The equation relating tension and speed of wave is given by:
v=sqrt(T/mu)v=√Tμ
where: T" is the tension"; mu" is the linear density of the string"T is the tension;μ is the linear density of the string
Since we know that v=flambdav=fλ, we can rewrite the equation into:
flambda=sqrt(T/mu)fλ=√Tμ
I)Tension increased by a factor of 4
Let TT be the initial tension of the string
:.4T will be the final tension of the string
4loops are formed,
L=2lambda
The string used is the same, thus the linear density, mu is a constant.
The frequency of the vibrator is also kept as the same, f is another constant.
Rearranging
flambda=sqrt(T/mu)
f^2mu=T/lambda^2
since f and mu are constants, their product is a constant too.
T_i/lambda_i^2=T_f/lambda_f^2
subscript i for initial, f for final
T/(L/2)^2=(4T)/lambda_f^2
(4T)/(4(L^2/4))=(4T)/lambda_f^2
4(L^2/4)=lambda_f^2
lambda_f^2=L^2
lambda_f=L
Therefore the number of loops formed will be 2.
II)Frequency increased by a factor of 4
Now, Tension instead of frequency becomes constant.
flambda=sqrt(T/mu)
Since Tension and Linear density remain the same, sqrt(T/mu) is constant.
Thus,
f_ilambda_i=f_flambda_f
Let f be the initial frequency,
4f will be the final frequency.
4loops are formed,
L=2lambda
lambda=L/2
f(L/2)=(4f)lambda_f
lambda_f=f(L/2)/(4f)
lambda_f=L/8
Thefore, the number of loops is 16.
