Mass-spring systems and simple pendulums

Mass on a spring

T = 2π sqrt(m / k)

where m is the oscillating mass and k is the spring constant (N/m). A larger mass gives a longer period; a stiffer spring gives a shorter period. The same equation works for horizontal and vertical springs.

Simple pendulum

T = 2π sqrt(l / g)

where l is the length from the pivot to the centre of mass of the bob. This holds only for small angles (less than about 10°), when the restoring force is close to proportional to displacement. The period does not depend on the mass of the bob.

Using the equations

• Neither period depends on the amplitude (for small oscillations).
• To find g, plot T² against l: the gradient is 4π² / g.
• To find k, plot T² against m: the gradient is 4π² / k.
• Time 10 or more oscillations and divide, to reduce the percentage uncertainty in T.

Worked example

A 0.50 kg mass hangs on a spring of stiffness 20 N/m. Find the period of oscillation.

1. T = 2π sqrt(0.50 / 20)
2. = 2π x 0.158

Answer: T = 0.993 s

Check yourself

1. Find the period of a 0.50 kg mass on a spring with k = 20 N/m, in s, to 3 s.f.
   Show answer

   0.993 s. T = 2π sqrt(0.50 / 20) = 2π x 0.158 = 0.993 s.

2. A 0.20 kg mass on a spring has a period of 0.60 s. Find k in N/m, to 3 s.f.
   Show answer

   21.9 N/m. k = 4π^2m / T² = 39.48 x 0.20 / 0.36 = 21.9 N/m.

3. Find the period of a simple pendulum of length 1.50 m, in s, to 3 s.f.
   Show answer

   2.46 s. T = 2π sqrt(1.50 / 9.81) = 2π x 0.391 = 2.46 s.

4. What length of simple pendulum has a period of 1.50 s? Answer in m, to 3 s.f.
   Show answer

   0.559 m. l = gT² / 4π² = 9.81 x 2.25 / 39.48 = 0.559 m.

5. The amplitude of a mass-spring oscillator is doubled. What happens to its period?
   • It doubles
   • It halves
   • It stays the same
   • It increases by a factor of sqrt(2)
   Show answer

   It stays the same. T = 2π sqrt(m/k) has no amplitude term.

6. The mass on a spring is made four times larger. What happens to the period?
   • It doubles
   • It becomes four times larger
   • It halves
   • It stays the same
   Show answer

   It doubles. T is proportional to sqrt(m), and sqrt(4) = 2.

7. A simple pendulum is taken to the Moon, where g is smaller. What happens to its period?
   • It decreases
   • It stays the same
   • It increases
   • The pendulum cannot oscillate
   Show answer

   It increases. T = 2π sqrt(l/g), so a smaller g gives a longer period.

Practise with instant feedback Print this lesson

Sources

• Pearson Edexcel International Advanced Subsidiary/Advanced Level in Physics specification (XPH11/YPH11), Issue 3
• Pearson qualifications: International Advanced Levels Physics (2018)
• Pearson Edexcel IAL Physics Unit 5 (WPH15) question paper, June 2023: list of data, formulae and relationships

Last updated 7 November 2026. Spotted a mistake? Tell us.