The transverse displacement of a wire (clamped at both its ends) is described as :
y (x, t) = 0.06 sin(2π/3x) cos(120πt)
The mass of the wire is 6 x 10-2 kg and its length is 3m. Provide answers to the following questions:
( I ) Is the function describing a stationary wave or a travelling wave?
( ii ) Interpret the wave as a superposition of two waves travelling in opposite directions. Find the speed, wavelength and frequency of each wave.
( iii ) Calculate the wire’s tension. [X and y are in meters and t in secs]
We know,
The standard equation of a stationary wave is described as:
y (x, t) = 2a sin kx cos ωt
Our given equation y (x, t) =0.06sin(2π/3x)cos(120πt) is similar to the general equation .
( i ) Thus, the given function describes a stationary wave.
( ii ) We know, a wave travelling in the positive x – direction can be represented as :
y1 = a sin( ωt – kx )
Also,
Wave travelling in the negative x – direction is represented as :
y2 = a sin( ωt + kx )
Super- positioning these two waves gives us :
y = y1 + y2
= a sin( ωt – kx ) – a sin( ωt + kx )
= asin(ωt)cos(kx) – asin(kx)cos(ωt) – asin(ωt)cos(kx) – asin(kx)cos(ωt)
= – 2asin(kx)cos(ωt)
= −2asin(2πλx)cos(2πvt) . . . . . . . . . . . . . . ( 1 )
The transverse displacement of the wires is described as :
0.06sin(2π3x)cos(120πt) . . . . . . . . . . . . . . . . .( 2 )
Comparing equations ( 1 ) and ( 2 ) , we get :
2π/ λ = 2π/ 3
Therefore, wavelength λ = 3m
Also, 2πv/λ = 120π
Therefore, speed v = 180 m/s
And, Frequency = v/λ = 180/3
= 60 Hz
( iii )Given, Velocity of the transverse wave, v = 180 m / s
The string’s mass, m = 6 × 10-2 kg
String length, l = 3 m
Mass per unit length of the string, μ = m/l = (6 x 10-2 )/3
= 2 x 10-2 kg/m
Let the tension in the wire be T
Therefore, T = v2 μ
= 1802 x 2 x 10-2
= 648 N.
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(f) 0.0006032 m2
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Fill in the blanks by suitable conversion of units:
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A physical quantity P is related to four observables a, b, c and d as follows :
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On an open ground, a motorist follows a track that turns to his left by an angle of 60° after every 500 m. Starting from a given turn, specify the displacement of the motorist at the third, sixth and eighth turn. Compare the magnitude of the displacement with the total path length covered by the motorist in each case.
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A transverse harmonic wave on a string is described by
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(b) What are its amplitude and frequency?
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On an open ground, a motorist follows a track that turns to his left by an angle of 60° after every 500 m. Starting from a given turn, specify the displacement of the motorist at the third, sixth and eighth turn. Compare the magnitude of the displacement with the total path length covered by the motorist in each case.
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