The mass of a string is 9.1 × 10-3 kg, and it is stretched so that the tension in it is 240 N. A transverse wave traveling on this string has a frequency of 240 Hz and a wavelength of 0.42 m. What is the length of the string?
The speed of the transverse wave is
V = (frequency) x (wavelength)
= 100.8 m/s
V = sqrt(T/sigma)
where T is the tension (240 N) and sigma is the mass per unit length. Solve for sigma and use it to compute the length.
Length = (Mass)/(sigma)
To find the length of the string, we can use the formula:
v = λf
Where:
v is the velocity of the wave,
λ is the wavelength, and
f is the frequency.
In this case, we are given the frequency (f = 240 Hz) and the wavelength (λ = 0.42 m). We need to calculate the velocity (v) first.
To calculate the velocity, we can use the formula:
v = √(T/μ)
Where:
v is the velocity of the wave,
T is the tension in the string, and
μ is the linear mass density of the string.
The linear mass density (μ) is defined as the mass per unit length of the string and is calculated by dividing the mass of the string by its length:
μ = m / L
Where:
μ is the linear mass density,
m is the mass of the string, and
L is the length of the string (which we are trying to find).
Given:
T = 240 N
m = 9.1 × 10^(-3) kg
f = 240 Hz
λ = 0.42 m
First, let's calculate the linear mass density (μ):
μ = m / L
μ = (9.1 × 10^(-3) kg) / L
Next, calculate the velocity (v):
v = √(T/μ)
v = √(240 N / ((9.1 × 10^(-3) kg) / L))
Now, we can substitute the velocity (v) into the formula v = λf to find the length (L):
λf = √(240 N / ((9.1 × 10^(-3) kg) / L))
(0.42 m) * (240 Hz) = √(240 N / ((9.1 × 10^(-3) kg) / L))
To solve for L, square both sides of the equation and isolate L:
(0.42 m) * (240 Hz) * (0.42 m) * (240 Hz) = 240 N / ((9.1 × 10^(-3) kg) / L)
Simplifying and rearranging the equation, we get:
L = (240 N / ((9.1 × 10^(-3) kg)) * (0.42 m) * (240 Hz) * (0.42 m) * (240 Hz)
Now, we can calculate the length of the string (L) using the given values in the equation.