String vibrates according to equation, `y= 5 sin (pix)/3 cos400pit`. At what time potential energy of string will be zero?

Answer

To determine at what time the potential energy of the string will be zero, we start with the given equation of motion for the vibrating string:

\[ y = 5 \sin\left(\frac{\pi x}{3}\right) \cos(400 \pi t) \]

### Step 1: Understand the relationship between potential energy and displacement
The potential energy (PE) in a vibrating string is related to the displacement of the string from its equilibrium position. The potential energy is zero when the displacement \( y \) is zero.

### Step 2: Set the displacement equation to zero
To find when the potential energy is zero, we need to set the displacement \( y \) to zero:

\[ 5 \sin\left(\frac{\pi x}{3}\right) \cos(400 \pi t) = 0 \]

### Step 3: Analyze the equation
This equation can be satisfied if either of the two factors is zero:

1. \( \sin\left(\frac{\pi x}{3}\right) = 0 \)
2. \( \cos(400 \pi t) = 0 \)

### Step 4: Solve for the first factor
The sine function is zero at integer multiples of \( \pi \):

\[ \frac{\pi x}{3} = n\pi \]
\[ x = 3n \]

where \( n \) is any integer (0, ±1, ±2, ...).

### Step 5: Solve for the second factor
The cosine function is zero at odd multiples of \( \frac{\pi}{2} \):

\[ 400 \pi t = \left(n + \frac{1}{2}\right) \pi \]
\[ t = \frac{(n + \frac{1}{2})}{400} \]

where \( n \) is any integer (0, 1, 2, ...).

### Step 6: Simplify the expression for time
Now we can simplify the expression for time:

\[ t = \frac{1}{400}(n + 0.5) \]

### Step 7: Calculate specific times
For different integer values of \( n \):

- For \( n = 0 \):
\[ t = \frac{0.5}{400} = \frac{1}{800} \text{ seconds} \]

- For \( n = 1 \):
\[ t = \frac{1.5}{400} = \frac{3}{800} \text{ seconds} \]

- For \( n = 2 \):
\[ t = \frac{2.5}{400} = \frac{5}{800} \text{ seconds} \]

Thus, the potential energy of the string will be zero at times \( t = \frac{1}{800} \), \( \frac{3}{800} \), \( \frac{5}{800} \), etc.

### Final Answer
The potential energy of the string will be zero at the times given by:

\[ t = \frac{(n + 0.5)}{400} \text{ seconds} \]

where \( n \) is an integer.

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