Source: PHYS/ENGR 216, PHYS 226 Lab Manual, Texas A&M University
Tags: simple harmonic motion, SHM, spring constant, Hooke's law, angular frequency, oscillation, period, amplitude, phase, mass-spring system, harmonic oscillator, tracking camera, pixel tracking, uncertainty
This lab is about finding the spring constant k for three different springs by watching them oscillate and analysing the motion. You hang a known mass on a spring, set it bouncing, record position vs. time with a tracking camera (in pixels, not centimetres), then extract the period from that data and work backwards through the SHM equations to solve for k. No ruler is available, so the method relies entirely on timing, not distance measurement.
Simple harmonic motion (SHM)
Periodic motion where the restoring force is proportional to displacement. The position follows a cosine (or sine) function over time. Often called "oscillatory motion" or just "harmonic motion."
Spring constant (k)
A measure of a spring's stiffness, in units of N/m. A larger k means the spring is harder to stretch. This is the quantity you are solving for in the lab.
Amplitude (A)
The maximum displacement from the equilibrium (rest) position. In the SHM equation, it sets the peak value of the cosine function.
Angular frequency (ω)
The rate of oscillation expressed in radians per second (rad/s). Related to both the spring constant and the period. This is the bridge between your time-domain measurement and the spring constant.
Period (T)
The time for one complete oscillation cycle, measured in seconds. This is what you actually measure from your position-vs-time plot.
Phase (ϕ)
The horizontal shift of the cosine wave, in radians. It accounts for where in the cycle the oscillation happens to start when you begin recording. It does not affect k or T.
Equilibrium position
The position where the net force on the mass is zero (the spring is neither compressed nor stretched beyond its loaded rest point). The oscillation happens symmetrically around this point.
The position of a harmonic oscillator as a function of time is:
x(t) = A cos(ωt + ϕ) — Equation (1)
A is amplitude
ω is angular frequency (rad/s)
ϕ is phase (rad)
t is time (s)
This is the waveform you should see in your position-vs-time plot. If damping is significant, the amplitude will decay over time, but for a short recording window the cosine model works well.
For a mass-spring system specifically, angular frequency depends on the spring constant and the hanging mass:
ω = √(k / m) — Equation (2)
k is the spring constant (N/m)
m is the total oscillating mass (kg), including the hanger
Rearranging to solve for the spring constant:
k = ω² · m
This is the key relationship for the lab. You measure ω from the data, you know m because you chose the masses, and you solve for k.
Angular frequency and period are related by:
ω = 2π / T — Equation (3)
Substituting into the expression for k:
k = (2π / T)² · m = 4π²m / T²
This version is often the most practical, because T is what you read directly off your position-vs-time graph.
Record position vs. time for a known mass on the spring
Identify the period T from the plot (time between successive peaks, or between any two identical points one cycle apart)
Calculate ω = 2π / T
Calculate k = ω² · m
No metre stick is provided, so you cannot calibrate the camera to give distances in centimetres
Use the Python scripts (6_track_motion_and_print.py or 4_track_and_print_with_camera_input.py) to record position in pixel units
Pixel position is perfectly fine for measuring the period. The time axis is what matters for finding T, and that is already in real units (seconds)
Amplitude will be in pixels rather than metres, but amplitude is not needed to find k
You need to report k with its associated uncertainty for each spring
Sources of uncertainty include timing resolution of the camera, slight variation in period from cycle to cycle, and the precision of the mass values
Measuring T over several cycles and averaging improves precision
Propagate uncertainty through k = 4π²m / T² using standard error propagation methods
Quantity | Formula | What you need |
|---|---|---|
Position | x(t) = A cos(ωt + ϕ) | Fit or read from data |
Angular frequency from spring | ω = √(k/m) | k and m |
Angular frequency from period | ω = 2π / T | T from plot |
Spring constant | k = 4π²m / T² | m (known), T (measured) |
⚠️ The period T does not depend on amplitude for ideal SHM. A larger pull does not change the oscillation frequency, only the peak displacement.
⚠️ Make sure m includes the mass of the hanger, not just the added masses. Forgetting the hanger mass is a common source of systematic error.
⚠️ You do not need distance in SI units to find k. The period is a time measurement and the mass is known, so pixel-based position data is sufficient.
⚠️ Phase ϕ shifts the wave left or right but does not change the period. Do not confuse phase with frequency.
⚠️ If your position-vs-time plot shows decaying amplitude, the spring is losing energy to friction or air resistance. The period should still be roughly constant, but note the damping in your report.
⚠️ Know how to rearrange ω = √(k/m) into k = ω²m and substitute ω = 2π/T. This algebra appears frequently in exam problems.
Q: A 0.25 kg mass oscillates on a spring with a period of 0.80 s. What is the spring constant?
A: k = 4π²m / T² = 4π²(0.25) / (0.80)² ≈ 15.4 N/m.
Q: Why is it acceptable to measure position in pixels rather than metres for this lab?
A: The spring constant is determined from the period T and the mass m. The period is a time measurement read from the horizontal axis of the position-vs-time plot, so it does not depend on the distance units used on the vertical axis.
Q: If you double the mass on the same spring, what happens to the period?
A: The period increases by a factor of √2. Since T = 2π√(m/k), doubling m multiplies T by √2 ≈ 1.41.
Q: What is the relationship between angular frequency and period?
A: ω = 2π / T. Angular frequency is in rad/s; the period is in seconds.
Q: A student measures the spring constant but forgets to include the mass of the hanger (50 g) in their calculation. They used 200 g of added mass. Will their reported k be too high or too low?
A: Too low. They used m = 0.200 kg instead of m = 0.250 kg, so k = 4π²(0.200)/T² is smaller than the true value 4π²(0.250)/T².
Q: Does changing the amplitude of oscillation change the spring constant you measure?
A: No. For ideal SHM the period is independent of amplitude, so the measured k should be the same regardless of how far you pull the mass before releasing it.
simple harmonic motion, SHM, harmonic oscillator, spring constant, Hooke's law, mass-spring system, angular frequency, omega, period, frequency, amplitude, phase, oscillation, restoring force, tracking camera, pixel tracking, uncertainty propagation, error analysis, PHYS 226, ENGR 216, Texas A&M, cosine fit, position vs time, spring stiffness, natural frequency