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Chapter 10: Problem 1467

A wave \(\mathrm{y}=\mathrm{a} \sin (\omega \mathrm{t}-\mathrm{kx})\) on a string meets with another wave producing a node at \(\mathrm{x}=0 .\) Then the equation of the unknown wave is \(\ldots \ldots \ldots\) (A) \(y=a \sin (\omega t+k x)\) (B) \(\mathrm{y}=-\mathrm{a} \sin (\omega \mathrm{t}+\mathrm{kx})\) (C) \(\mathrm{y}=\mathrm{a} \sin (\omega \mathrm{t}-\mathrm{kx})\) (D) \(\mathrm{y}=-\mathrm{a} \sin (\omega \mathrm{t}-\mathrm{kx})\)

Short Answer

Expert verified
The equation of the unknown wave is \(y = -a \sin(\omega t - kx)\) (option D).

Step by step solution

01

Superposition of waves

When two waves interfere, their amplitudes add together by the principle of superposition. Let the unknown wave be Y(x, t), then the resulting wave after the interference will be y(x, t) + Y(x, t). #Step 2: Apply condition of node at x = 0#
02

Node condition

If there is a node at x = 0, then the total displacement at this point must be zero. We can write this as y(0, t) + Y(0, t) = 0. #Step 3: Substitute x = 0 into the condition and solve for Y(0, t)#
03

Substitute and solve

Using the equation of the first wave, we can write y(0, t) = a sin(ωt - k * 0) = a sin(ωt). Substitute this into the node condition: a sin(ωt) + Y(0, t) = 0 ⇒ Y(0, t) = -a sin(ωt) #Step 4: Identify the matching wave equation#
04

Compare with the given options

Now we compare our result, Y(0, t) = -a sin(ωt), with the given options to find the most suitable option. (A) y=a sin(ωt + kx): The sine term is not inverted, so this is not the correct option. (B) y=-a sin(ωt + kx): The sine term is inverted, but the wave is traveling in the same direction as the first wave, so this is not the correct option. (C) y=a sin(ωt - kx): The equation is the same as the first wave, so this is not the correct option. (D) y=-a sin(ωt - kx): The sine term is inverted, and the wave travels in the same direction as the first wave, so this is the correct option. Thus, the equation of the unknown wave is \(y = -a \sin(\omega t - kx)\) (option D).

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Most popular questions from this chapter

If the equation for displacement of two particles executing S.H.M. is given by \(\mathrm{y}_{1}=2 \sin (10 \mathrm{t}+\theta)\) and $\mathrm{y}_{2}=3 \cos 10 \mathrm{t}$ respectively, then the phase difference between the velocity of two particles will be \(\ldots \ldots \ldots\) (A) \(-\theta\) (B) \(\theta\) (C) \(\theta-(\pi / 2)\) (D) \(\theta+(\pi / 2)\).

A closed organ pipe has fundamental frequency \(100 \mathrm{~Hz}\). What frequencies will be produced if its other end is also opened? (A) \(200,400,600,800 \ldots \ldots\) (B) \(200,300,400,500 \ldots \ldots\) (C) \(100,300,500,700 \ldots \ldots\) (D) \(100,200,300,400 \ldots \ldots\)

The amplitude for a S.H.M. given by the equation $\mathrm{x}=3 \sin 3 \mathrm{pt}+4 \cos 3 \mathrm{pt}\( is \)\ldots \ldots \ldots \ldots \mathrm{m}$ (A) 5 (B) 7 (C) 4 (D) \(3 .\)

When a block of mass \(\mathrm{m}\) is suspended from the free end of a massless spring having force constant \(\mathrm{k}\), its length increases by y. Now when the block is slightly pulled downwards and released, it starts executing S.H.M with amplitude \(\mathrm{A}\) and angular frequency \(\omega\). The total energy of the system comprising of the block and spring is \(\ldots \ldots \ldots\) (A) \((1 / 2) \mathrm{m} \omega^{2} \mathrm{~A}^{2}\) (B) \((1 / 2) m \omega^{2} A^{2}+(1 / 2) \mathrm{ky}^{2}\) (C) \((1 / 2) \mathrm{ky}^{2}\) (D) \((1 / 2) m \omega^{2} A^{2}-(1 / 2) k y^{2}\)

The equation for displacement of a particle at time \(t\) is given by the equation \(\mathrm{y}=3 \cos 2 \mathrm{t}+4 \sin 2 \mathrm{t}\). The periodic time of oscillation is \(\ldots \ldots \ldots \ldots\) (A) \(2 \mathrm{~s}\) (B) \(\pi \mathrm{s}\) (C) \((\pi / 2) \mathrm{s}\) (D) \(2 \pi \mathrm{s}\)
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