A string vibrates according to the equation `y=5 sin ((2 pi x)/(3)) cos 20 pi t`, where x and y are in cm and t in sec . The distance between two adjacent nodes is

Options

A. 3 cm
B. `4.5 cm`
C. 6 cm
D. ` 1.5 cm`

Answer

To find the distance between two adjacent nodes in the given vibrating string, we can follow these steps:

### Step 1: Understand the given equation
The equation of the vibrating string is given as:
\[ y = 5 \sin\left(\frac{2\pi x}{3}\right) \cos(20\pi t) \]
Here, \( x \) and \( y \) are in centimeters, and \( t \) is in seconds.

### Step 2: Identify the wave parameters
The general form of the wave equation for a vibrating string is:
\[ y = 2a \sin\left(\frac{2\pi x}{\lambda}\right) \cos\left(2\pi f t\right) \]
From the given equation, we can identify:
- The coefficient of \( x \) in the sine function gives us the wavelength \( \lambda \).

### Step 3: Extract the wavelength
From the sine term \( \sin\left(\frac{2\pi x}{3}\right) \), we can see that:
\[ \frac{2\pi}{\lambda} = \frac{2\pi}{3} \]
Thus, we can equate and solve for \( \lambda \):
\[ \lambda = 3 \text{ cm} \]

### Step 4: Calculate the distance between adjacent nodes
The distance between two adjacent nodes is given by the formula:
\[ \text{Distance between adjacent nodes} = \frac{\lambda}{2} \]
Substituting the value of \( \lambda \):
\[ \text{Distance} = \frac{3}{2} = 1.5 \text{ cm} \]

### Step 5: Conclusion
The distance between two adjacent nodes is:
\[ \text{Distance} = 1.5 \text{ cm} \]

### Final Answer
The distance between two adjacent nodes is **1.5 cm**.

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