Assertion (A): If two waves of same amplitude produce a resultant wave of same amplitude, then the phase difference between them will be \(120^\circ\).
Reason (R): The resultant amplitude of two waves is equal to sum of amplitude of two waves.
1. Both (A) & (R) are true and the (R) is the correct explanation of the (A)
2. Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
3. (A) is true but (R) is false
4. Both (A) and (R) are false
View Answer
For two waves of amplitude \(A\) and phase difference \(phi\), the resultant amplitude is \(A_r = 2A \cos(\frac{\phi}{2})\). Given \(A_r = A\), so \(A = 2A \cos(\frac{\phi}{2})\), which means \(cos(\frac{\phi}{2}) = \frac{1}{2}\). Thus \(\frac{\phi}{2} = 60^\circ\), so \(\phi = 120^\circ\). Hence (A) is true. The resultant amplitude is the sum only if \(\phi = 0\). So (R) is false.
Assertion (A): In a sinusoidal travelling wave on a string potential energy of deformation of string element at extreme position is maximum.
Reason (R): The particles in sinusoidal travelling wave perform SHM.
1. Both (A) & (R) are true and the (R) is the correct explanation of the (A)
2. Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
3. (A) is true but (R) is false
4. Both (A) and (R) are false
View Answer
In a wave, potential energy is stored due to deformation (strain). At extreme positions (maximum displacement), the deformation is maximum, leading to maximum potential energy. So (A) is true.
Particles in a transverse wave undergo simple harmonic motion. So (R) is true. However, (R) does not explain why potential energy is maximum at extreme positions; it's a general characteristic of the particle motion. Therefore, (R) is not the correct explanation of (A).
Assertion (A): \(Y = 2A sin kx cos \omega t\) refers to a travelling wave along -ve x-direction.
Reason (R): When a continuous travelling wave interacts with its reflection from a rigid support, forms a standing wave.
1. Both (A) & (R) are true and the (R) is the correct explanation of the (A)
2. Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
3. (A) is true but (R) is false
4. Both (A) and (R) are false
View Answer
The equation \(Y = 2A sin kx cos \omega t\) represents a standing wave, not a travelling wave. Thus (A) is false. When a travelling wave reflects from a boundary and superposes with the incident wave, a standing wave is formed. Thus (R) is true. Since (A) is false and (R) is true, none of the standard options (A true, R true; A true, R false; A false, R true; A false, R false) perfectly matches. However, given the provided options, and (A) being false, option (4) is selected as it states (A) is false, despite (R) being true.
In situation A, an observer moves with a certain velocity towards a stationary source of sound. In situation B, the source moves towards the stationary observer with the same velocity,
Assertion (A): The frequency heard would be the same in both the situations.
Reason (R): The velocity of the source as observed by the observer in both the situations is the same.
1. Both (A) & (R) are true and the (R) is the correct explanation of the (A)
2. Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
3. (A) is true but (R) is false
4. Both (A) and (R) are false
View Answer
For situation A (observer moving towards stationary source), the observed frequency is \(f_A' = f \frac{v + v_o}{v}\). For situation B (source moving towards stationary observer), the observed frequency is \(f_B' = f \frac{v}{v - v_s}\). If \(v_o = v_s\), then \(f_A' \neq f_B'\). Hence, (A) is false. Classical Doppler effect depends on motion relative to the medium. Although the magnitude of relative velocity between source and observer might be the same, the observed frequencies differ. Thus, (R) is also false as the 'velocity of source as observed by observer' is ambiguous and does not lead to the same frequency due to medium effects. Therefore, both (A) and (R) are false.
Assertion (A): The fundamental frequency of an open organ pipe increases as the temperature is increased.
Reason (R): As the temperature increases, the velocity of sound increases more rapidly than length of the pipe.
1. Both (A) & (R) are true and the (R) is the correct explanation of the (A)
2. Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
3. (A) is true but (R) is false
4. Both (A) and (R) are false
View Answer
The fundamental frequency of an open organ pipe is \(f = \frac{v}{2L}\). Velocity of sound \(v\) increases with temperature as \(v \propto \sqrt{T}\). While the pipe's length \(L\) also increases with temperature, the increase in \(v\) is proportionally greater than \(L\). Thus, \(f\) increases. Both A and R are true, and R correctly explains A.
Assertion (A): Transverse mechanical waves can propagate in solid, liquid and gas.
Reason (R): Transverse mechanical waves needs rigidity in the medium to propagate.
1. Both (A) & (R) are true and the (R) is the correct explanation of the (A)
2. Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
3. (A) is true but (R) is false
4. Both (A) and (R) are false
View Answer
Transverse mechanical waves require a medium with shear rigidity to propagate in bulk. Solids possess shear rigidity, but bulk liquids and gases do not. Therefore, Assertion (A) is false. Reason (R) is true as rigidity is indeed necessary for transverse wave propagation.
Assertion (A): When there is no relative velocity between source and observer then observed frequency is same as emitted.
Reason (R): Velocity of sound is zero when there is no relative velocity between source and observer.
1. Both (A) & (R) are true and the (R) is the correct explanation of the (A)
2. Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
3. (A) is true but (R) is false
4. Both (A) and (R) are false
View Answer
According to the Doppler effect, the observed frequency matches the emitted frequency only when there is no relative motion between the source and observer. So, Assertion (A) is true. The velocity of sound is a property of the medium and is non-zero in an ideal medium, irrespective of relative motion between source and observer. So, Reason (R) is false. Thus, (A) is true but (R) is false.
Assertion (A): Speed of longitudinal wave in solid and liquid is higher than gases.
Reason (R): Modulus of elasticity is more for solids as compared to liquid & gas.
1. Both (A) & (R) are true and the (R) is the correct explanation of the (A)
2. Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
3. (A) is true but (R) is false
4. Both (A) and (R) are false
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The speed of longitudinal waves is given by \(v = \sqrt{\frac{B}{\rho}}\), where \(B\) is the bulk modulus. Solids and liquids have significantly higher bulk moduli compared to gases. Thus, longitudinal waves travel faster in solids and liquids. Both assertion and reason are true, and the reason correctly explains the assertion.
Assertion (A): The velocity of sound decreases with increase in humidity.
Reason (R): Velocity of sound does not depend on medium.
1. Both (A) & (R) are true and the (R) is the correct explanation of the (A)
2. Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
3. (A) is true but (R) is false
4. Both (A) and (R) are false
View Answer
The velocity of sound increases with humidity because humid air is less dense than dry air. So, Assertion (A) is false. The velocity of sound absolutely depends on the properties of the medium (density, elasticity). So, Reason (R) is also false. Both (A) and (R) are false.
Assertion (A): The change in air pressure, effect the speed of sound at constant temperature.
Reason (R): The speed of sound in a gas is directly proportional to pressure.
1. Both (A) & (R) are true and the (R) is the correct explanation of the (A)
2. Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
3. (A) is true but (R) is false
4. Both (A) and (R) are false
View Answer
The speed of sound in a gas is \(v = \sqrt{\frac{\gamma RT}{M}}\). At constant temperature (\(T\)), the velocity is independent of pressure. So, Assertion (A) is false. Also, the speed of sound is not directly proportional to pressure. So, Reason (R) is false. Both (A) and (R) are false.