A mass-spring oscillator is one of the simplest systems that shows repeating motion. A block attached to a spring moves back and forth around an equilibrium position when it is pulled or pushed and released. This model matters because it explains vibrations in machines, molecules, car suspensions, and many measuring devices.
It also gives a clear example of how force, energy, and motion are connected.
Key Facts
- Hooke's law for a spring is F = -kx, where k is the spring constant and x is displacement from equilibrium.
- For an ideal mass-spring oscillator, the period is T = 2π√(m/k).
- The frequency is f = 1/T = (1/2π)√(k/m).
- Position can be modeled by x(t) = A cos(ωt + φ), where ω = √(k/m).
- Maximum speed occurs at equilibrium and equals vmax = Aω.
- Total mechanical energy is E = (1/2)kA^2 and stays constant if friction is negligible.
Vocabulary
- Equilibrium position
- The position where the spring is neither stretched nor compressed and the net force on the mass is zero.
- Restoring force
- A force that points back toward equilibrium and tries to return the system to its balanced position.
- Amplitude
- The maximum displacement of the mass from equilibrium during oscillation.
- Period
- The time required for one complete back-and-forth cycle of motion.
- Angular frequency
- A measure of how quickly the oscillator cycles, given by ω = √(k/m) for an ideal mass-spring system.
Common Mistakes to Avoid
- Using F = kx instead of F = -kx, which misses that the spring force points opposite the displacement from equilibrium.
- Thinking the mass moves fastest at the endpoints, which is wrong because its speed is zero there and maximum at equilibrium.
- Assuming a larger amplitude changes the ideal period, which is wrong because T = 2π√(m/k) does not include amplitude for a linear spring.
- Confusing spring constant k with mass m in the period formula, which gives the wrong trend because increasing m increases T while increasing k decreases T.
Practice Questions
- 1 A 0.50 kg block is attached to a spring with k = 200 N/m on a low-friction surface. Find the period T and frequency f of the oscillation.
- 2 A spring oscillator has amplitude A = 0.12 m and spring constant k = 80 N/m. Find the total mechanical energy stored in the oscillation.
- 3 A block on a spring is released from maximum stretch to the right. Describe the direction of the force, the speed, and the acceleration as it moves through equilibrium and reaches maximum compression.