A standing wave on a string is given by `y = ( 4 cm) cos [ x pi] sin [ 50 pi t]`, where `x` is in metres and t is in seconds. The velocity of the string section at `x = 1//3 m at t = 1//5 s`, is

Options

A. zero
B. ` pi m//s`
C. `840 pi m//s`
D. noneof these

Explanation

To find the velocity of the string section at \( x = \frac{1}{3} \, \text{m} \) and \( t = \frac{1}{5} \, \text{s} \) for the standing wave given by the equation \( y = 4 \cos(\pi x) \sin(50 \pi t) \), we will follow these steps:

### Step 1: Write down the wave equation
The wave equation is given as:
\[ y = 4 \cos(\pi x) \sin(50 \pi t) \]

### Step 2: Find the velocity of the string
The velocity \( v \) of the string is given by the partial derivative of \( y \) with respect to time \( t \):
\[ v = \frac{\partial y}{\partial t} \]

### Step 3: Differentiate the wave equation with respect to \( t \)
Using the product rule, we differentiate:
\[ \frac{\partial y}{\partial t} = 4 \cos(\pi x) \cdot \frac{\partial}{\partial t}(\sin(50 \pi t)) \]
The derivative of \( \sin(50 \pi t) \) is:
\[ \frac{\partial}{\partial t}(\sin(50 \pi t)) = 50 \pi \cos(50 \pi t) \]
Thus, we have:
\[ \frac{\partial y}{\partial t} = 4 \cos(\pi x) \cdot 50 \pi \cos(50 \pi t) \]
This simplifies to:
\[ v = 200 \pi \cos(\pi x) \cos(50 \pi t) \]

### Step 4: Substitute \( x = \frac{1}{3} \) and \( t = \frac{1}{5} \)
Now we substitute \( x = \frac{1}{3} \) and \( t = \frac{1}{5} \) into the velocity equation:
\[ v = 200 \pi \cos\left(\pi \cdot \frac{1}{3}\right) \cos\left(50 \pi \cdot \frac{1}{5}\right) \]

### Step 5: Calculate \( \cos\left(\pi \cdot \frac{1}{3}\right) \) and \( \cos\left(50 \pi \cdot \frac{1}{5}\right) \)
Calculating the cosine values:
\[ \cos\left(\pi \cdot \frac{1}{3}\right) = \cos\left(\frac{\pi}{3}\right) = \frac{1}{2} \]
\[ \cos\left(50 \pi \cdot \frac{1}{5}\right) = \cos(10 \pi) = 1 \]

### Step 6: Substitute the cosine values back into the velocity equation
Now substituting these values back:
\[ v = 200 \pi \cdot \frac{1}{2} \cdot 1 = 100 \pi \, \text{cm/s} \]

### Step 7: Convert to meters per second
Since the answer is in centimeters per second, we convert it to meters per second:
\[ v = 100 \pi \, \text{cm/s} = \frac{100 \pi}{100} \, \text{m/s} = \pi \, \text{m/s} \]

### Final Answer
The velocity of the string section at \( x = \frac{1}{3} \, \text{m} \) and \( t = \frac{1}{5} \, \text{s} \) is:
\[ \boxed{\pi \, \text{m/s}} \]