Locate the point of discontinuity of the function
Question: Locate the point of discontinuity of the function $f(x)=\left\{\begin{aligned}\left(x^{3}-x^{2}+2 x-2\right), \text { if } x \neq 1 \\ 4, \text { if } x=0 \end{aligned}\right.$ Solution:...
Read More →A 40 cm wire having a mass of 3.2g
Question: A $40 \mathrm{~cm}$ wire having a mass of $3.2 \mathrm{~g}$ is stretched between two fixed supports $40.05 \mathrm{~cm}$ apart. In its fundamental mode, the wire vibrates at $220 \mathrm{~Hz}$. If the area of cross-section of the wire is $1.0 \mathrm{~mm}^{2}$, find the Young's modulus. Solution:...
Read More →The equation of a standing wave,
Question: The equation of a standing wave, produced on a string fixed at both ends, is $\left.\mathrm{y}=(0.4 \mathrm{~cm}) \sin \left[\left(0.314 \mathrm{~cm}^{-1}\right) \mathrm{x}\right] \cos \left[600^{\pi \mathrm{s}^{-1}}\right) \mathrm{t}\right]$ What could be the smallest length of the string? Solution:...
Read More →Solve this following
Question: Show that $f(x)=\left\{\begin{array}{ll}x, \text { if } x \neq 0 ; \\ 1, \text { if } x=0\end{array}\right.$ is continuous at each point except $0 .$ Solution:...
Read More →The equation for the vibration of a string,
Question: The equation for the vibration of a string, fixed at both ends vibrating in its third harmonic is given by $\mathrm{y}=(0.4 \mathrm{~cm}) \sin \left[\left(0.314 \mathrm{~cm}^{-1}\right) \mathrm{x}\right] \cos \left[\left(600 \pi \mathrm{s}^{-1}\right) \mathrm{t}\right]$ a. At what frequency of vibration? b. What are the position of the nodes? c. What is the length of the string? d. What is the wavelength and the speed of two travelling waves that can interfere to give this vibration? S...
Read More →Solve this following
Question: Show that $f(x)=\left\{\begin{array}{cl}(2 x-1), \text { if } x2 \\ \frac{3 x}{2}, \text { if } x \geq 2\end{array}\right.$ is continuous. Solution:...
Read More →A 2m long string fixed at both ends is set into vibrations in its first overtone.
Question: A $2 \mathrm{~m}$ long string fixed at both ends is set into vibrations in its first overtone. The wave speed on the string is $200 \mathrm{~m} / \mathrm{s}$ and the amplitude is $0.5 \mathrm{~cm}$ a. Find the wavelength and frequency b. Write the equation giving the displacement of different points as a function of time. Choose the $X$-axis along the string with the origin at one end and $t=0$ at the instant when the point $x=50 \mathrm{~cm}$ has reached its maximum displacement. Solu...
Read More →Solve this following
Question: Discuss the continuity of $f(x)=[x]$. Solution: Therefore, $f(x)=[x]$ is discontinuous at $x=n$....
Read More →A string of length L fixed at both ends vibrates in its fundamental
Question: A string of length $L$ fixed at both ends vibrates in its fundamental mode at a frequency $v$ and a maximum amplitude A. a. Find the wavelength and the wave number $k$. b. Take the origin at one end of the string and the $X$-axis along the string. Take the $Y$-axis along the direction of the displacement. Take $\mathrm{t}=0$ at the instant when the middle point of the string passes through its mean position and is going towards the positive $y$-direction. Write the equation describing ...
Read More →The given figure shows an aluminum wire of length
Question: The given figure shows an aluminum wire of length $60 \mathrm{~cm}$ joined to a steel wire of length $80 \mathrm{~cm}$ and stretched between two fixed supports. The tension produced is $40 \mathrm{~N}$. The cross-sectional area of the steel wire is $1.0 \mathrm{~mm}{ }^{2}$ and that of the aluminum wire is $3.001 .0 \mathrm{~mm}^{2}$. What could be the minimum frequency of a tuning fork which can produce standing waves in the Solution:...
Read More →Solve this following
Question: Show that function $f(x)=\left\{\begin{array}{c}(2-x), \text { when } x \geq 1 ; \\ x, \text { when } 0 \leq x \leq 1\end{array}\right.$ is continuous. Solution:...
