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PHXI15:WAVES

355135 The ends of a stretched wire of length $L$ are fixed at $x = 0$ and $x = L$ . In one experiment, the displacement of the wire is $y_{1} = A \sin (\frac{π x}{L}) \sin$ $(ω t)$ and energy is $E_{1}$ and in another experiment its displacement is $y_{2} = A \sin (\frac{2 π x}{L}) \sin (2 ω t)$ and energy is $E_{2}$ . Then

1

E_{2} = E_{1}

2

E_{2} = 2 E_{1}

3

E_{2} = 4 E_{1}

4

E_{2} = 16 E_{1}

Explanation:

Consider a standing wave
$y = A \sin k x \sin ω t$
Consider a small element of mass $d m$ in the string. The kinetic energy of the
Element is $d k = \frac{1}{2} d m v_{p}^{2}$
$v_{p} = \frac{\partial y}{\partial t} = A ω \sin k x \cos ω t, d m = μ d x$
$d k = \frac{1}{2} (μ d x) A^{2} ω^{2} \sin^{2} k x \cos^{2} ω t$
$(μ = linear mass density)$
$d k = \frac{1}{2} μ A^{2} ω^{2} \sin^{2} (k x) d x (At t = 0)$
$k = \frac{1}{2} μ A^{2} ω^{2} \int_{0}^{L} \sin^{2} k x d x$
$= \frac{1}{4} μ A^{2} ω^{2} [L - \frac{1}{2 k} \sin 2 k L]$
$T . E = K \cdot E + P \cdot E$
$= \frac{1}{2} μ A^{2} ω^{2} [L - \frac{1}{2 k} \sin 2 k L] (P . E = K . E)$
For the first wave
$E_{1} = \frac{1}{2} μ A^{2} ω^{2} L (k = \frac{π}{L})$
$E_{2} = \frac{1}{2} μ A^{2} (2 ω)^{2} L (k = \frac{2 π}{L})$
$\frac{E_{1}}{E_{2}} = \frac{1}{4} \Rightarrow E_{2} = 4 E_{1}$

PHXI15:WAVES

355136 A stretched wire will produce a note of very high frequency, then the nature of the wire is

1 Thin and short wire of light material under high tension

2 Thick and short wire of light material under high tension

3 Thin and long wire of light material under high tension

4 Thin and short wire of heavy material under high tension.

Explanation:

Frequency of a stretched string is
$f = \frac{n}{2 L} \sqrt{\frac{T}{μ}}$
For $f$ to be high $L$ is small, $T$ is large and $μ$ is also small.

PHXI15:WAVES

355137 For superposition of two waves, the following is correct

1 They must have the same frequency and wavelength

2 They must have equal frequencies but may have unequal wavelengths

3 They must have the same wavelength, but may have different frequencies

4 They may have different wavelength and different frequencies

PHXI15:WAVES

355138 Which of the following is a standing wave equation?

1

y = a x^{2} - b t^{2}

2

y = A \sin ω t \cos k x

3

y = \cos^{2} (k x - ω t)

4

y = (x - v t)^{3}

PHXI15:WAVES

355135 The ends of a stretched wire of length $L$ are fixed at $x = 0$ and $x = L$ . In one experiment, the displacement of the wire is $y_{1} = A \sin (\frac{π x}{L}) \sin$ $(ω t)$ and energy is $E_{1}$ and in another experiment its displacement is $y_{2} = A \sin (\frac{2 π x}{L}) \sin (2 ω t)$ and energy is $E_{2}$ . Then

1

E_{2} = E_{1}

2

E_{2} = 2 E_{1}

3

E_{2} = 4 E_{1}

4

E_{2} = 16 E_{1}

Explanation:

