A 2.49-m-long string, fixed at both ends, has a mass of 7.17 g. If you want to set up a standing wave in this string having a frequency of 455 Hz and 5 antinodes, what tension should you put the string under?

String mass per unit length is

w = 0.00717/2.49 = 2.88*10^-4 kg/m

With five antinodes, the wavelength is
L = 2/5 * 2.49 = 0.996 m

The frequency is f = 455 Hz

The wave speed is C = L*f = 453.2 m/s

The tension T must provide that wave speed.

sqrt(T/w) = 453.2 m/s

Solve for T, in Newtons

T = w*(453.2)^2 = 59.2 N

To find the tension in the string, we can use the formula for the speed of waves on a string:

v = √(T/μ)

where:
v is the velocity of the wave,
T is the tension in the string,
and μ (pronounced mu) is the linear mass density of the string.

First, we need to find the linear mass density:

μ = m/L

where:
m is the mass of the string, and
L is the length of the string.

Given that the length of the string is 2.49 m and its mass is 7.17 g (or 0.00717 kg), we can substitute these values into the equation to find the linear mass density μ:

μ = 0.00717 kg / 2.49 m

Now, we can move on to finding the velocity of the wave. The velocity of a wave is related to its frequency (f) and wavelength (λ) through the equation:

v = f * λ

Since we are dealing with a standing wave with 5 antinodes, the wavelength (λ) can be found using the formula:

λ = 2L/n

where:
L is the length of the string, and
n is the number of antinodes.

Given that L = 2.49 m and n = 5, we can calculate the wavelength (λ) using this formula.

Next, we know the frequency of the wave is 455 Hz. So, we can now substitute the values of frequency (f) and wavelength (λ) into the velocity equation to find the velocity (v).

Finally, by rearranging the equation for wave velocity (v = √(T/μ)), we can solve for T:

T = v^2 * μ

Substituting the values for velocity (v) and linear mass density (μ) into the equation, we will get the tension (T) in the string.