A standing wave `y=A sin((20pix)/(3))cos (1000pit)` is set up in a taut string where x and y are in meter. The distance between two successive points oscillating with the amplitude `(A)/(2)` can be equal to (x)cm. Find the value of x.

Answer

To solve the problem, we will analyze the standing wave given by the equation:

\[ y = A \sin\left(\frac{20\pi x}{3}\right) \cos(1000\pi t) \]

### Step 1: Understand the standing wave equation

The standing wave can be expressed in the form of a sine function, where the spatial part is given by \( \sin\left(\frac{20\pi x}{3}\right) \). The amplitude of oscillation varies with the sine function.

### Step 2: Determine the amplitude condition

We need to find the distance between two successive points that oscillate with an amplitude of \( \frac{A}{2} \). This means we need to find the points where:

\[ A \sin\left(\frac{20\pi x}{3}\right) = \frac{A}{2} \]

Dividing both sides by \( A \) (assuming \( A \neq 0 \)) gives:

\[ \sin\left(\frac{20\pi x}{3}\right) = \frac{1}{2} \]

### Step 3: Solve for \( x \)

The sine function equals \( \frac{1}{2} \) at specific angles. The general solutions for \( \sin(\theta) = \frac{1}{2} \) are:

\[ \theta = \frac{\pi}{6} + 2n\pi \quad \text{and} \quad \theta = \frac{5\pi}{6} + 2n\pi \]

where \( n \) is any integer.

Setting \( \theta = \frac{20\pi x}{3} \), we get:

1. For the first case:
\[ \frac{20\pi x}{3} = \frac{\pi}{6} + 2n\pi \]

2. For the second case:
\[ \frac{20\pi x}{3} = \frac{5\pi}{6} + 2n\pi \]

### Step 4: Solve the first case

For \( n = 0 \):
\[ \frac{20\pi x}{3} = \frac{\pi}{6} \]
Multiplying both sides by \( 3 \):
\[ 20\pi x = \frac{\pi}{2} \]
Dividing by \( 20\pi \):
\[ x = \frac{1}{40} \text{ meters} \]

### Step 5: Solve the second case

For \( n = 0 \):
\[ \frac{20\pi x}{3} = \frac{5\pi}{6} \]
Multiplying both sides by \( 3 \):
\[ 20\pi x = \frac{15\pi}{2} \]
Dividing by \( 20\pi \):
\[ x = \frac{3}{40} \text{ meters} \]

### Step 6: Find the distance between two successive points

The two points oscillating with amplitude \( \frac{A}{2} \) are at \( x = \frac{1}{40} \) m and \( x = \frac{3}{40} \) m. The distance \( X \) between these two points is:

\[ X = \left(\frac{3}{40} - \frac{1}{40}\right) = \frac{2}{40} = \frac{1}{20} \text{ meters} \]

### Step 7: Convert to centimeters

To convert meters to centimeters:
\[ X = \frac{1}{20} \text{ m} \times 100 = 5 \text{ cm} \]

### Final Answer

The value of \( x \) is \( 5 \text{ cm} \).

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