Mass and springs laboratory

Learning objectives

• Recognize what makes a motion periodic.
• Relate elastic force to Hooke’s law.
• Predict how the period and equilibrium change when parameters change.

What can you investigate?

Increasing mass makes the system oscillate more slowly. Increasing stiffness makes it oscillate faster. Gravity shifts the equilibrium position without changing the basic oscillation rule by itself.

Periodic motion

A motion is periodic when it repeats its state after a time interval. In an ideal mass-spring system, the mass moves away from equilibrium, passes through it, and repeats the cycle. The repetition time is called the period (T).

Hooke’s law

While the spring stays within its elastic range, the restoring force is proportional to its deformation and points toward equilibrium: Fₑ = −k·x. The constant k measures stiffness: a larger value resists the same deformation more strongly.

Mass, stiffness, and period

For an ideal model, T = 2π√(m/k). More mass increases the period; a stiffer spring reduces it. Use this relationship to make predictions before moving the controls.

What changes with gravity?

Gravity adds a constant force and moves the equilibrium point downward. In the ideal model, it does not change the period of small oscillations around that new equilibrium by itself.

Energy during the oscillation

Elastic potential energy is greatest when the spring is most deformed. As the mass passes through equilibrium, much of that energy has become kinetic energy. Without friction, total mechanical energy would remain constant.

Simplified model

We use a lightweight numerical integration of F = −kx + mg. It is an educational approximation: it does not model friction, permanent deformation, or a real laboratory setup.

How to explore

1. Choose initial mass, stiffness, and gravity values.
2. Predict whether the period will increase or decrease when one variable changes.
3. Open the simulation, pause it, and compare the equilibrium position.
4. Reset the values and repeat the test with another hypothesis.

Examples to reason about

• If you double the mass while keeping k constant, the period does not double: it increases by a factor of √2.
• If you quadruple stiffness with the same mass, the period is reduced by half.
• With greater gravity, equilibrium moves downward, but the ideal period stays approximately the same.