A pulse is started at a time `t = 0` along the `+x` directions an a long, taut string. The shaot of the puise at `t = 0` is given by funcation `y` with
`y = {{:((x)/(4)+1fo r-4ltxle0),(-x+1fo r0ltxlt1),("0 otherwise"):}`
here `y` and `x` are in centimeters. The linear mass density of the string is `50 g//m` and it is under a tension of `5N`,
The vertical displacement of the particle of the string at `x = 7 cm` and `t = 0.01 s` will be

Options

A. `0.75 cm`
B. `0.5 cm`
C. `0.25 cm`
D. zero

Answer

To solve the problem step by step, we will follow the given information and apply the relevant physics concepts.

### Step 1: Understand the Pulse Function
The pulse function is given as:
- \( y = \frac{x}{4} + 1 \) for \( -4 < x < 0 \)
- \( y = -x + 1 \) for \( 0 < x < 1 \)
- \( y = 0 \) otherwise

We need to find the vertical displacement of the particle at \( x = 7 \, \text{cm} \) and \( t = 0.01 \, \text{s} \).

### Step 2: Determine the Wave Velocity
The wave velocity \( v \) can be calculated using the formula:
\[ v = \sqrt{\frac{T}{\mu}} \]
where \( T \) is the tension and \( \mu \) is the linear mass density.

Given:
- \( T = 5 \, \text{N} \)
- \( \mu = 50 \, \text{g/m} = 0.05 \, \text{kg/m} \)

Now substituting the values:
\[ v = \sqrt{\frac{5}{0.05}} = \sqrt{100} = 10 \, \text{m/s} = 1000 \, \text{cm/s} \]

### Step 3: Calculate the Position of the Pulse at \( t = 0.01 \, \text{s} \)
The pulse travels in the positive x-direction. The distance traveled by the pulse in \( t = 0.01 \, \text{s} \) is:
\[ \text{Distance} = v \cdot t = 1000 \, \text{cm/s} \cdot 0.01 \, \text{s} = 10 \, \text{cm} \]

### Step 4: Determine the New Position of the Pulse
At \( t = 0.01 \, \text{s} \), the pulse that started at \( x = 0 \) will have moved to:
\[ x = 0 + 10 = 10 \, \text{cm} \]

### Step 5: Find the Displacement at \( x = 7 \, \text{cm} \)
Since \( x = 7 \, \text{cm} \) is less than \( 10 \, \text{cm} \), we need to evaluate the displacement at this position using the original pulse function.

At \( x = 7 \, \text{cm} \):
- The pulse has not yet reached this position since it only reaches \( x = 10 \, \text{cm} \) at \( t = 0.01 \, \text{s} \).
- Therefore, the displacement at \( x = 7 \, \text{cm} \) is still \( 0 \).

### Final Answer
The vertical displacement of the particle of the string at \( x = 7 \, \text{cm} \) and \( t = 0.01 \, \text{s} \) is:
\[ \text{Displacement} = 0 \, \text{cm} \]

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