Consider a standing wave
$y = A \sin k x \sin ω t$
Consider a small element of mass $d m$ in the string. The kinetic energy of the
Element is $d k = \frac{1}{2} d m v_{p}^{2}$
$v_{p} = \frac{\partial y}{\partial t} = A ω \sin k x \cos ω t, d m = μ d x$
$d k = \frac{1}{2} (μ d x) A^{2} ω^{2} \sin^{2} k x \cos^{2} ω t$
$(μ = linear mass density)$
$d k = \frac{1}{2} μ A^{2} ω^{2} \sin^{2} (k x) d x (At t = 0)$
$k = \frac{1}{2} μ A^{2} ω^{2} \int_{0}^{L} \sin^{2} k x d x$
$= \frac{1}{4} μ A^{2} ω^{2} [L - \frac{1}{2 k} \sin 2 k L]$
$T . E = K \cdot E + P \cdot E$
$= \frac{1}{2} μ A^{2} ω^{2} [L - \frac{1}{2 k} \sin 2 k L] (P . E = K . E)$
For the first wave
$E_{1} = \frac{1}{2} μ A^{2} ω^{2} L (k = \frac{π}{L})$
$E_{2} = \frac{1}{2} μ A^{2} (2 ω)^{2} L (k = \frac{2 π}{L})$
$\frac{E_{1}}{E_{2}} = \frac{1}{4} \Rightarrow E_{2} = 4 E_{1}$

PHXI15:WAVES

355136 A stretched wire will produce a note of very high frequency, then the nature of the wire is

1 Thin and short wire of light material under high tension

2 Thick and short wire of light material under high tension

3 Thin and long wire of light material under high tension

4 Thin and short wire of heavy material under high tension.

Explanation:

Frequency of a stretched string is
$f = \frac{n}{2 L} \sqrt{\frac{T}{μ}}$
For $f$ to be high $L$ is small, $T$ is large and $μ$ is also small.

PHXI15:WAVES

355137 For superposition of two waves, the following is correct

1 They must have the same frequency and wavelength

2 They must have equal frequencies but may have unequal wavelengths

3 They must have the same wavelength, but may have different frequencies

4 They may have different wavelength and different frequencies

PHXI15:WAVES

355138 Which of the following is a standing wave equation?

1

y = a x^{2} - b t^{2}

2

y = A \sin ω t \cos k x

3

y = \cos^{2} (k x - ω t)

4

y = (x - v t)^{3}

PHXI15:WAVES

355135 The ends of a stretched wire of length $L$ are fixed at $x = 0$ and $x = L$ . In one experiment, the displacement of the wire is $y_{1} = A \sin (\frac{π x}{L}) \sin$ $(ω t)$ and energy is $E_{1}$ and in another experiment its displacement is $y_{2} = A \sin (\frac{2 π x}{L}) \sin (2 ω t)$ and energy is $E_{2}$ . Then

1

E_{2} = E_{1}

2

E_{2} = 2 E_{1}

3

E_{2} = 4 E_{1}

4

E_{2} = 16 E_{1}

Explanation:

Consider a standing wave
$y = A \sin k x \sin ω t$
Consider a small element of mass $d m$ in the string. The kinetic energy of the
Element is $d k = \frac{1}{2} d m v_{p}^{2}$
$v_{p} = \frac{\partial y}{\partial t} = A ω \sin k x \cos ω t, d m = μ d x$
$d k = \frac{1}{2} (μ d x) A^{2} ω^{2} \sin^{2} k x \cos^{2} ω t$
$(μ = linear mass density)$
$d k = \frac{1}{2} μ A^{2} ω^{2} \sin^{2} (k x) d x (At t = 0)$
$k = \frac{1}{2} μ A^{2} ω^{2} \int_{0}^{L} \sin^{2} k x d x$
$= \frac{1}{4} μ A^{2} ω^{2} [L - \frac{1}{2 k} \sin 2 k L]$
$T . E = K \cdot E + P \cdot E$
$= \frac{1}{2} μ A^{2} ω^{2} [L - \frac{1}{2 k} \sin 2 k L] (P . E = K . E)$
For the first wave
$E_{1} = \frac{1}{2} μ A^{2} ω^{2} L (k = \frac{π}{L})$
$E_{2} = \frac{1}{2} μ A^{2} (2 ω)^{2} L (k = \frac{2 π}{L})$
$\frac{E_{1}}{E_{2}} = \frac{1}{4} \Rightarrow E_{2} = 4 E_{1}$

PHXI15:WAVES

355136 A stretched wire will produce a note of very high frequency, then the nature of the wire is

1 Thin and short wire of light material under high tension

2 Thick and short wire of light material under high tension

3 Thin and long wire of light material under high tension

4 Thin and short wire of heavy material under high tension.

Explanation:

Frequency of a stretched string is
$f = \frac{n}{2 L} \sqrt{\frac{T}{μ}}$
For $f$ to be high $L$ is small, $T$ is large and $μ$ is also small.

