In the arrangement shown in the given figure,
Question: In the arrangement shown in the given figure, the string has a mass of $4.5 \mathrm{~g}$. How much time will it take for a transverse disturbance produced at the floor to reach the pulley? take $g=10 \mathrm{~m} / \mathrm{s}^{2}$ Solution:...
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Question: Prove that $f(x)=\left\{\begin{array}{l}-x, \text { when } x0 \\ 1, \text { when } x=0 ; \text { is discontinuous at } x=0 \\ x, \text { when } x0\end{array}\right.$ Solution:...
Read More →Two blocks each having a mass of 3.2 kg are connected
Question: Two blocks each having a mass of $3.2 \mathrm{~kg}$ are connected by a wire CD and the system is suspended from the ceiling by another wire $A B$. The linear mass density of the wire $A B$ is $10 \mathrm{~g} / \mathrm{m}$ and that of $C D$ is $8 \mathrm{~g} / \mathrm{m}$. Find the speed of transverse wave pulse produced in $A B$ and in $C D$ Solution:...
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Question: Prove that $f(x)=\left\{\begin{array}{l}2 x, \text { when } x2 \\ 2, \text { when } x=2 ; \text { is discontinuous at } x=2 \\ x^{2}, \text { when } x2\end{array}\right.$ Solution:...
Read More →A string of length 40 cm and weighing 10 g is attached
Question: A string of length $40 \mathrm{~cm}$ and weighing $10 \mathrm{~g}$ is attached to a spring at one end and to a fixed wall at the other end. The spring has a spring constant of $160 \mathrm{~N} / \mathrm{m}$ and is stretched by $1.0 \mathrm{~cm}$. If a wave pulse is produced on the string near the wall, how much time will it take to reach the spring? Solution:...
Read More →A travelling wave is produced on a long horizontal string
Question: A travelling wave is produced on a long horizontal string by vibrating an end up and down sinusoidally. The amplitude of vibration is $1.0 \mathrm{~cm}$ and the displacement becomes zero 200 times per second. The linear mass density of the string is $0.10$ $\mathrm{kg} / \mathrm{m}$ and it is kept under a tension of $90 \mathrm{~N}$. a. Find the speed and the wavelength of the wave b. Assume that the wave moves in the positive $x$ direction and at $t=0$ the end $x=0$ is at its positive...
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Question: Prove that $f(x)=\left\{\begin{array}{c}\sin \frac{1}{x}, \text { when } x \neq 0 ; \\ 0, \text { when } x=0 ;\end{array}\right.$ is discontinuous at $x=0$ Solution:...
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Question: Prove that $f(x)=\left\{\begin{array}{c}\frac{1}{2}(x-|x|), \text { when } x \neq 0 ; \\ 2, \text { when } x=0\end{array}\right.$ is discontinuous at $x=0$ Solution:...
Read More →A transverse wave described by
Question: A transverse wave described by $y=(0.02 m) \sin \left[\left(1.0 \mathrm{~m}^{-1}\right) x+\left(30^{s^{-1}}\right) t\right]$ Propagates on a stretched string having a linear mass density of $1.2 \times 10 \mathrm{~m}^{-4} \mathrm{~kg} /$ $\mathrm{kg} / \mathrm{m}$. Find the tension in the string. Solution:...
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Question: Prove that $f(x)=\left\{\begin{aligned} \frac{|x-a|}{x-a}, \text { when } x \neq a ; \\ 1, \text { when } x=a \end{aligned}\right.$ is discontinuous at $x=a$ Solution:...
Read More →The wires of different densities but same area of cross
Question: The wires of different densities but same area of cross section are soldered together at one end and are stretched to a tension $\mathrm{T}$. The velocity of a transverse wave in the first wire is double of that in the second wire. Find the ratio of the density of the first wire to that of the second wire. Solution:...
Read More →A string of linear mass density 0.5 g/cm and a total length 30 cm
Question: A string of linear mass density $0.5 \mathrm{~g} / \mathrm{cm}$ and a total length $30 \mathrm{~cm}$ is tied to a fixed wall at one end and to a frictionless ring at the other end. The ring can move on a vertical rod. A wave pulse is produced on the string which moves towards the ring at a speed of $20 \mathrm{~cm} / \mathrm{s}$. The pulse is symmetric about its maximum which is located at a distance of $20 \mathrm{~cm}$ from the end joined to the ring. a. Assuming that the wave is ref...
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Question: Prove that $f(x)=\left\{\begin{aligned} \cos x, \text { when } x \geq 0 ; \\-\cos x, \text { when } x0 \end{aligned}\right.$ is discontinuous at $x=0$ Solution:...
