Sunday, October 2, 2011

Standing Waves on a Wire

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Standing Waves on a Wire

Aim

To understand about resonance, standing waves, wavelength, and superposition principle.


Introduction

Wave motion can accomplish the transfer of energy without the transport of matter between locations.

The distance between two adjacent crests on the wave is equal to the distance called wavelength. The time taken for a complete vibration of a point in the path of the wave is called period of the wave and is related and is represented T. The frequency of the wave is related to its period through f =1/T. Furthermore the frequency is equal to the number of wave crests passing through a specific point each second. The wavelength is the distance (d) that a wave travels during the period, the time (t) taken. Therefore,

v (speed of the wave) = 7(wavelength)/T (From v=d/t)

Thus

v (speed of the wave) = 7(wavelength) f (f =1/T)

The waves that we consider in this experiment are called transverse waves. The particles of the transverse waves move across to the direction in which the wave is propagating. The speed of propagation of transverse waves on a string (or wire) can be defined as

V (wave speed) = T(Tension)/8(linear density)

By using resonance(standing wave) method, we can calculate the velocity of the wave quite accurately by measuring the frequency and the length of the wave. By doing a set of experiments with different tensions on the string, we are able to see the relations between the velocity of the wave and the tension of the string and also can determine the mass per unit length of the string.


Equipment

• 1 V 50 Hz AC source

• Mounted magnet, lamp and wire guide setup

• Pulley with clamp attachment

• Mass holder with masses

• Set of white leads with spade connections


Method

1. Set up the apparatus as figure 1 and showing.

2. With a fixed tension (the mass holder with the masses) changing the positions of the bridge obtain

- Standing wave with one loop

- Standing wave with two loops

- Standing wave with three loops

3. Changing the tension find again the position of the loops shown above.

4. Find the wavelength and spend of propagation of the waves on each tension.

5. Obtain the mass per unit length of the wire (linear density)


Results

Quantity 1 loop loops loops wavelegth/ Wavelength speed of

propagation ¼V mass on end of wire

Tension in wire

unit m (+-0.001m) m (+-0.001m) m (+-0.001m) m (+-0.00m) m (+-0.00m) m/s (m/s) kg N

¼1 0.400 0.75 1.175 0.5 0.7 .5 1056.5 0.15 1.47

¼ 0.470 0.11 1.60 0.45 0.14 45.68 086.7 0.0 1.6

¼ 0.4 0.655 1.77 0. 0.658 . 108.40 0.10 0.8

¼4 0.8 0.478 0.70 0.5 0.4 4.5 6.50 0.05 0.4

¼5 0.51 1.150 1.484 0.54 1.04 5.4 74.5 0.5 .45


The speed of propagation is found from the equation

v (speed of the wave) = 7(wavelength) f


The relationship between speed of the wave and Tension is V (wave speed) = T(Tension)/8(linear density). If we say that V = y and T = x the equation can be written as y= 1/ 8 x, which is a linear equation. The graph we plot does not include any errors since we didn’t input errors in the graphing software when we did the experiment by using the graphing software ‘General Linear Plot’, we can plot a v/T graph, which would be linear and clear shows the relation ship between the wave’s velocity and the tension of the string. And we can calculate the mass per unit length of the wire by calculate the gradient of the line of best fit.

By inserting the data into the software, we are able to get the equation

y=.6E-04X+-.1E-0

The gradient of the graph is the reciprocal of the linear density. According to the equation, the gradient is .610¯$

G= .610¯$ Therefore 8 = .610¯$, since T=uV

Ie. Mass per unit length of wire (8) = .610¯$kg/m


Conclusion

From this experiment, we have measured wave length of the standing waves and known the frequency, so we are able to calculate the velocity of the wave. By adding masses on one side of the string, we are able to get the tension of the string thus we can calculate the mass per unit length of the wire by finding the relations between V and T, since T=uV, and it would give us a linear graph and by calculate the gradient of the line of best fit, we determined the mass per unit length (u).


Discussion and analysis

The experiment we have conducted has followed carefully designed procedure, which end up proven to be quite effective, and with great care from our group members.

The final results we were getting, shows the mass per unit length of the wire we have tested, which is .610¯$ kg/m (0.6g/m), is very reasonable. From the experiment results we have collected, we know that the greater tension of the wire, the greater velocity the wave travels, and longer the wave length. since the period of the waves are all 50Hz. Which has proved the relations between wave’s propagation speed and wave length mentioned in the introduction part is reasonable according the equation v (speed of the wave) = 7(wavelength) f.

Although the experiment we have conducted has followed carefully designed procedure, which end up proven to be quite effective, and with great care from our group members, there are still errors within the experiment and still can be improved by using more effective techniques during the experiment, they are:

1. Systematic errors there were no markings on the end of the ruler which can not be read accurately; the ruler we were using can’t measure anything smaller than 1mm.

2. Human errors people from other groups have been constantly hitting the pulley with the weight clamped on the other end of the bench, which might change the tension of the string during the experiment; the string might not be straight due to the bridge in the middle is not stationary; the wave we were measuring could not be the biggest wave due to errors made by our bear eyes; the technique we were using to measure the length of the wave may not be exact accurate since the person who hold the ruler might have moved while the other person’s doing the reading.

3. We did not include errors in the experiment which should not be happened again in the future

Overall, the experiment we have conducted is successful and we are satisfied with the result.



References

Physics Spectrum (Peter H. Eastwell) McGRAW-HILL BOOK COMPANY, Sydney

First Year Physics Laboratory Manual


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