A 1.88-m long rope has a mass of 0.139 kg. The tension is 55.4 N. An oscillator at one end sends a harmonic wave with an amplitude of 1.09 cm down the rope. The other end of the rope is terminated so all of the energy of the wave is absorbed and none is reflected. What is the frequency of the oscillator if the power transmitted is 118 W?

You need to know that the power in a vibrating string is

P = (1/2) ì ù^2 A^2 v .

Reference: http://hyperphysics.phy-astr.gsu.edu/hbase/waves/powstr.html

In that equation, ì (mu) is the mass per unit length, ù (omega) is the angular frequency of oscillation, A is the amplitude and v is the wave speed,

v = sqrt (T/ì). T is the tension.

With those equations, you can solve the problem yourself. Your value of mu (ì) is
ì = 0.0739 kg/m.
The wave speed is
v = sqrt(55.4/0.0739) = 27.4 m/s

Using the information provided about the power, solve for the angular frequency omega (ù). Divide that by 2 pi for the frequency in Hz.

My symbols for omega and mu did not come out in Greek like they should, but I hope you get the idea.

To find the frequency of the oscillator, we can use the equation for the power transmitted by a wave on a rope:

Power = (1/8) * π^2 * f^2 * ρ * A^2 * v

Where:
- Power is the power transmitted by the wave (118 W)
- π is a mathematical constant approximately equal to 3.14159
- f is the frequency of the oscillator that generated the wave (what we need to find)
- ρ is the linear mass density of the rope (mass per unit length, given as the mass divided by the length)
- A is the amplitude of the wave (1.09 cm, convert to meters by dividing by 100)
- v is the velocity of the wave along the rope

First, let's find the linear mass density (ρ):

ρ = mass / length
= 0.139 kg / 1.88 m

Next, we need to find the velocity of the wave (v). We can use the equation for wave velocity:

v = √(Tension / ρ)

Where:
- Tension is the tension in the rope (55.4 N)
- ρ is the linear mass density we found earlier

Now, let's calculate the velocity:

v = √(55.4 N / ρ)

With the velocity calculated, we can rearrange the equation for power to solve for the frequency (f):

f = √((8 * Power) / (π^2 * ρ * A^2 * v))

Plug in the known values:

f = √((8 * 118 W) / (π^2 * ρ * A^2 * v))

Now, substitute the previously calculated values for ρ and v:

f = √((8 * 118 W) / (π^2 * (0.139 kg / 1.88 m) * (1.09 cm / 100 m)^2 * √(55.4 N / ρ)))

Simplify and solve for f:

f = √((8 * 118 W) / (π^2 * (0.139 kg / 1.88 m) * (0.0109 m)^2 * √(55.4 N / ρ)))

Calculate the value of f using a calculator, and that will give you the frequency of the oscillator.