\sqrt{0.004} Tp≈2π×0.0632≈0.397 sT_p \approx 2\pi \times 0.0632 \approx 0.397 \text{ s}

In series, the same force passes through both springs, but each stretches by a different amount. The total extension is the sum, so:

1keff=1k1+1k2=1200+1300=5600\tfrac{1}{k_{\text{eff}}} = \tfrac{1}{k_1} + \tfrac{1}{k_2} = \tfrac{1}{200} + \tfrac{1}{300} = \tfrac{5}{600} keff=6005=120 N/mk_{\text{eff}} = \tfrac{600}{5} = 120 \text{ N/m} Ts=2π2120=2π0.01667T_s = 2\pi\sqrt{\tfrac{2}{120}} = 2\pi\sqrt{0.01667} Ts≈2π×0.1291≈0.811 sT_s \approx 2\pi \times 0.1291 \approx 0.811 \text{ s}

Final answers: Tp≈0.40 sT_p \approx \mathbf{0.40 \text{ s}}, Ts≈0.81 sT_s \approx \mathbf{0.81 \text{ s}}.

Why This Works

The stiffness combination rules look like the opposite of capacitor combinations because of how the same quantity is shared:

  • Parallel springs: same extension → forces add → keffk_{\text{eff}} adds.
  • Series springs: same force → extensions add → 1/keff1/k_{\text{eff}} adds.

A series combination is always softer than either spring alone, hence a longer period.

Alternative Method

Energy approach: write total spring PE as 12k1x12+12k2x22\tfrac{1}{2}k_1 x_1^2 + \tfrac{1}{2}k_2 x_2^2 and use the constraints (parallel: x1=x2x_1 = x_2; series: k1x1=k2x2k_1 x_1 = k_2 x_2) to express in terms of total extension xx. The coefficient of x2/2x^2/2 is keffk_{\text{eff}}. Recovers both formulas.

Common Mistake

Mixing up parallel and series rules with capacitor formulas. For springs: parallel adds stiffness (like resistors in series for resistance). Get the analogy right by asking “what is shared?” — same extension means parallel; same force means series.

Want to master this topic?

Read the complete guide with more examples and exam tips.

Go to full topic guide →

Try These Next

A mass-spring system has k=200 N/m and m=0.5 kg — find period and frequency

easy

A pendulum clock gains 5 min/day — how should length be changed

hard

Show that the projection of uniform circular motion is SHM

hard

Simple Harmonic Motion: Application Problems (5)

medium

Simple Harmonic Motion: Common Mistakes and Fixes (1)

easy