Read More →A uniform horizontal rod of length 40 cm and mass 1.2kg
Question: A uniform horizontal rod of length $40 \mathrm{~cm}$ and mass $1.2 \mathrm{~kg}$ is supported by two identical wires as shown in figure. Where should a mass of $4.8 \mathrm{~kg}$ be placed on rod so that the same tuning fork may excite the wire on left into its fundamental vibrations and that on right into its first overtone? Take $\mathrm{g}=10 \mathrm{~m} / \mathrm{s}^{2}$ Solution:...
Read More →Two wires are kept tight between the same pair of supports.
Question: Two wires are kept tight between the same pair of supports. The tensions in the wires are in the ratio $2: 1$, the radii are in the ratio $3: 1$ and the densities are in the ratio $1: 2$. Find the ratio of their fundamental frequencies. Solution:...
Read More →Three resonant frequencies of a string are
Question: Three resonant frequencies of a string are $90,150,210 \mathrm{~Hz}$. a. Find the highest possible fundamental frequency of vibration of this string. b. Which harmonics of the fundamental are the given frequencies? c. Which overtones are these frequencies d. If the length of the string is $80 \mathrm{~cm}$, what would be the speed of the transverse wave on this string? Solution:...
Read More →A steel wire fixed at both ends has a fundamental frequency
Question: A steel wire fixed at both ends has a fundamental frequency of $200 \mathrm{~Hz}$. Aa person can hear sound of maximum frequency 14Khz. What is the highest harmonic that can be played on this string which is audible to the person? Solution:...
Read More →A particular guitar wire is 30.0cm long and vibrates
Question: A particular guitar wire is $30.0 \mathrm{~cm}$ long and vibrates at a frequency of $196 \mathrm{~Hz}$ when no finger is placed on it. The next higher notes on the scale are $220 \mathrm{~Hz}, 247 \mathrm{~Hz}, 262 \mathrm{~Hz}$ and $294 \mathrm{~Hz}$. How far from the end of the string must the finger be placed to play these notes? Solution:...
Read More →A 660 Hz tuning fork sets up vibration in a string clamped
Question: A $660 \mathrm{~Hz}$ tuning fork sets up vibration in a string clamped at both ends. The wave speed for a transverse wave on this string is $220 \mathrm{~m} / \mathrm{s}$ and the string vibrates in here loops a. Find the length of the string b. If the maximum amplitude of a particle is $0.5 \mathrm{~cm}$, write a suitable equation describing the motion. Solution:...
Read More →A string, fixed at both ends,
Question: A string, fixed at both ends, vibrates in a resonant mode with a separation of $20 \mathrm{~cm}$ between the consecutive nodes. For the next higher resonant frequency, this separation is reduced by $16 \mathrm{~cm}$. Find the length of the string. Solution:...
Read More →A wire, fixed at both ends is seen to vibrate at a resonant frequency
Question: A wire, fixed at both ends is seen to vibrate at a resonant frequency of $240 \mathrm{~Hz}$ and also at $300 \mathrm{~Hz}$ a. What could be the maximum value of the fundamental frequency? b. If transverse waves can travel on this string at a speed $0 f 40 \mathrm{~m} / \mathrm{s}$, what is its length? Solution:...
Read More →Show that sec $|x|$ is a continuous function
Question: Show that sec $|x|$ is a continuous function. Solution:...
Read More →A one meter long stretched string having a mass of
Question: A one meter long stretched string having a mass of $40 \mathrm{~g}$ is attached to a tuning fork. The fork vibrates 128 Hz in a direction perpendicular to the string. What should be the tension in the string if it is to vibrate in four loops? Solution:...
Read More →The length of the wire shown in figure between
Question: The length of the wire shown in figure between the pulleys is $1.5 \mathrm{~m}$ and its mass is $120 \mathrm{~g}$. Find the frequency of vibration with which the wire vibrates in two loops leaving the middle point of the wire between the pulleys at rest. Solution:...
Read More →Solve this following
Question: Show that $\sec \mathrm{x}$ is a continuous function. Solution:...
Read More →A sonometer wire having a length of 1.50m
Question: A sonometer wire having a length of $1.50 \mathrm{~m}$ between the bridges vibrates in its second harmonic resonance with a tuning fork of frequency $256 \mathrm{~Hz}$. What is the speed of the transverse wave on the wire? Solution:...
Read More →A piano wire weighing 6.00g and having a length of 90.0cm
Question: A piano wire weighing $6.00 \mathrm{~g}$ and having a length of $90.0 \mathrm{~cm}$ emits a fundamental frequency of vibration is 100 Hz, find its linear mass density. Solution:...
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