The equation of a stationary wave in a metal rod is given by `y=0.92 sin.(pix)/(3)sin1000t`, where x is in cm and t is in second. The maximum tensile stress at a point x = 1 cm is `(npi)/(3)xx10^(8)" dyne cm"^(-2)`. What is the value of n? [Young's modulus of the material of rod is `=8xx10^(11)" dyne cm"^(-2)`]

Answer

To solve the problem, we need to find the value of \( n \) in the expression for maximum tensile stress given by:

\[ \text{Maximum Tensile Stress} = \frac{n\pi}{3} \times 10^8 \, \text{dyne cm}^{-2} \]

The equation of the stationary wave is given by:

\[ y = 0.92 \sin\left(\frac{\pi x}{3}\right) \sin(1000t) \]

### Step 1: Find \(\frac{dy}{dx}\)

To find the maximum tensile stress, we first need to calculate the derivative \(\frac{dy}{dx}\).

Taking the derivative of \( y \) with respect to \( x \):

\[ \frac{dy}{dx} = 0.92 \cdot \cos\left(\frac{\pi x}{3}\right) \cdot \frac{d}{dx}\left(\frac{\pi x}{3}\right) \cdot \sin(1000t) \]

Calculating \(\frac{d}{dx}\left(\frac{\pi x}{3}\right)\):

\[ \frac{d}{dx}\left(\frac{\pi x}{3}\right) = \frac{\pi}{3} \]

So, we have:

\[ \frac{dy}{dx} = 0.92 \cdot \cos\left(\frac{\pi x}{3}\right) \cdot \frac{\pi}{3} \cdot \sin(1000t) \]

### Step 2: Evaluate \(\frac{dy}{dx}\) at \( x = 1 \, \text{cm} \)

Now we will evaluate \(\frac{dy}{dx}\) at \( x = 1 \, \text{cm} \):

\[ \frac{dy}{dx}\bigg|_{x=1} = 0.92 \cdot \cos\left(\frac{\pi \cdot 1}{3}\right) \cdot \frac{\pi}{3} \cdot \sin(1000t) \]

Calculating \(\cos\left(\frac{\pi}{3}\right)\):

\[ \cos\left(\frac{\pi}{3}\right) = \frac{1}{2} \]

Thus:

\[ \frac{dy}{dx}\bigg|_{x=1} = 0.92 \cdot \frac{1}{2} \cdot \frac{\pi}{3} \cdot \sin(1000t) \]

### Step 3: Find the maximum value of \(\frac{dy}{dx}\)

The maximum value of \(\sin(1000t)\) is 1, so:

\[ \left(\frac{dy}{dx}\right)_{\text{max}} = 0.92 \cdot \frac{1}{2} \cdot \frac{\pi}{3} = \frac{0.92 \pi}{6} \]

### Step 4: Calculate the maximum tensile stress

The maximum tensile stress \(\sigma\) is given by:

\[ \sigma = Y \cdot \frac{dy}{dx} \]

Where \( Y \) is Young's modulus:

\[ Y = 8 \times 10^{11} \, \text{dyne cm}^{-2} \]

Substituting the values:

\[ \sigma = 8 \times 10^{11} \cdot \frac{0.92 \pi}{6} \]

Calculating this:

\[ \sigma = \frac{8 \cdot 0.92 \cdot \pi}{6} \times 10^{11} \]

Calculating \( 8 \cdot 0.92 \):

\[ 8 \cdot 0.92 = 7.36 \]

So:

\[ \sigma = \frac{7.36 \pi}{6} \times 10^{11} \]

### Step 5: Express \(\sigma\) in the required form

We need to express \(\sigma\) in the form:

\[ \sigma = \frac{n\pi}{3} \times 10^8 \]

To do this, we can rewrite:

\[ \sigma = \frac{7.36 \pi}{6} \times 10^{11} = \frac{7.36 \pi}{6} \times 10^8 \times 10^3 \]

This gives:

\[ \sigma = \frac{7.36 \cdot 10^3 \cdot \pi}{6} \times 10^8 \]

### Step 6: Compare and solve for \( n \)

Now we compare:

\[ \frac{n\pi}{3} = \frac{7.36 \cdot 10^3 \cdot \pi}{6} \]

Cancelling \(\pi\) from both sides:

\[ \frac{n}{3} = \frac{7.36 \cdot 10^3}{6} \]

Multiplying both sides by 3:

\[ n = \frac{7.36 \cdot 10^3 \cdot 3}{6} = \frac{7.36 \cdot 10^3}{2} = 3.68 \cdot 10^3 \]

Thus, the value of \( n \) is:

\[ n = 3680 \]

### Final Answer

The value of \( n \) is \( 3680 \).

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