the tension in the wire is decreased by 19%. The percentage decrease in the frequency will be?
A vibrating wire can have many different modes. I assume you are talking about the fundamental vibration frequency.
The frequency of any mode is proportional to the square root of tension.
sqrt(0.81) = 0.90
The decrease is therefore exactly 10%
To calculate the percentage decrease in the frequency, we need to use the relationship between tension and frequency in a vibrating wire. According to the formula for the frequency of a vibrating wire,
frequency α √(tension / linear mass density)
Where:
- "α" means "proportional to"
- "tension" is the tension in the wire
- "linear mass density" is the mass per unit length of the wire
Now, let's consider the given information. The tension in the wire is decreased by 19%, which means the new tension is 81% (100% - 19%) of the original tension. So, the new tension can be represented as:
new tension = original tension × (1 - 0.19)
= original tension × 0.81
Since we want to find the percentage decrease in the frequency, it's easier to work with the inverse relationship of the frequency. In other words, we'll calculate the percentage increase in the square of the frequency.
Now, substituting the new tension into the equation, we have:
new frequency² α (new tension / linear mass density)
Substituting the expressions for new tension and original tension:
new frequency² α (original tension × 0.81 / linear mass density)
The original frequency squared is proportional to (original tension / linear mass density). Therefore, we can write:
(original frequency)² α (original tension / linear mass density)
Now, let's simplify the equation by dividing both sides by (original tension / linear mass density):
(new frequency)² / (original frequency)² = (original tension × 0.81 / linear mass density) / (original tension / linear mass density)
Simplifying further:
(new frequency)² / (original frequency)² = 0.81
Taking the square root of both sides to find the ratio of the new frequency to the original frequency:
new frequency / original frequency = √0.81
= 0.9
Now, let's calculate the percentage decrease in the frequency:
percentage decrease = (1 - new frequency / original frequency) × 100
= (1 - 0.9) × 100
= 10%
Therefore, the percentage decrease in the frequency will be 10%.
To find the percentage decrease in the frequency, we can use the formula:
Percentage decrease = (Initial value - Final value) / Initial value * 100
In this case, the initial value is the tension in the wire and the final value is the frequency. Let's assume the tension is T and the frequency is f.
Given that the tension in the wire is decreased by 19%, we can find the final tension by subtracting 19% of the initial tension from the initial tension:
Final tension = T - (0.19 * T)
= T - 0.19T
= 0.81T
Now, let's consider the relationship between tension and frequency for a wire under tension, which is given by:
Frequency = k * sqrt(T / m)
Where k is a constant and m is the mass per unit length of the wire. Assuming these values remain constant, we can rewrite the equation as:
Frequency = k' * sqrt(T)
Where k' is the adjusted constant combining k and sqrt(m).
So, we can say that the frequency is directly proportional to the square root of the tension.
To find the percentage decrease in the frequency, we'll use the same formula as before with the new values:
Percentage decrease = (Initial value - Final value) / Initial value * 100
= (f - f') / f * 100
Where f' is the final frequency.
Since the frequency is directly proportional to the square root of the tension, we can write:
(f - f') / f = (sqrt(T) - sqrt(0.81T)) / sqrt(T)
Now, let's simplify the expression:
(f - f') / f = (sqrt(T) - sqrt(0.81T)) / sqrt(T)
= (sqrt(T) - sqrt(0.9) * sqrt(T)) / sqrt(T)
= (1 - sqrt(0.9))
Therefore, the percentage decrease in the frequency will be approximately equal to the percentage decrease in the tension, which is 19%.