Answer :
a) The speed of the wave will increase by a factor [tex]\sqrt{2}[/tex].
b) The speed of the wave will halve
c) The speed of the wave will double
Explanation:
a)
The speed of a standing wave on a string is given by
[tex]v=\sqrt{\frac{T}{m/L}}[/tex]
where
T is the tension in the string
m is the mass of the string
L is the length of the string
In this part of the problem, the tension in the string is doubled, so that the new tension is
T' = 2T
Substituting into the equation, we find the new speed of the wave in the string:
[tex]v'=\sqrt{\frac{T'}{m/L}}=\sqrt{\frac{2T}{m/L}}=\sqrt{2}\sqrt{\frac{T}{m/L}}=\sqrt{2}v[/tex]
So, the speed will increase by a factor [tex]\sqrt{2}[/tex].
b)
We can solve also this part by referring to the formula
[tex]v=\sqrt{\frac{T}{m/L}}[/tex]
where
T is the tension
m is the mass
L is the length
In this case, the string mass is quadrupled, so the new mass is:
m' = 4m
Substituting into the equation, we find what happens to the speed of the wave:
[tex]v'=\sqrt{\frac{T}{m'/L}}=\sqrt{\frac{T}{4m/L}}=\frac{1}{\sqrt{4}}\sqrt{\frac{T}{m/L}}=\frac{1}{2}v[/tex]
So, the speed of the wave will halve.
c)
Again, we can solve this part by referring to the same equation
[tex]v=\sqrt{\frac{T}{m/L}}[/tex]
where
T is the tension
m is the mass
L is the length
In this case, the length of the string is quadrupled, so the new length is:
L' = 4L
Substituting into the equation, we find that the new speed is:
[tex]v'=\sqrt{\frac{T}{m/L'}}=\sqrt{\frac{T}{m/(4L)}}=\sqrt{4}\sqrt{\frac{T}{m/L}}=2v[/tex]
So, the speed of the wave will double.
Learn more about waves:
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