Stationary waves on a string
Investigate how the resonant frequency of stationary waves on a string depends on length, tension and mass per unit length.
Apparatus
- Signal generator and mechanical vibration transducer
- String or nylon thread of known mass per unit length
- Pulley, slotted masses and hanger
- Metre rule
- Stands and clamps
Safety
- Secure the pulley and stand so hanging masses cannot fall onto anyone below.
- Use a low signal generator amplitude to prevent the string breaking under excessive vibration.
Method
- Thread the string from the vibration transducer, over a pulley and attach slotted masses to provide tension T = mg.
- Switch on the signal generator and slowly increase the frequency until a stable fundamental (first harmonic) standing wave forms with one loop.
- Record the resonant frequency f and measure the vibrating length L between the transducer and pulley.
- Repeat at different lengths (move the pulley) to investigate f vs L at constant T and mass per unit length.
- Repeat with different hanging masses (to vary T) at constant L; repeat with strings of different linear density.
Key Variables
Independent
String length L (or tension T, or mass per unit length mu)
Dependent
Resonant frequency f
Controlled
Tension (when varying L or mu); Length (when varying T or mu); Mass per unit length (when varying L or T)
Analysis and Results
- The fundamental frequency is given by $f = \frac{1}{2L}\sqrt{\frac{T}{\mu}}$.
- Plot f against 1/L at constant T and mu: expect a straight line through the origin with gradient $\frac{1}{2}\sqrt{\frac{T}{\mu}}$.
- Plot f squared against T at constant L and mu: straight line through origin, gradient $= \frac{1}{4L^2\mu}$.
- Wave speed on the string: $v = \sqrt{T/\mu}$. Check consistency using v = f multiplied by 2L.
Common Errors
- Measuring L beyond the pulley rather than between the vibrator and pulley.
- Not waiting for the wave to stabilise before recording frequency.
- Plotting f against L (a curve) instead of f against 1/L (a straight line).
- Ignoring the mass of the string when calculating tension from hanging masses.
Exam-style questions on this practical. Click Show mark scheme to reveal the answer after attempting each question.
Q13 marks
A string of length 0.60 m and mass per unit length $4.0 \times 10^{-3}$ kg/m is held under a tension of 16 N. Calculate the fundamental frequency of vibration.
Q24 marks
Describe an experiment to investigate how the resonant frequency of a stretched string varies with tension. Include how you would display and interpret your results.
Q32 marks
State two differences between a stationary wave and a progressive wave.