Oscillator period
Discover how mass and spring stiffness influence the oscillation period with the FizziQ Web Spring Oscillator simulation.
Activity overview:
The student uses the FizziQ Web Spring Oscillator simulation to measure the oscillation period by systematically varying the mass (at fixed stiffness) then the stiffness (at fixed mass). By plotting T² versus m, they verify the relationship T = 2π√(m/k) and determine the spring constant from the graph slope.
Level:
Author:
High school
FizziQ Web
Duration (minutes) :
35
What students will do :
- Measure the period of a spring-mass oscillator from a position-time graph
- Identify the influence of mass on the period (T increases as m increases)
- Identify the influence of stiffness on the period (T decreases as k increases)
- Verify the relationship T = 2π√(m/k) by plotting T² versus m
- Verify that the period is independent of amplitude
Scientific concepts:
- Harmonic oscillator
- Oscillation period
- Spring constant (k)
- Relationship T = 2π√(m/k)
- Natural frequency
- Period-amplitude independence
Sensors:
- FizziQ Web Spring Oscillator simulation
Material needed:
- Computer, tablet, or smartphone with FizziQ Web
Experimental procedure:
Open the Spring Oscillator simulation in FizziQ Web (Experiment → Simulations → Spring Oscillator).
Part 1 — Effect of mass: set the stiffness to 20 N/m, amplitude to 0.3 m, and damping to 0. Set the mass to 0.5 kg.
Start the simulation with REC. Record a few oscillations, then stop. In the experiment notebook, measure the period T (time between two consecutive maxima).
Repeat for masses 1.0 kg, 1.5 kg, 2.0 kg, 3.0 kg, and 4.0 kg. Record the period T for each mass in a table.
Plot T versus m. Is the curve a straight line? Then plot T² versus m. This time, you should get a straight line through the origin.
The slope of T²(m) equals 4π²/k. Calculate k from the slope and compare with the set value (20 N/m).
Part 2 — Effect of stiffness: set the mass to 1.0 kg and amplitude to 0.3 m. Vary the stiffness: 5, 10, 20, 40, and 80 N/m.
For each value of k, measure the period T. Record the results in a table. How does T change when k increases?
Part 3 — Effect of amplitude: set m = 1.0 kg and k = 20 N/m. Measure the period for amplitudes 0.1 m, 0.3 m, and 0.5 m.
Does the period change with amplitude? State your general conclusion: T = 2π√(m/k), independent of amplitude.
Expected results:
Part 1: T increases with mass. For k = 20 N/m: T ≈ 0.99 s (m = 0.5 kg), 1.40 s (1.0 kg), 1.72 s (1.5 kg), 1.99 s (2.0 kg), 2.43 s (3.0 kg), 2.81 s (4.0 kg). The graph T(m) is a curve, but T²(m) is a straight line through the origin with slope 4π²/k ≈ 1.97 s²/kg, giving k ≈ 20 N/m. Part 2: T decreases when k increases. Part 3: T does not depend on amplitude (within measurement precision).
Scientific questions:
- If you double the mass, does the period double? Why?
- If you double the stiffness, how does the period change?
- Why does the period not depend on the amplitude?
- How could you use a spring-mass system to measure an unknown mass?
- What happens to the period if you use two identical springs in parallel?
- Why is the T²(m) graph more useful than the T(m) graph for verification?
Scientific explanations:
A spring-mass oscillator is a system where a mass is subjected to a restoring force proportional to its displacement from equilibrium: F = -kx. This force always pulls the mass back toward equilibrium, creating oscillations.
The period of oscillations is given by the formula T = 2π√(m/k). It depends on only two parameters: the mass m and the spring constant k. It does not depend on the amplitude of the oscillations.
To verify this relationship, one plots T² versus m: if it is a straight line through the origin, the relationship T ∝ √m is confirmed. The slope equals 4π²/k, allowing k to be determined experimentally.
The spring constant k is expressed in N/m. A stiff spring (large k) exerts an intense restoring force and produces fast, short-period oscillations. A soft spring (small k) produces slow oscillations.
The independence of the period from amplitude is a remarkable property of the harmonic oscillator (linear restoring force). If the force is not proportional to displacement, the period generally depends on amplitude.
Extension activities:
- If you double the mass, does the period double? Why?
- If you double the stiffness, how does the period change?
- Why does the period not depend on the amplitude?
- How could you use a spring-mass system to measure an unknown mass?
- What happens to the period if you use two identical springs in parallel?
- Why is the T²(m) graph more useful than the T(m) graph for verification?
Frequently asked questions:
Q: How do I measure the period on the graph?
R: Identify two consecutive maxima on the position-time curve. The time difference between these two maxima is the period T. For better accuracy, measure the time for 5 or 10 oscillations and divide by the number.
Q: The measured period does not exactly match the formula. Why?
R: If damping is not set to zero, the pseudo-period is slightly longer than the theoretical period. Verify that damping is set to exactly 0.
Q: Why plot T² instead of T?
R: The relationship T = 2π√(m/k) means T is proportional to √m, which is a curve. By squaring, T² = 4π²m/k, which is linear — much easier to verify visually.
Q: Does the spring constant depend on the mass attached to it?
R: No, k is an intrinsic property of the spring. It does not change when you change the mass. Only the period changes.