The formula for the frequency of vibrations of a vibrating string is:

$f = \frac{1}{2L} \sqrt{\frac{T}{\rho}}$

where:

• $f$ is the frequency,
• $L$ is the length of the string,
• $T$ is the tension,
• $\rho$ is the linear density.

We need to find the rate of change of the frequency $f$ with respect to:

1. $L$ (when $T$ and $\rho$ are constant),
2. $T$ (when $L$ and $\rho$ are constant),
3. $\rho$ (when $L$ and $T$ are constant).

(i) Rate of change of frequency with respect to $L$:

We differentiate $f$ with respect to $L$, treating $T$ and $\rho$ as constants:

$f = \frac{1}{2L} \sqrt{\frac{T}{\rho}}$

Differentiate with respect to $L$:

$\frac{df}{dL} = -\frac{1}{2L^2} \sqrt{\frac{T}{\rho}}$

So, the rate of change of frequency with respect to $L$ is:

$\frac{df}{dL} = -\frac{f}{L}$

(ii) Rate of change of frequency with respect to $T$:

Differentiate $f$ with respect to $T$, treating $L$ and $\rho$ as constants:

$f = \frac{1}{2L} \sqrt{\frac{T}{\rho}}$

Differentiate with respect to $T$:

$\frac{df}{dT} = \frac{1}{2L} \cdot \frac{1}{2\sqrt{T\rho}} \cdot \frac{1}{\rho}$

Simplify:

$\frac{df}{dT} = \frac{1}{4L} \cdot \frac{1}{\sqrt{T\rho}} = \frac{f}{2T}$

So, the rate of change of frequency with respect to $T$ is:

$\frac{df}{dT} = \frac{f}{2T}$

(iii) Rate of change of frequency with respect to $\rho$:

Differentiate $f$ with respect to $\rho$, treating $L$ and $T$ as constants:

$f = \frac{1}{2L} \sqrt{\frac{T}{\rho}}$

Differentiate with respect to $\rho$:

$\frac{df}{d\rho} = \frac{1}{2L} \cdot \frac{-1}{2\sqrt{T\rho}} \cdot \frac{T}{\rho^2}$

Simplify:

$\frac{df}{d\rho} = -\frac{f}{2\rho}$

So, the rate of change of frequency with respect to $\rho$ is:

$\frac{df}{d\rho} = -\frac{f}{2\rho}$

Summary of results:

• $\frac{df}{dL} = -\frac{f}{L}$
• $\frac{df}{dT} = \frac{f}{2T}$
• $\frac{df}{d\rho} = -\frac{f}{2\rho}$

Let me know if you would like further clarification or details!

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5 related questions:

1. How would the frequency change if the string length is doubled?
2. What happens to the frequency if the tension is increased by 50%?
3. How does decreasing the linear density by half affect the frequency?
4. Can you derive the rate of change of frequency with respect to tension in a similar wave equation?
5. How would varying both tension and length together affect the frequency?

Tip: When working with partial derivatives, always treat other variables as constants to focus on the change with respect to the variable of interest.