Physics teacher support material
Investigation 8
The mass per unit length of a wire measured
by two different methods.
The aim of this experiment is to compare the experimental values from two different methods of
determining the value the mass (m) per unit length (L) of a wire, where µ =
m
.
L
The first method determines a value by direct measures of mass and length. In this case, I
measured the mass of the wire as m ± ∆m = (0.0026 ± 0.0005 )kg with a top pan balance and the
length of the wire as L ± ∆L = (1.710 ± 0.002 )m with a metre rule.
The first method yields a value of µOne =
m 0.0026 kg
=
= 1.52 × 10 −3 kg m −1 . The uncertainty in
L
1.710 m
this is about ∆µOne = ± 0.3 × 10 −3 kg m −1 .
The second method is based on the variation of the speed v of a wave in the wire with the
tension T in the wire. The equipment is set up as shown in the sketch.
© International Baccalaureate Organization 2007 Physics teacher support material
Investigation 8
For a given load, the length of the wire is adjusted until it vibrates in its fundamental modes (first
harmonic). Theory tells us that the frequency f, length L, the speed v, and mass per unit length µ
are related as follows:
T
f =
µ
v
=
Ÿ
2L 2L
T
µ
= 2L f Ÿ
T = 2 µ f L Ÿ T = 4µ f 2L 2
The slope of a graph of tension T against length squared L2 is 4 µ f 2 for the first harmonic.
Hence µ Two =
slope
4f2
The raw data collect from the experiment is listed here.
Load (kg) ±0.001 kg
1.000
1.500
2.000
2.500
3.000
1st Harmonic
L (m) ±0.02m
0.42
0.58
0.61
0.70
0.79
By using the above formulas I calculated the following information based on the raw data.
Tension (N) ±0.01N
9.8
14.7
19.6
24.5
29.4
L2 ±0.05m
0.18
0.34
0.37
0.49
0.62
© International Baccalaureate Organization 2007
Physics teacher support material
Investigation 8
Graph of Tension against the Square of the Length for the First harmonic.
The slope here is given as 46.13N m −2 . Hence µ Two =
slope
= 1.2 × 10 −3 kg m −1 .
4f2
My conclusion is tabulated in the table shown here.
Method One
µOne
(1.5 ± 0.3) × 10 −3 kg m −1
Method Two
µ Two
1.2 × 10 −3 kg m −1
As we can see the values of µ for both methods are nearly the same if uncertainties are taken
into account.
As we can see from the graph, the trend line matches the point with a few errors. A cause of error
in this investigation is that it is hard to judge the exact value of the length that corresponds to the
resonance. Also, the wire used might not be uniform, causing error to the data collected.
Some ways to improve the experiment are to take more measurements and take different length
sample from the same type of wire. By doing that the experimental data should reduce errors.
© International Baccalaureate Organization 2007