Spring constant
Determine the spring constant of a spring by measuring the oscillation period of a spring-mass system with the smartphone accelerometer.
When you hang an object from a spring and let it oscillate, the period depends on the object's mass and the spring stiffness. This fundamental relationship, T = 2π√(m/k), allows you to measure the spring constant with nothing more than a smartphone, a spring, and a little ingenuity. By recording the oscillation with the accelerometer and measuring the period precisely, you can determine k without any static force measurement.
Activity overview:
The student attaches their smartphone to a suspended spring, sets it oscillating vertically, and records the acceleration with FizziQ. By measuring the oscillation period and knowing the phone's mass, they calculate the spring constant.
Level:
High School
FizziQ
Author:
Duration (minutes) :
30
What students will do :
- Measure the oscillation period of a spring-mass system with the accelerometer
- Calculate the spring constant from the period and mass
- Verify experimentally that the period is independent of amplitude
- Compare the dynamic method (oscillation) with the static method (extension under load)
- Use the autocorrelation function for precise period measurement
Scientific concepts:
- Spring constant
- Oscillation period
- Simple harmonic motion
- Hooke's law (F = -kx)
- Autocorrelation
- Natural angular frequency (ω = 2π/T)
Sensors:
- Accelerometer (linear acceleration, without gravity)
Material needed:
- Smartphone or tablet with FizziQ
- A spring (laboratory spring or Slinky)
- A rigid support (door handle, curtain rod, lab stand)
- Optional: additional known masses
Experimental procedure:
Build a support for your smartphone: take a cardboard roll (toilet paper type), cut two slits at each end, and slide the phone in. Attach a hook or loop of string at the top.
Weigh your smartphone (with its case and support) using a balance. Note this mass m in kilograms.
Open FizziQ and select the Accelerometer instrument (linear acceleration, without gravity).
Hang the spring from a solid support (door handle, curtain rod, lab stand). Suspend the phone in its holder from the bottom of the spring.
Gently pull the phone downward (a few centimeters is enough) then release. Be careful not to pull too far — the phone must not fall!
Start recording in FizziQ while the system oscillates. Record at least 10 complete oscillations.
Stop recording and observe the acceleration versus time graph. You should see a clean sinusoidal oscillation with slowly decreasing amplitude.
Measure the period T by counting the number of peaks over a given time interval, or use the FizziQ autocorrelation function for higher precision.
Calculate the spring constant with the formula: k = (2π/T)² × m. Compare with the manufacturer's value for the spring if available.
Repeat the experiment by adding an extra mass (for example, a small book attached to the phone). The period should increase. Recalculate k and verify consistency.
Expected results:
The acceleration graph shows a regular sinusoidal oscillation with slowly decreasing amplitude due to damping. The period T is constant (independent of amplitude, as long as oscillations remain small). Typical values: for a standard lab spring with k = 10-50 N/m and a smartphone of 0.2 kg, T ≈ 0.4-0.9 s. The calculated k should agree with the static method within 5-10%.
Scientific questions:
- Why does the oscillation period not depend on the amplitude?
- What happens if you pull the spring very far: does the period change? Why?
- How does adding mass to the system affect the period?
- Why is the autocorrelation method more precise than counting peaks?
- How does damping affect the measured period?
- What are the advantages of the dynamic method over the static method for measuring k?
Scientific explanations:
Simple harmonic motion (SHM) is periodic motion where the restoring force is proportional to displacement: F = -kx. The acceleration is sinusoidal: a(t) = -Aω²cos(ωt), where ω = 2π/T.
The smartphone accelerometer measures the linear acceleration at each instant. For SHM, the acceleration is sinusoidal with the same period T as the position.
The oscillation period is given by T = 2π√(m/k). This relationship shows that the period depends only on the mass and the spring constant, not on the amplitude.
Rearranging this formula gives k = (2π/T)² × m = 4π²m/T². Knowing the smartphone mass and the measured period, the spring constant is calculated directly.
The autocorrelation method for determining the period consists of calculating the correlation of the signal with a shifted copy of itself. The first peak of the autocorrelation function gives the period T.
This method is more robust than simply reading peaks because it averages over the entire signal and reduces the effect of noise.
By plotting T² versus m for several masses, one obtains a straight line with slope 4π²/k. This graphical method gives a more precise value of k.
The observed damping (decreasing amplitude) is due to air friction and internal losses in the spring. It slightly lengthens the pseudo-period compared to the ideal undamped period.
Extension activities:
- Why does the oscillation period not depend on the amplitude?
- What happens if you pull the spring very far: does the period change? Why?
- How does adding mass to the system affect the period?
- Why is the autocorrelation method more precise than counting peaks?
- How does damping affect the measured period?
- What are the advantages of the dynamic method over the static method for measuring k?
Frequently asked questions:
Q: Why use acceleration without g rather than with g?
R: The accelerometer 'with g' measures gravity in addition to the motion acceleration. The oscillation signal is superimposed on a constant 9.81 m/s² offset, making analysis harder. Without g, the signal oscillates around zero.
Q: The oscillation amplitude decreases rapidly. Is this a problem?
R: Some damping is normal. As long as you can identify at least 5-10 clear oscillations, the period measurement is reliable. Use a stiffer spring or lighter holder to reduce damping.
Q: My calculated k does not match the manufacturer's value.
R: Account for the mass of the spring itself (add about 1/3 of the spring mass to the oscillating mass). Also check that the spring is in its linear range (not overstretched).
Q: Can I use a rubber band instead of a spring?
R: Rubber bands have non-linear elasticity and significant hysteresis. They do not follow Hooke's law well, so the period will depend on amplitude.