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When a wave traverses a medium, the displacement of a particle located at x at time t is given by y = a sin (bt - cx) where a, b and c are constants of the wave. The dimensions of b/c are the same as those of
- a)Wave velocity
- b)Amplitude
- c)Wavelength
- d)Wave frequency
Correct answer is option 'A'. Can you explain this answer?
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Verified Answer
When a wave traverses a medium, the displacement of a particle located...
Argument of sin is dimensionless and using principle of dimensional homogeneity, we get
∣bt∣=∣cx∣=[M0L0T0]
Thus, ∣b∣=T−1 and ∣c∣=[L−1]
Thus, dimensions of b/c are [LT−1], which is the same as the velocity of wave.
∣bt∣=∣cx∣=[M0L0T0]
Thus, ∣b∣=T−1 and ∣c∣=[L−1]
Thus, dimensions of b/c are [LT−1], which is the same as the velocity of wave.
Most Upvoted Answer
When a wave traverses a medium, the displacement of a particle located...
The given equation y = a sin (bt - cx) represents a wave traveling through a medium. To understand the dimensions of the constants a, b, and c, let's analyze the equation and its physical interpretation.
Understanding the Equation:
- The equation represents the displacement (y) of a particle located at position x at time t.
- The displacement is given as a*sin(bt - cx), where a is the amplitude of the wave, b represents the angular frequency, and c represents the wave number.
Analyzing the Dimensions:
1. Amplitude (a):
- The amplitude (a) represents the maximum displacement of the particle from its equilibrium position.
- Since displacement is a physical quantity, the amplitude (a) also has dimensions.
- The dimensions of amplitude are the same as the dimensions of displacement, which are usually expressed in units such as meters (m) or any other appropriate unit.
- Therefore, the correct option cannot be the amplitude (b).
2. Angular Frequency (b):
- The term inside the sine function, (bt - cx), represents the phase of the wave.
- The angular frequency (b) determines how quickly the wave oscillates.
- The dimensions of angular frequency (b) are radians per unit time (rad/s).
- It represents the change in phase per unit time.
- Therefore, the correct option cannot be the wave frequency (d), which is the reciprocal of the period.
3. Wave Number (c):
- The wave number (c) represents the spatial frequency of the wave.
- It is defined as the number of wavelengths per unit distance.
- The dimensions of wave number (c) are radians per unit distance (rad/m).
- It represents the change in phase per unit distance traveled by the wave.
- Therefore, the correct option cannot be the wavelength (c).
4. Wave Velocity (v):
- Wave velocity (v) is the speed at which the wave propagates through the medium.
- In the given equation, the velocity (v) can be determined by the ratio of b (angular frequency) to c (wave number), as v = b/c.
- The dimensions of wave velocity are distance per unit time (m/s), which are the same as the dimensions of b/c.
- Therefore, the correct option is the wave velocity (a).
In conclusion, the dimensions of b/c in the given equation y = a sin (bt - cx) are the same as the dimensions of wave velocity.
Understanding the Equation:
- The equation represents the displacement (y) of a particle located at position x at time t.
- The displacement is given as a*sin(bt - cx), where a is the amplitude of the wave, b represents the angular frequency, and c represents the wave number.
Analyzing the Dimensions:
1. Amplitude (a):
- The amplitude (a) represents the maximum displacement of the particle from its equilibrium position.
- Since displacement is a physical quantity, the amplitude (a) also has dimensions.
- The dimensions of amplitude are the same as the dimensions of displacement, which are usually expressed in units such as meters (m) or any other appropriate unit.
- Therefore, the correct option cannot be the amplitude (b).
2. Angular Frequency (b):
- The term inside the sine function, (bt - cx), represents the phase of the wave.
- The angular frequency (b) determines how quickly the wave oscillates.
- The dimensions of angular frequency (b) are radians per unit time (rad/s).
- It represents the change in phase per unit time.
- Therefore, the correct option cannot be the wave frequency (d), which is the reciprocal of the period.
3. Wave Number (c):
- The wave number (c) represents the spatial frequency of the wave.
- It is defined as the number of wavelengths per unit distance.
- The dimensions of wave number (c) are radians per unit distance (rad/m).
- It represents the change in phase per unit distance traveled by the wave.
- Therefore, the correct option cannot be the wavelength (c).
4. Wave Velocity (v):
- Wave velocity (v) is the speed at which the wave propagates through the medium.
- In the given equation, the velocity (v) can be determined by the ratio of b (angular frequency) to c (wave number), as v = b/c.
- The dimensions of wave velocity are distance per unit time (m/s), which are the same as the dimensions of b/c.
- Therefore, the correct option is the wave velocity (a).
In conclusion, the dimensions of b/c in the given equation y = a sin (bt - cx) are the same as the dimensions of wave velocity.
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When a wave traverses a medium, the displacement of a particle located at x at time t is given by y = a sin (bt - cx) where a, b and c are constants of the wave. The dimensions of b/c are the same as those ofa)Wave velocityb)Amplitudec)Wavelengthd)Wave frequencyCorrect answer is option 'A'. Can you explain this answer?
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