The frequency of the first overtone of a closed pipe of length \(L_1\), is equal to that of the first overtone of an open pipe of length \(L_2\). The ratio of their lengths \((L_1 : L_2)\) is:
1. 2 : 3
2. 4 : 5
3. 3 : 5
4. 3 : 4
View Answer
The first overtone of a closed pipe of length \(L_1\) has frequency \(f_{c,1} = \frac{3v}{4L_1}\) and that of an open pipe of length \(L_2\) is \(f_{o,1} = \frac{v}{L_2}\). Equating the two frequencies gives \(\frac{3v}{4L_1} = \frac{v}{L_2}\), which simplifies to \(\frac{L_1}{L_2} = \frac{3}{4}\).
Given below are two statements:
Assertion (A): Sound would travel faster on a hot summer day than on a cold winter day.
Reason (R): Velocity of sound is directly proportional to the square root of its absolute temperature.
1. Both (A) and (R) are true and (R) is the correct explanation of (A).
2. Both (A) and (R) are true but (R) is not the correct explanation of (A).
3. (A) is true but (R) is false.
4. (A) is false but (R) is true.
View Answer
The speed of sound in a gas is \(v = \sqrt{\frac{\gamma RT}{M}}\), meaning \(v \propto \sqrt{T}\). Since temperature is higher on a hot summer day than a cold winter day, sound travels faster in summer. Both statements are true and Reason is the correct explanation.
The fundamental frequency of a sonometer wire increases by 6 Hz. If its tension is increased by 44%, keeping the length constant. Then find this fundamental frequency:
1. 28 Hz
2. 30 Hz
3. 33 Hz
4. 42 Hz
View Answer
Since frequency \(f \propto \sqrt{T}\), increasing tension by 44% makes \(f' = f \sqrt{1.44} = 1.2f\). Thus, the change in frequency is \(0.2f = 6 \text{ Hz}\), which gives \(f = 30 \text{ Hz}\).
Given below are two statements: one is labelled as Assertion (A) and the other is labelled as Reason (R).
Assertion (A): The presence of moisture increases the velocity of sound in air.
Reason (R): Density of moist air is more than the density of dry air.
In the light of the above statements, the correct option is
1. Both (A) and (R) are true and (R) is the correct explanation of (A)
2. Both (A) and (R) are true but (R) is not the correct explanation of (A)
3. (A) is true but (R) is false
4. Both (A) and (R) are false
View Answer
Velocity of sound \(v = \sqrt{\frac{\gamma P}{\rho}}\). Since water vapor has a lower density than dry air, moist air has a lower density, raising the sound velocity. Thus, (A) is true but (R) is false.
A string is fixed at both ends and the vibrations of string is given by the equation \(y = 10sin(2x)cos(2t)\) where \(x, y\) are in cm and \(t\) is in second. Nearby node from left end at \(x = 0\), is at a distance
1. \(\frac{\pi}{2}\text{ cm}\)
2. \(\pi\text{ cm}\)
3. 2 cm
4. 4 cm
View Answer
Nodes occur where the spatial amplitude term \(sin(2x) = 0\), which implies \(2x = n\pi\) or \(x = \frac{n\pi}{2}\). The closest node to the left end \(x=0\) (where \(n=1\)) is at \(x = \frac{\pi}{2}\text{ cm}\).
A wave travelling in the positive x-direction having displacement amplitude along y-direction as 1 m, wavelength \(2\pi\text{ m}\) and frequency of \(\frac{1}{\pi}\text{ Hz}\) is represented by
1. \(y = \sin(10\pi x - 20\pi t)\)
2. \(y = \sin(2\pi x + 2\pi t)\)
3. \(y = \sin(x - 2t)\)
4. \(y = \sin(2\pi x - 2\pi t)\)
View Answer
Wave equation is \(y = A\sin(kx - \omega t)\). Here, \(A = 1\text{ m}\), \(k = \frac{2\pi}{\lambda} = 1\text{ m}^{-1}\), and \(\omega = 2\pi f = 2\text{ rad/s}\). Thus, \(y = \sin(x - 2t)\).