Solve this following
Question: Show that function $f(x)=\left\{\begin{array}{c}\frac{x^{n}-1}{x-1}, \text { when } x \neq 1 ; \\ n, \text { when } x=1\end{array}\right.$ is continuous. Solution:...
Read More →A steel wire of mass 40g and length 80 cm is fixed
Question: A steel wire of mass $40 \mathrm{~g}$ and length $80 \mathrm{~cm}$ is fixed at the two ends. The tension in the wire is $50 \mathrm{~N}$. Find the frequency and wavelength of the fourth harmonic of the fundamental. Solution:...
Read More →A wire of 2oom is stretched in a tension of 160 N.
Question: A wire of $200 \mathrm{~m}$ is stretched in a tension of $160 \mathrm{~N}$. If the fundamental frequency of vibration is $100 \mathrm{~Hz}$, find its linear mass density. Solution:...
Read More →Solve this following
Question: Show that function $f(x)=\left\{\begin{array}{c}\sin x, \text { if } x0 ; \\ x, \text { if } x \geq 0\end{array}\right.$ is continuous. Solution:...
Read More →If the speed of a transverse wave on a stretched
Question: If the speed of a transverse wave on a stretched string of length $1 \mathrm{~m}$ is $60 \mathrm{~m} / \mathrm{s}$, what is the fundamental frequency of vibration? Solution:...
Read More →Two waves each having a frequency of
Question: Two waves each having a frequency of $100 \mathrm{~Hz}$ and a wavelength of $2.0 \mathrm{~cm}$, are travelling in the same direction on a string. What is the phase difference between the waves (a) If the second wave was produced $0.015 \mathrm{~s}$ later than the first one at the same place (b) If the two waves were produced at the same instant but the first one was produced at the same instant but the first one was produced at a distance $4.0 \mathrm{~cm}$ behind the second one? (c) I...
Read More →The given figure shows two wave pulses
Question: The given figure shows two wave pulses at $t=0$ travelling on a string in opposite direction with the same wave speed 50 $\mathrm{cm} / \mathrm{s}$. Sketch the shape of the string at $\mathrm{t}-4 \mathrm{~ms}, 6 \mathrm{~ms}, 8 \mathrm{~ms}$ and $12 \mathrm{~ms}$. Solution:...
Read More →Solve this following
Question: Show that function $f(x)=\left\{\begin{array}{l}(7 x+5), \text { when } x \geq 0 ; \\ (5-3 x), \text { when } x0\end{array}\right.$ is continuous function. Solution:...
Read More →Two waves, travelling in the same direction through the same region,
Question: Two waves, travelling in the same direction through the same region, have equal frequencies, wavelengths and amplitudes. If the amplitude pf each wave is $4 \mathrm{~mm}$ and the phase difference between the waves is $90^{\circ}$, what is the resultant amplitude? Solution:...
Read More →A tuning fork of frequency 400 Hz is attached
Question: A tuning fork of frequency $400 \mathrm{~Hz}$ is attached to a long string of linear mass density $0.01 \mathrm{~kg} / \mathrm{m} \mathrm{kept}$ under a tension $49 \mathrm{~N}$. The fork produces transverse waves of amplitude 0-50 mm on the string. (a) Find the wave speed and wavelength of the waves. (b) Find the maximum speed and acceleration of a particle of the string. (c) At what average rate is the tuning fork transmitting energy to the string? Solution:...
Read More →Prove that the function f given
Question: Prove that the function $f$ given $f(x)=|x-3|, x \in R$ is continuous but not differentiable at $x=3$ Solution:...
Read More →A transverse wave of 0.50mm and frequency
Question: A transverse wave of $0.50 \mathrm{~mm}$ and frequency of $100 \mathrm{~Hz}$ is produced on a wire stretched to a tension of $100 \mathrm{~N}$. If the wave speed is $100 \mathrm{~m} / \mathrm{s}$, what average power is the source transmitting to the wire? Solution:...
