The tension in a 2.6-m-long, 1.1-cm-diameter steel cable (ρ = 7800 kg/m3) is 860 N. What is the fundamental frequency of vibration of the cable?

The answer is 6.55 Hz.

I know that I'm going to use the formula:
f=nsqrt(T/4L^2u)
However, I'm not sure how to get the u for this equation. Any help is appreciated.

You need to find the mass of the cable and divide it by the length to get the mass per unit length.

Volume of cable = pi r^2 L
Mass of cable = pi r^2 L * 2.6
so
mass/length = 7800 * pi * (.011)^2

Thank you! I was able to figure out the correct answer. However, in this case shouldn't (.011^2) be (.011/2)^2?

To find the value of u in the formula f = n * sqrt(T / (4L^2u)), where f is the fundamental frequency of vibration, n is the mode number, T is the tension in the cable, L is the length of the cable, and u is the linear mass density, you need to calculate the linear mass density of the steel cable first.

The linear mass density (u) is defined as the mass of the cable per unit length. To calculate it, you need to find the mass of the steel cable.

The volume of the cable can be calculated using the formula for the volume of a cylinder:
V = πr^2h
where r is the radius (half the diameter) of the cable and h is the length of the cable.

Given that the diameter of the cable is 1.1 cm, the radius (r) would be 0.55 cm (or 0.0055 m) and the length (h) is given as 2.6 m.

Now, we can calculate the volume (V) of the cable:
V = π * (0.0055)^2 * 2.6

The density (ρ) of the steel cable is given as 7800 kg/m^3, which means that the mass per unit volume of the cable is 7800 kg/m^3.

To find the mass (m) of the cable, we multiply the volume (V) by the density (ρ):
m = V * ρ

Now that we have the mass of the cable, we can calculate the linear mass density (u):
u = m / L

Finally, substitute the known values (T = 860 N, L = 2.6 m) along with the calculated value of u into the formula f = n * sqrt(T / (4L^2u)), and solve for f.

Once you have calculated the value of f, you will get the fundamental frequency of vibration of the cable to be 6.55 Hz.