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Sagot :
The velocity is 21.203 m/s
What is tension ?
Tension is the force that is sent through a rope, string, or wire when two opposing forces pull on it. Along the whole length of the wire, the tension force pulls energy equally on the bodies at the ends. Every physical object that comes into contact with another one exerts force on it.
According to the given information
Cross sectional Area, A = 2.83 [tex]\times 10^{-3} \mathrm{~m}^2$[/tex]
Tension on the string, T = 1 [tex]\times 10^4[/tex] N
Density of steel, [tex]$\rho[/tex] = 7860 [tex]\mathrm{~kg} / \mathrm{m}^3$[/tex]
let [tex]$\psi(x)$[/tex] be the equation of the wave on the cable.
then Net force on the cable, [tex]$F_{n e t}[/tex] = T [tex]\left[\psi^{\prime}(x+d x)-\psi^{\prime}(x)\right][/tex]
= [tex]T d x\left(d^2 \psi(x) / d^2 x\right)$[/tex]
where we have assumed that [tex]$d x$[/tex] is infinitesimal
Now mass density of cable per unit length can be calculated as following
[tex]& \mu=A \rho \\[/tex]
[tex]& \mu=2.83 \times 10^{-3} \times 7860=22.243 \mathrm{~kg} / \mathrm{m}[/tex]
Using equation 1 ,
[tex]& F_{\text {net }}[/tex] = [tex]T d x\left(d^2 \psi(x) / d^2 x\right)[/tex] = m a =[tex]\mu d x\left(d^2 \psi(x) / d^2 t\right) \text { where } m[/tex][tex]=\mu d x \\& d^2 \psi(x) / d^2 t=(T / \mu)\left(d^2 \psi(x) / d^2 x\right)[/tex]
Comparing with the above equation,
[tex]$d^2 \psi(x) / d^2t[/tex] = [tex]\left(v^2\right)\left(d^2 \psi(x) / d^2 x\right)$[/tex] where V is the speed of the wave. then
v = [tex]\sqrt{T / \mu}$[/tex]
Using equation 2 and given value of tension,
v = [tex]\sqrt{1 \times 10^4 / v}=\sqrt{1 \times 10^4 / 22.243}[/tex]
= [tex]\sqrt{449.57}=21.203 \mathrm{~m} / \mathrm{s}[/tex]
The velocity is 21.203 m/s
To know more about Tension
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