Read More →A circular loop of string rotates about its axis on a frictionless horizontal
Question: A circular loop of string rotates about its axis on a frictionless horizontal plane at a uniform rate so that the tangential speed of any particle of the string is $\mathrm{U}$. If a small transverse disturbance is produced at a point of the loop, with what speed (relative to the string) will this disturbance travel on the string? Solution:...
Read More →A heavy ball is suspended from the ceiling of a motor car
Question: A heavy ball is suspended from the ceiling of a motor car through a light string. A transverse pulse travels at a speed of 60 $\mathrm{cm} / \mathrm{s}$ on the string when the car is at rest and $62 \mathrm{~cm} / \mathrm{s}$ when the car accelerates on a horizontal road. Find the acceleration of the car. Tale $\mathrm{g}=10 \mathrm{~m} / \mathrm{s}^{2}$. Solution:...
Read More →Find the values of a and b such that the following functions f, defined as
Question: Find the values of $a$ and $b$ such that the following functions $f$, defined as $\left\{\begin{array}{l}\operatorname{asin} \frac{\pi}{2}(x+1), x \leq 0 \\ \frac{\tan x-\sin x}{x^{3}}, x0\end{array}\right.$ is continuous at $x=0$ Solution: $=\frac{1}{2}$...
Read More →Find the values of a and b such that the following functions continuous
Question: Find the values of $a$ and $b$ such that the following functions continuous Solution:...
Read More →Solve this following
Question: Show that: $f(x)=\left\{\begin{array}{l}x^{3}-3, \text { if } x \leq 2 \\ x^{2}+1, \text { if } x2\end{array}\right.$ is continuous at $x=2$ Solution:...
Read More →Solve this following
Question: Show that: $f(x)=\left\{\begin{array}{ll}x+1, \text { if } x \geq 1 \\ x^{2}+1, \text { if } x1\end{array}\right.$ is continuous at $x=1$ Solution:...
Read More →Show that function:
Question: Show that function: $f(x)=\left\{\begin{array}{c}x^{2} \sin \frac{1}{x}, \text { if } x \neq 0 ; \\ 0, \text { if } x=0\end{array}\right.$ is continuous at $x=0$ Solution:...
Read More →For what valve of k is the following function
Question: For what valve of $k$ is the following function $f(x)=\left\{\begin{array}{c}\frac{k \cos x}{\pi-2 x}, \text { if } x \neq \frac{\pi}{2} \\ 3, \text { if } x=\frac{\pi}{2}\end{array}\right.$ is continuous at $x=\frac{\pi}{2}$ Ans. $k=6$ Solution:...
Read More →For what valve of k is the following function
Question: For what valve of $k$ is the following function $f(x)=\left\{\begin{aligned} \frac{x^{2}-9}{x-3}, \text { when } x \neq 3 ; \text { is continuous at } x=3 \\ k, \text { when } x=3 \end{aligned}\right.$ Ans. $k=6$ Solution:...
Read More →For what valve of k is the following function continuous at
Question: For what valve of $k$ is the following function continuous at $x=2$ $f(x)=\left\{\begin{array}{r}2 x+1, \text { when } x2 \\ k, \text { when } x=2 \\ 3 x-1, \text { when } x2\end{array}\right.$ Solution:...
Read More →Find the value of for which
Question: Find the value of $\lambda$ for which Solution:...
Read More →Find the value of k for which
Question: Find the value of $k$ for which $f(x)=\left\{\begin{array}{c}\frac{\sin 2 x}{5 x}, \text { when } x \neq 0 ; \\ \lambda, \text { when } x=0\end{array}\right.$ is continuous at $x=0$ Solution:...
Read More →A 4.0 kg block is suspended from the ceiling of an elevator through
Question: A $4.0 \mathrm{~kg}$ block is suspended from the ceiling of an elevator through a string having a linear mass density of $19.2 \times 10^{-3} \mathrm{~kg} / \mathrm{m}$. Find the speed (with respect to the string) with which a wave pulse can proceed on the string if the elevator accelerates up at the rate of $2.0 \mathrm{~m} / \mathrm{s}^{2}$. Solution:...
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