To determine the linear mass density ([tex]\( \mu = \frac{m}{L} \)[/tex]) of the string given the speed of the transverse wave and the tension, we can follow these steps:
1. Understand the given formula:
The speed of the wave [tex]\( v \)[/tex] on a string is given by the formula:
[tex]\[
v = \sqrt{\frac{T}{\mu}}
\][/tex]
where:
- [tex]\( v \)[/tex] is the speed of the wave,
- [tex]\( T \)[/tex] is the tension in the string,
- [tex]\( \mu \)[/tex] is the linear mass density ([tex]\( \frac{m}{L} \)[/tex]).
2. Given values:
From the problem, we know:
- The tension [tex]\( T \)[/tex] is [tex]\( 10.0 \, N \)[/tex].
- The speed of the wave [tex]\( v \)[/tex] is [tex]\( 5.0 \, m/s \)[/tex]. This is described as the wave speed in the simulation.
3. Rearrange the formula to solve for [tex]\( \mu \)[/tex]:
Starting with the given formula:
[tex]\[
v = \sqrt{\frac{T}{\mu}}
\][/tex]
Square both sides of the equation to eliminate the square root:
[tex]\[
v^2 = \frac{T}{\mu}
\][/tex]
Solve for [tex]\( \mu \)[/tex] by rearranging the equation:
[tex]\[
\mu = \frac{T}{v^2}
\][/tex]
4. Substitute the known values:
- [tex]\( T = 10.0 \, N \)[/tex]
- [tex]\( v = 5.0 \, m/s \)[/tex]
Substitute these values into the equation:
[tex]\[
\mu = \frac{10.0 \, N}{(5.0 \, m/s)^2}
\][/tex]
5. Calculate the result:
[tex]\[
\mu = \frac{10.0}{25.0}
\][/tex]
[tex]\[
\mu = 0.4 \, \frac{kg}{m}
\][/tex]
Thus, the linear mass density of the string is [tex]\( 0.4 \, \frac{kg}{m} \)[/tex].
