To solve the problem of finding the time taken for a wave pulse to travel up a uniform string of length 10 m, we can follow these steps:
### Step 1: Understand the problem
We have a uniform string of length \( L = 10 \, \text{m} \) suspended from a rigid support. A wave pulse is introduced at the lower end of the string, and we need to determine the time it takes for the pulse to reach the support.
### Step 2: Determine the wave velocity
The velocity \( v \) of a wave traveling along a string is given by the formula:
\[
v = \sqrt{\frac{T}{\mu}}
\]
where \( T \) is the tension in the string and \( \mu \) is the mass per unit length of the string.
### Step 3: Calculate the tension in the string
At a distance \( x \) from the lower end of the string, the tension \( T \) in the string due to the weight of the string below that point is given by:
\[
T = \mu g x
\]
where \( g \) is the acceleration due to gravity (given as \( g = 10 \, \text{m/s}^2 \)).
### Step 4: Substitute the tension into the wave velocity formula
Substituting the expression for tension into the wave velocity formula, we get:
\[
v = \sqrt{\frac{\mu g x}{\mu}} = \sqrt{g x}
\]
### Step 5: Relate velocity to distance and time
We know that wave velocity can also be expressed as:
\[
v = \frac{dx}{dt}
\]
Thus, we can write:
\[
dx = v \, dt
\]
Substituting \( v = \sqrt{g x} \) into this equation gives:
\[
dx = \sqrt{g x} \, dt
\]
### Step 6: Rearranging and integrating
Rearranging the equation gives:
\[
\frac{dx}{\sqrt{x}} = \sqrt{g} \, dt
\]
Integrating both sides, we have:
\[
\int \frac{dx}{\sqrt{x}} = \sqrt{g} \int dt
\]
The left side integrates to \( 2\sqrt{x} \), and the right side integrates to \( \sqrt{g} t + C \).
### Step 7: Setting the limits of integration
When \( x = 0 \), \( t = 0 \) (the wave pulse starts at the bottom), and when \( x = 10 \, \text{m} \), we want to find \( t \):
\[
2\sqrt{10} - 0 = \sqrt{g} t
\]
### Step 8: Solve for time \( t \)
Substituting \( g = 10 \, \text{m/s}^2 \):
\[
2\sqrt{10} = \sqrt{10} t
\]
Dividing both sides by \( \sqrt{10} \):
\[
2 = t
\]
Thus, the time taken for the wave pulse to reach the support is \( t = 2 \, \text{seconds} \).
### Final Answer
The time taken to reach the support is **2 seconds**.
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