Lab Equipment:
- Frequency wave driver
- long string
- Function Generator
- Weight hanger and slotted weight set
- 2 table clamps
- short rod
- pendulum clamp
- pulley
- digital multimeter
- meter stick
Lab Set-up:
1) measure and record the mass and length of the string
2) tie two loops in the string more than 1 meter apart
3) set up the pendulum clamp and pulley clamp slightly more than 0.7 cm apart
4) attach the string to the pendulum clamp and thread it through the pulley. attach the mass hanger to the hanging end of the string.
5) thread the string through the wave driver. measure and record the distance from the wave driver to the pulley.
6) connect the wave driver to the function generator
Data Collection:
1) adjust the frequency of the function generator until the string oscillates in the fundamental mode. Record the oscillation frequency, the number of nodes, and the length of string participationg in the oscillation. Determine the distance between nodes and wavelength of oscillation.
2) repeat for at least the first 7 harmonics
3) change the tension in the string by changing the hanging mass and repeat steps 1 and 2 for this new tension.
mass of string = 7.5 g (error = 0.02 g)
length of string = 90.6 cm (error = 0.2 cm)
mass per unit length = 0.00826 kg/m (error = 0.0004 kg/m)
Case 1
hanging mass = 0.55 kg (error = 0.001 kg)
Length of vibrating string = 0.61 m (error = 0.005 m)
Tension = 5.4 N (error = 0.01 N)
wave speed = 25.6 m/s (error = 1.0 m/s)
f = frequency, l = wavelength, x = # of nodes, n = x - 1
1) f = 21 hz, x = 2, l = 1.22 m, n = 1
2) f = 43 hz, x = 3, l = 0.61 m, n = 2
3) f = 64 hz, x = 4, l = 0.407 m, n = 3
4) f = 85 hz, x = 5, l = 0.305 m, n = 4
5) f = 108 hz, x = 6, l = 0.244 m, n = 5
6) f = 129 hz, x = 7, l = 0.203 m, n = 6
7) f = 152 hz, x = 8, l = 0.174 m, n = 7
8) f = 174 hz, x = 9, l = 0.153 m, n = 8
9) f = 195 hz, x = 10, l = 0.136 m, n = 9
Plot f vs 1/l and find the slope of the linear fit. compare this value to the predicted wave speed.
Using a stat plot and least squares regression line we find:
y = 26.69x - 1.39 (y = f, x = 1/l)
the slope of the line (26.7) is within the margin of error of our prediction for the wave speed.
Case 2
hanging mass = 0.3 kg (error = 0.001 kg)
Length of vibrating string = 0.61 m (error = 0.005 m)
Tension = 2.94 N (error = 0.01 N)
wave speed = 18.9 m/s (error = 0.9 m/s)
1) f = 16 hz, x = 2, l = 1.22 m, n = 1
2) f = 32 hz, x = 3, l = 0.61 m, n = 2
3) f = 48 hz, x = 4, l = 0.407 m, n = 3
4) f = 64 hz, x = 5, l = 0.305 m, n = 4
5) f = 80 hz, x = 6, l = 0.244 m, n = 5
6) f = 96 hz, x = 7, l = 0.203 m, n = 6
7) f = 112 hz, x = 8, l = 0.174 m, n = 7
8) f = 128 hz, x = 9, l = 0.153 m, n = 8
9) f = 145 hz, x = 10, l = 0.136 m, n = 9
plot f vs 1/l and find the slope of the linear fit. compare this value to the predicted wave speed.
Using a stat plot and least squares regression line we find:
y = 19.65x - 0.367
the line slope (19.7) is within the margin of error of our predicted wave speed.
Additional Analysis:
- the ratio wave speeds obtained from the graphs for the first and second case is 1.36. The ratio of the predicted wave speeds is 1.35. These values are very close.
- The standing wave frequency for each case was approximately equal to n*(fundamental frequency) for each of the harmonics. There were very slight deviations since the frequency generator only increased by unit values.
- compare the corresponding harmonics for the two cases
ratio 1 = 21/16 = 1.31
ratio 2 = 43/32 = 1.34
ratio 3 = 64/48 = 1.33
ratio 4 = 85/64 = 1.33
ratio 5 = 108/80 = 1.35
ratio 6 = 129/96 = 1.34
ratio 7 = 152/112 = 1.36
ratio 8 = 174/128 = 1.36
ratio 9 = 195/145 = 1.34
each of these ratios is very close to the ratio of the wave speeds of the two cases.
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