Read More →A string of length 20 cm and linear mass density 0.40 g/cm
Question: A string of length $20 \mathrm{~cm}$ and linear mass density $0.40 \mathrm{~g} / \mathrm{cm}$ is fixed at both ends and is kept under a tension of $16 \mathrm{~N}$. A wave pulse is produced at $t=0$ near an end as shown in figure. Which travels towards the other end. (a) When will the string have the shape shown in the figure again? (b) Sketch the shape of the string at a time half that found in part (a) Solution:...
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Question: Prove that $f(x)=\left\{\begin{array}{c}x-1, \text { when } 1 \leq x2 ; \\ 2 x-3, \text { when } 2 \leq x \leq 3\end{array}\right.$ is continuous at $x=2$ Solution:...
Read More →14 A steel wire of length 64 cm weighs 5 g.
Question: A steel wire of length $64 \mathrm{~cm}$ weighs $5 \mathrm{~g}$. If it is stretched by a force of $8 \mathrm{~N}$, what would be the speed of transverse wave passing on it? Solution:...
Read More →A wave travelling on a string at a speed of 10 m/s causes
Question: A wave travelling on a string at a speed of $10 \mathrm{~m} / \mathrm{s}$ causes each particle of the string to oscillate with a time period of 20 ms. a. What is the wavelength of the wave b. If the displacement of a particle is $1.5 \mathrm{~mm}$ at a certain instant, what will be the displacement of a particle $10 \mathrm{~cm}$ away from it at the same instant? Solution:...
Read More →Figure shows a plot of the transverse displacements of
Question: Figure shows a plot of the transverse displacements of the particles of a string at $t=0$ through which a travelling wave is passing in the positive $x$-direction. The wave speed is $20 \mathrm{~cm} / \mathrm{s}$. Find (a) the amplitude (b) The wavelength (c) The wave number and (d) The frequency of the wave. Solution:...
Read More →A particle on a stretched string supporting a travelling
Question: A particle on a stretched string supporting a travelling wave takes $5.0$ ms to move its mean position to the extreme position. The distance between two consecutive particles which are at their mean position is $2.0 \mathrm{~cm}$. Find the frequency, the wavelength and the wave speed. Solution:...
Read More →A wave is described by the equation
Question: A wave is described by the equation $y=(1.0 \mathrm{~mm}) \sin \pi\left[\frac{\mathrm{x}}{2.0 \mathrm{~cm}}-\frac{\mathrm{t}}{0.01 \mathrm{~s}}\right]$ a. Fin the time period and the wave length. b. Write the equation for the velocity of the particles. Find the speed of the particle at $x=1 . p \mathrm{~cm}$ at the time $t=0.01 \mathrm{~s}$ c. What are the speeds of the particles at $x=3.0 \mathrm{~cm}, 5.0 \mathrm{~cm}$ and $7.0 \mathrm{~cm}$ at $t=0.01 \mathrm{~s}$ ? d. What are the ...
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Question: Prove that $f(x)=\left\{\begin{array}{l}5 x-4, \text { when } 0x \leq 1 \\ 4 x^{2}-3 x, \text { when } 1x2\end{array}\right.$ is continuous at $x=1$ Solution:...
Read More →A wave travels along the positive x-direction with
Question: A wave travels along the positive $x$-direction with a speed of $20 \mathrm{~m} / \mathrm{s}$. The amplitude of the wave is $0.20 \mathrm{~cm}$ and the wavelength is $2.0 \mathrm{~cm}$ (a) Write suitable wave equation which describes this wave (b) What is the displacement and velocity of the particle at $x=2.0 \mathrm{~cm}$ at time $t=0$ according to the wave equation written? Can you get different values of this quantity if the wave equation is written in a different fashion? Solution...
Read More →The equation of a wave travelling on a string is
Question: The equation of a wave travelling on a string is $\mathrm{y}=(0.10 \mathrm{~mm}) \sin \left[\left(31.4 \mathrm{~m}^{-1}\right) \mathrm{x}+\left(314 \mathrm{~s}^{-1}\right) \mathrm{t}\right]$ (a) In which direction does the wave travel (b) Find the wave speed the wavelength and the frequency of the wave. (c) What is the maximum displacement and the maximum speed of a portion of the string? Solution:...
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Question: Prove that $f(x)=\left\{\begin{array}{c}3-x, \text { when } x \leq 0 ; \\ x^{2}, \text { when } x0\end{array}\right.$ is discontinuous at $x=0$ Solution:...
Read More →A wave propagates on a string with a speed v.
Question: A wave propagates on a string with a speed $v$. The shape of the string at $t=t_{0}$ is given by $g\left(x, t_{0}\right)=A$ sin $(x / a)$. Write the wave equation for a general time t. Solution:...
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