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Objective:__

To experimentally verify the formula for the dependence of wave speed on the properties of the medium by putting a stretched string into oscillation

__Equipment:__

An electric string vibrator, a mass scale, string, a pulley with attachments, two sets of rod and C-clamp, a weight hanger, slotted weights, a meter stick, and a sheet of Cartesian graph paper

__
Theory:__

In a medium with constant physical properties, mechanical waves
travel at a certain constant speed**.** The formula for this constant speed is

v = f λ ,

where, in SI units, f is
the frequency in Hz and λ is the
wavelength in meters**. ** If the properties of the medium change, then the wave
speed, v, changes**.** For a string, the
mass and length remain constant to a high degree if the string is put under
tension**.** Putting a string under tension affects the speed of waves in it**.** The
formula for this dependence is

where, in SI
units, F is the tension in the string in newtons, v is the wave speed in m/s,
and μ is the mass per unit length of the string in kg/m**.**
μ = M_{t} / L_{t }
= total mass/total length**.**

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Procedure:__

1.
Measure the mass of the string using a mass scale**.** (It is okay to use a
string that is much longer than is necessary**.** Do not cut the string.)

2.
Measure the length of the string, and calculate its mass per unit length,
μ, SI units**.**

3.
Mount a rod and C-clamp on the edge of the table, and use a skew clamp to
connect the string vibrator to the vertical rod**.**

4.
Mount another rod and C-clamp on the same edge (preferably at the corner
of the table), and attach the pulley system to it**.**

5.
Attach one end of the string to the string vibrator. Pass the other end
over the pulley and connect it to the weight hanger**.** See Fig.1
below**:**

6.
Adjust the elevations of the string vibrator and the pulley system such
that the string is level or parallel to the edge of the table**.** Also, make sure
that both the vibrator and pulley are at their lowest possible position to
minimize the torque they exert on the rods**.** This guarantees a firmer set up**.**
You may have to adjust the horizontal distance between the vibrator and the
pulley from some 70 cm to over 150 cm for more favorable results**.**

7.
Plug in the string vibrator to the electric socket and turn it on**.**

8.
Adjust the amount of weight suspended from the string until you obtain a
good standing-wave pattern. Try to develop patterns ranging from one to eight
loops if possible. (The meaning of the word “loop” as used here needs to be
clarified. A loop is a section of string between nodes, but the point at which
the string is attached to the vibrator is not strictly a node. Therefore, for
the example shown in figure 1 above, count **three** loops, **not four**.
Sections AB, BC, and CD are the three loops. The section from point D to the
point of attachment at the vibrator is not counted)**.**

9.
In the table shown below, record the **value of the suspended mass M**
and the **effective length L** of the string for each case. (The “effective
length” is the distance from the top of the pulley, where the string has a node,
to the last “true” node near the vibrator. For example, the effective length
for the case shown in figure 1 is the distance from point A to point D)**.**

10.
Complete the calculations for the other columns in the table under Data**.**

11. If your
measurements are correct, the measured value of frequency in the last column should be
close to 120Hz**.** This should be the case
for each trial or each line of the table**.**

12.
As shown in the Theory section, v = fλ,
and since v = (F/μ)^{0.5} ;
therefore,

fλ = (F/μ)^{0.5} .
Solving for λ
results in

This equation has the form y = mx + b, the equation of a straight line, where y corresponds to λ, and x corresponds to . The slope is therefore given by

.

Plot the
wavelength λ
versus
and fit a straight line to the data**.** Find the slope of the graph, and
calculate the frequency f, using the above equation**.** This is **the measured
value for frequency.** Compare it with the **accepted value **of 2 x 60**.**0
Hz = **120. Hz**, which is the frequency of oscillations of the string
vibrator, and calculate the percent error**.** The frequency of the vibrator is
120 Hz for the following reason**.** The frequency of the city electric outlet is
60**.**0 Hz, and in each **half-cycle** the coil (electromagnet) of the
vibrator attracts and releases the blade. In each **cycle,** the blade is
attracted and released twice**.** Therefore, the blade oscillates 120**.** times
per second**.**

Suspended Mass
(kg) |
Tension in the String F = M |
# of Loops
number |
Effective
Length (L) (m) |
Wavelength λ =2 (L/n)
(m) |
(N) |
v =
(F/μ)^{0.5}(m/s) |
f = v/λ Hz |

1 | |||||||

2 | |||||||

3 | |||||||

4 | |||||||

5 | |||||||

6 | |||||||

7 | |||||||

8 |

**Data:**

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__Given:__

f_{accepted}
= 120**.** Hz.

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__Measured:__

The measured values are recorded in the table.

__Calculations:__

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Calculations are performed in the table.

** **

__Comparison of the
Results:__

__Conclusion:__

To be explained by students

__Discussion:__

To be explained by students