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If the tension and diameter of a sonometer wire of fundamental frequency n are doubled and density is halved, then its fundamental frequency will become
- a)n/4
- b)√2n
- c)n
- d)n/√2
Correct answer is option 'C'. Can you explain this answer?
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If the tension and diameter of a sonometer wire of fundamental frequen...
Sonometer Wire and its Fundamental Frequency
A sonometer wire is a type of musical instrument used to produce sound waves. It consists of a wire stretched between two points, with a weight attached to one end. When the wire is plucked or struck, it vibrates and produces sound waves.
The fundamental frequency of a sonometer wire is the frequency at which it vibrates when it is plucked or struck. It depends on various factors like tension, diameter, density and length of the wire.
Effect of Doubling Tension and Diameter
When the tension and diameter of a sonometer wire are doubled, the following changes occur:
- Tension: The tension in the wire increases, which makes it harder to vibrate.
- Diameter: The diameter of the wire increases, which increases its mass and makes it harder to vibrate.
Effect of Halving Density
When the density of the wire is halved, the following change occurs:
- Density: The density of the wire decreases, which reduces its mass and makes it easier to vibrate.
Overall Effect on Fundamental Frequency
The overall effect of these changes on the fundamental frequency of the wire can be determined using the following formula:
f = (1/2L) * √(T/μ)
where f is the fundamental frequency, L is the length of the wire, T is the tension in the wire, and μ is the linear density of the wire (mass per unit length).
Using this formula, we can see that:
- Doubling the tension and diameter of the wire will double the value of T in the formula, which will cause the frequency to increase by a factor of √2.
- Halving the density of the wire will halve the value of μ in the formula, which will cause the frequency to decrease by a factor of √2.
Therefore, the net effect of doubling the tension and diameter and halving the density on the fundamental frequency of the sonometer wire will be:
f' = (1/2L) * √(2T/(1/2μ))
= (1/2L) * √(4T/μ)
= √2 * (1/2L) * √(T/μ)
= √2 * f
where f' is the new fundamental frequency of the wire.
Thus, the fundamental frequency of the wire will be multiplied by a factor of √2, which means the correct answer is option 'C' - nd/√2.
A sonometer wire is a type of musical instrument used to produce sound waves. It consists of a wire stretched between two points, with a weight attached to one end. When the wire is plucked or struck, it vibrates and produces sound waves.
The fundamental frequency of a sonometer wire is the frequency at which it vibrates when it is plucked or struck. It depends on various factors like tension, diameter, density and length of the wire.
Effect of Doubling Tension and Diameter
When the tension and diameter of a sonometer wire are doubled, the following changes occur:
- Tension: The tension in the wire increases, which makes it harder to vibrate.
- Diameter: The diameter of the wire increases, which increases its mass and makes it harder to vibrate.
Effect of Halving Density
When the density of the wire is halved, the following change occurs:
- Density: The density of the wire decreases, which reduces its mass and makes it easier to vibrate.
Overall Effect on Fundamental Frequency
The overall effect of these changes on the fundamental frequency of the wire can be determined using the following formula:
f = (1/2L) * √(T/μ)
where f is the fundamental frequency, L is the length of the wire, T is the tension in the wire, and μ is the linear density of the wire (mass per unit length).
Using this formula, we can see that:
- Doubling the tension and diameter of the wire will double the value of T in the formula, which will cause the frequency to increase by a factor of √2.
- Halving the density of the wire will halve the value of μ in the formula, which will cause the frequency to decrease by a factor of √2.
Therefore, the net effect of doubling the tension and diameter and halving the density on the fundamental frequency of the sonometer wire will be:
f' = (1/2L) * √(2T/(1/2μ))
= (1/2L) * √(4T/μ)
= √2 * (1/2L) * √(T/μ)
= √2 * f
where f' is the new fundamental frequency of the wire.
Thus, the fundamental frequency of the wire will be multiplied by a factor of √2, which means the correct answer is option 'C' - nd/√2.
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If the tension and diameter of a sonometer wire of fundamental frequency n are doubled and density is halved, then its fundamental frequency will becomea)n/4b)√2nc)nd)n/√2Correct answer is option 'C'. Can you explain this answer?
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