A spring-mass system has k = 50 N/m and m = 2 kg. What is the period?

The correct answer is: T ≈ 1.26 s. You can work it out this way: use T = 2π√(m/k). Period is independent of amplitude.

If you double the mass, how does the period change?

The correct answer is: Period increases by √2 ≈ 1.41×. T ∝ √m — doubling mass multiplies period by √2 ≈ 1.41.

At what position is kinetic energy maximum? Why?

The correct answer is: At equilibrium (x = 0). At equilibrium (x=0), all potential energy converts to kinetic.

A system has k = 80 N/m, m = 0.5 kg, A = 0.3 m. Find Eₜₒₜₐₗ and vₘₐₓ.

The correct answer is: E = 3.6 J, vₘₐₓ = 3.79 m/s. E = ½kA², then vₘₐₓ = √(2E/m).

How does adding damping affect energy and amplitude over time?

The correct answer is: Amplitude decays exponentially; energy decays exponentially. Enable the damping slider and observe the envelope decay.

Pro 🔒~20 min

Simple Harmonic Motion

Springs, pendulums, and oscillation

How it works

Simple Harmonic Motion (SHM) occurs when a restoring force is proportional to displacement: F = -kx. The period depends only on mass and spring constant, not on amplitude — a key insight that confuses many students. Energy constantly transforms between kinetic (maximum at equilibrium) and potential (maximum at extreme positions), with total mechanical energy conserved in the absence of damping.

If you double the amplitude of a pendulum, does it take longer to swing?

Most people guess yes, but the answer is no — period is completely independent of amplitude in simple harmonic motion.

What you'll learn

The Restoring Force. In SHM, the restoring force is always proportional to displacement and directed toward equilibrium: F = -kx. This linear relationship is what makes the motion sinusoidal. Any system with this property — springs, pendulums at small angles, molecules in a lattice — oscillates harmonically.
Energy Exchange. Energy continuously transforms between kinetic and potential forms. At maximum displacement, all energy is potential (½kA²). At equilibrium, all energy is kinetic (½mv²max). Total mechanical energy remains constant when there is no damping.
Period Independence from Amplitude. The period T = 2π√(m/k) depends only on mass and spring constant, not on how far you pull the spring. This counterintuitive result means a clock pendulum keeps the same time whether it swings wide or narrow — the principle behind mechanical clocks for centuries.
Damping and Energy Loss. Real oscillators lose energy to friction and air resistance. The damping force opposes velocity (F_d = -bv), causing amplitude to decay exponentially while the frequency shifts slightly lower. Critical damping returns the system to equilibrium fastest without oscillating.

Step-by-step

Adjust the spring constant k and mass m to observe how period changes.
Notice that changing amplitude does NOT affect the period — try it.
Watch the energy bar at the bottom show Ek and Ep cycling.
Use the damping slider (Pro) to explore energy dissipation.

Key formulas

$T = 2\pi\sqrt{\dfrac{m}{k}}$ Period of oscillation
$x(t) = A\cos(\omega t + \phi)$ Position as a function of time
$\omega = \sqrt{\dfrac{k}{m}}$ Angular frequency
$E_{total} = \frac{1}{2}kA^2 = \frac{1}{2}mv^2 + \frac{1}{2}kx^2$ Total mechanical energy (conserved)
$v_{max} = A\omega = A\sqrt{\dfrac{k}{m}}$ Maximum velocity (at equilibrium)

Frequently asked questions

A spring-mass system has k = 50 N/m and m = 2 kg. What is the period?: The correct answer is: T ≈ 1.26 s. You can work it out this way: use T = 2π√(m/k). Period is independent of amplitude.
If you double the mass, how does the period change?: The correct answer is: Period increases by √2 ≈ 1.41×. T ∝ √m — doubling mass multiplies period by √2 ≈ 1.41.
At what position is kinetic energy maximum? Why?: The correct answer is: At equilibrium (x = 0). At equilibrium (x=0), all potential energy converts to kinetic.
A system has k = 80 N/m, m = 0.5 kg, A = 0.3 m. Find Eₜₒₜₐₗ and vₘₐₓ.: The correct answer is: E = 3.6 J, vₘₐₓ = 3.79 m/s. E = ½kA², then vₘₐₓ = √(2E/m).
How does adding damping affect energy and amplitude over time?: The correct answer is: Amplitude decays exponentially; energy decays exponentially. Enable the damping slider and observe the envelope decay.