4. Results and Discussion
4.3 Characterization of Temperature Effects on T5 110% Response
4.3.2 THz detection Vs Temperature
You now know that in this part of the experiment you have to keep mass per unit length of the wire and its frequency of vibration constant. The former of these can be accomplished by working with a wire of known material. To achieve the latter you can use either a tuning fork or an electromagnet. Of these two, an electromagnet is preferred because with its help the wire can be made to execute sustained vibrations.
Fig. 5.3: Experimental arrangement for setting up transverse stationary waves in a sonometer wire
The experimental arrangement is shown in Fig. 5.3. Connect the electromagnet to a 6 V transformer and place the electromagnet near the middle of the wire. When an alternating current (ac) is sent through the electromagnet, in each cycle the core is magnetised twice with opposite polarities. As a result, the sonometer wire is attracted by the electromagnet twice in each cycle and it begins to vibrate. Since the frequency of ac is 50 Hz, the wire will vibrate with a fixed frequency of 100 Hz.
SELF ASSESSMENT EXERCISE 1
Suppose that the electromagnet is connected to a source of direct current. Will the wire vibrate? If yes, what will be its frequency of vibration?
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Stretch the wire by putting weight of 0.5 kg in the hanger. (If a weight of M kg is used for stretching the wire then the tension in the wire will be T=mgNewton, where g is acceleration due to gravity. You can use g
= 10 m/s2.) Keep the bridges of the sonometer at a distance of about 25 cm. As soon as the current is switched on, the electromagnet is energised and you will observe that the wire begins to vibrate. This means that the apparatus is in working order and you can begin your investigations. When the sonometer wire vibrates in the fundamental mode, the distance between the two nodes is equal to half the wavelength of the stationary wave in the wire. The vibrating length of the wire will therefore be a measure of the wavelength of stationary waves set up in the wire. That is why we are interested in determining that length of the wire which vibrates in the fundamental mode with a frequency of 100 Hz. First of all, make the wire vibrate in one single
loop. Then to achieve unison, you first place a rider on the wire. Fix one of the bridges, say B1 and move the other bridge B2 towards it. What do you observe? Does the amplitude of vibration of the wire decrease? If so, then move the bridge B2 away fromB1. Continue to move it away from B1 till the amplitude becomes maximum. In this position, the rider will fall down. Measure the distance between the bridges accurately and record it in Observation Table 5.1. Next, you repeat the above procedure by keeping the bridges closer, separated by 10 cm. Move bridge B2 away from bridge B1 and note the length of the wire between the bridges at which the rider is again thrown off. Enter your reading in Observation Table 5.1,
Observation Table 5.1: Dependence of Wavelength on Tension Frequency of Vibration of the wire = 100 Hz Least count of metre scale = ...cm.
S.
No. Weight placed on the hanger
(kg)
Tension T=Mg
(N)
Length (l) of the wire between two bridges in unison with
electromagnet (cm)
Meanl
(cm)
Wav e-length
l
=2 λ
(m) ln T ln
λ
load increasing load decreasing when
bridges are far
apart
when bridges
are closer
when bridges
are far apart
when bridges
are closer 12
34 5
Now you change the tension in the wire by adding weights on the hanger in equal steps of, say, 0.5 kg and measure the resonating lengths of the wire in each case. Enter your data in Observation Table 5.1. You should not load the wire beyond its elastic limit.
To check that you are working within the permissible range, you should repeat the above — said procedure by unloading the wire in equal steps.
Tabulate your observations in each case. Do these lengths differ from those measured while loading the wire? We expect these to be almost
the same. If they differ significantly, you should discuss with your Counsellor. Calculate the mean length for a given tension.
From the table you will observe that λ changes with T. The variation in
λ suggests that it is related to tension. Mathematically, we can write
∝T λ
Can you give an exact relation between these variables by looking at your observations? Probably you cannot. To discover the exact relationship between λ and T, you can proceed along the lines suggested in the experiment on simple pendulum. That is, you may plot λ vs T1/2,
λ vs. T , λ vs T2 and soon. One of these plots will be a straight line.
For example, if λ vs T1/2 plot is a straight line passing through the origin and the slope of the line is k1, the exact relation between λ and T is given by λ =k1T1/2. Alternatively you can arrive at this relation as follows: Letλ ∝Tα ,
or λ=k1Ta (5.1) where k1 is the constant of proportionality and a is another constant.
Taking logarithms to the base e on both sides, we get
T a k ln ln
lnλ = 1 + (5.2)
So if you plot lnλ along the y-axis and ln T along the x-axis, the graph will be a straight line.
On comparing Eq. (5.2) with the equation of a straight line namely y = mx + c,
we find that the intercept on the y-axis gives lnk1, while the slope gives the value of a. Calculate the slope by using two well separated points on the straight line. We expect the value of a to be 1/2. What is your result?
Calculate the error in the slope by drawing lines of maximum and minimum slope.
Then the relation between λ and T is:
T k1
λ= (5.3)
SELF ASSESSMENT EXERCISE 2
Plot a graph between λ andT1/2. For T1= 64 N and T2= 324 N, calculate the ratio of wavelengths from your graph.
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