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Simple Pendulum & Spring–Mass System Worksheet

Class 11 Physics & JEE · about 45 minutes · open the simulation at jugnix.com/simulations/pendulum-spring

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By the end you will: find what the period of a pendulum and of a spring depends on, use T = 2π√(L/g) and T = 2π√(m/k), and measure g from a graph.

A. Explore with the simulation

  1. On Earth, set the start angle to 10°. Change the pendulum length and record the period. Work out T² and T² ÷ L.

    Length L (m)Period T (s)T² (s²)T² ÷ L (s²/m)
    0.25
    0.50
    1.00
    2.00

  2. Use your table: T² ÷ L = 4π² ÷ g. Calculate g.

  3. Keep L = 1 m. Change the bob mass from 0.1 kg to 5 kg. Then raise the start angle from 10° to 90°. What happens to the period each time?

  4. Set the spring to k = 10 N/m. Record its period with masses of 1 kg and 4 kg. Then switch to the Moon. What changes?

B. Calculate (then check in the simulation)

  1. Find the length of a seconds pendulum (period 2 s) on Earth (g = 9.8 m/s²).

  2. A pendulum clock keeps perfect time on Earth. Would it run fast or slow on the Moon (g = 1.62 m/s²)? By what factor does its period change?

  3. A 0.5 kg mass on a spring of k = 50 N/m is pulled down and let go. Find its period and frequency.

  4. With L = 1 m and a 1 kg spring mass, what spring constant makes the spring swing in step with the pendulum? Check it in the simulation.

C. Think it through

  1. Sketch the graph of T² against L for a simple pendulum. What is its shape, and how do you get g from it?

Answer key (for teachers and parents)

  1. 0.25 m: 1.005 s, 1.01, 4.04 · 0.50 m: 1.422 s, 2.02, 4.04 · 1.00 m: 2.011 s, 4.04, 4.04 · 2.00 m: 2.844 s, 8.09, 4.04. T² ÷ L stays about 4.04: T² is proportional to L.
  2. g = 4π² ÷ 4.04 = 39.48 ÷ 4.04 ≈ 9.8 m/s². (The 10° swing makes the period about 0.2% longer than the formula, so the answer comes out a touch low.)
  3. Mass: no change at all. Angle: the period grows slowly, about 2.01 s at 10° to about 2.37 s at 90°, so T = 2π√(L/g) is only exact for small swings.
  4. 1 kg: 1.99 s. 4 kg: 3.97 s, twice as long (T ∝ √m). On the Moon the spring's period doesn't change at all, but the pendulum's gets longer.
  5. L = gT² ÷ 4π² = 9.8 × 4 ÷ 39.48 ≈ 0.99 m.
  6. Slow. T ∝ 1/√g, so the period becomes √(9.8 ÷ 1.62) ≈ 2.46 times longer.
  7. T = 2π√(0.5 ÷ 50) = 2π × 0.1 ≈ 0.63 s. f = 1 ÷ T ≈ 1.6 Hz.
  8. Equal periods: 2π√(L/g) = 2π√(m/k), so k = mg ÷ L = 1 × 9.8 ÷ 1 = 9.8 N/m. The nearest slider setting, 10 N/m, shows “In step? Yes”.
  9. A straight line through the origin. Its slope is 4π² ÷ g, so g = 4π² ÷ slope.

Free to print and share for teaching. Made with the interactive simulation at jugnix.com/simulations/pendulum-spring.