PHXI15:WAVES

355137 For superposition of two waves, the following is correct

1 They must have the same frequency and wavelength

2 They must have equal frequencies but may have unequal wavelengths

3 They must have the same wavelength, but may have different frequencies

4 They may have different wavelength and different frequencies

PHXI15:WAVES

355138 Which of the following is a standing wave equation?

1

y = a x^{2} - b t^{2}

2

y = A \sin ω t \cos k x

3

y = \cos^{2} (k x - ω t)

4

y = (x - v t)^{3}

PHXI15:WAVES

355135 The ends of a stretched wire of length $L$ are fixed at $x = 0$ and $x = L$ . In one experiment, the displacement of the wire is $y_{1} = A \sin (\frac{π x}{L}) \sin$ $(ω t)$ and energy is $E_{1}$ and in another experiment its displacement is $y_{2} = A \sin (\frac{2 π x}{L}) \sin (2 ω t)$ and energy is $E_{2}$ . Then

1

E_{2} = E_{1}

2

E_{2} = 2 E_{1}

3

E_{2} = 4 E_{1}

4

E_{2} = 16 E_{1}

Explanation:

Consider a standing wave
$y = A \sin k x \sin ω t$
Consider a small element of mass $d m$ in the string. The kinetic energy of the
Element is $d k = \frac{1}{2} d m v_{p}^{2}$
$v_{p} = \frac{\partial y}{\partial t} = A ω \sin k x \cos ω t, d m = μ d x$
$d k = \frac{1}{2} (μ d x) A^{2} ω^{2} \sin^{2} k x \cos^{2} ω t$
$(μ = linear mass density)$
$d k = \frac{1}{2} μ A^{2} ω^{2} \sin^{2} (k x) d x (At t = 0)$
$k = \frac{1}{2} μ A^{2} ω^{2} \int_{0}^{L} \sin^{2} k x d x$
$= \frac{1}{4} μ A^{2} ω^{2} [L - \frac{1}{2 k} \sin 2 k L]$
$T . E = K \cdot E + P \cdot E$
$= \frac{1}{2} μ A^{2} ω^{2} [L - \frac{1}{2 k} \sin 2 k L] (P . E = K . E)$
For the first wave
$E_{1} = \frac{1}{2} μ A^{2} ω^{2} L (k = \frac{π}{L})$
$E_{2} = \frac{1}{2} μ A^{2} (2 ω)^{2} L (k = \frac{2 π}{L})$
$\frac{E_{1}}{E_{2}} = \frac{1}{4} \Rightarrow E_{2} = 4 E_{1}$

PHXI15:WAVES

355136 A stretched wire will produce a note of very high frequency, then the nature of the wire is

1 Thin and short wire of light material under high tension

2 Thick and short wire of light material under high tension

3 Thin and long wire of light material under high tension

4 Thin and short wire of heavy material under high tension.

Explanation:

Frequency of a stretched string is
$f = \frac{n}{2 L} \sqrt{\frac{T}{μ}}$
For $f$ to be high $L$ is small, $T$ is large and $μ$ is also small.

PHXI15:WAVES

355137 For superposition of two waves, the following is correct

1 They must have the same frequency and wavelength

2 They must have equal frequencies but may have unequal wavelengths

3 They must have the same wavelength, but may have different frequencies

4 They may have different wavelength and different frequencies

PHXI15:WAVES

355138 Which of the following is a standing wave equation?

1

y = a x^{2} - b t^{2}

2

y = A \sin ω t \cos k x

3

y = \cos^{2} (k x - ω t)

4

y = (x - v t)^{3}

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