Oscillator period
Discover how mass and spring stiffness influence the oscillation period with the FizziQ Web Spring Oscillator simulation.
Attach a mass to a spring, pull it down and release: the mass oscillates. But what determines the speed of these oscillations? If you use a stiffer spring, are the oscillations faster or slower? And what happens if you change the mass? The Spring Oscillator simulation in FizziQ Web lets you explore these questions systematically by changing one parameter at a time while measuring the period precisely from the position-time graph. You will discover a remarkably simple formula that connects the period to just two quantities: the mass and the spring constant.
Visão geral da atividade:
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.
Nível:
High school
FizziQ Web
Autor:
Duração (minutos):
35
O que os alunos farão:
- 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
Conceitos científicos:
- Harmonic oscillator
- Oscillation period
- Spring constant (k)
- Relationship T = 2π√(m/k)
- Natural frequency
- Period-amplitude independence
Sensores:
- FizziQ Web Spring Oscillator simulation
Materiais necessários:
- Computer, tablet, or smartphone with FizziQ Web
Procedimento experimental:
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.
Resultados esperados:
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).
Questões científicas:
- 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?
Explicações científicas:
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.
Atividades de extensão:
- 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?
Perguntas frequentes:
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.