### Video Transcript

A string of length 5.0 meters and
mass of 90 grams is held under a tension of 1.0 times 10 to the two newtons. A wave travels down the string that
is modeled as ๐ฆ as a function of ๐ฅ and ๐ก equals 0.010 meters times the sin of
0.40 ๐ฅ minus 1170.12 times ๐ก, where ๐ฅ is measured in meters and ๐ก is measured in
seconds. What is the power of the wave over
one wavelength?

Letโs begin our solution by
highlighting some of the vital information given. Our string has a length of 5.0
meters. Weโll refer to that as capital
๐ฟ. And it has a mass of 90 grams. Weโll refer to that as ๐. The string is held under a tension
of 1.0 times 10 to the two newtons. Weโll call that value capital ๐น
sub ๐ก. Weโre also given an equation, ๐ฆ as
a function of ๐ฅ and ๐ก, which tells us the position of the string over time. We are asked to solve for the power
of the wave over one wavelength. Weโll call this capital ๐ sub avg
for average power.

Letโs begin our solution by looking
at this equation, ๐ฆ as a function of ๐ฅ and ๐ก, a little more closely. Here, weโve written down everything
weโve been given about the problem on the right. But we can go even further by
looking into this equation, ๐ฆ as a function of position and time. Whenever we have an equation of
this form, certain parts of the equation represent particular values of our
wave.

The prefactor before the sine
function, which in this case is 0.010 meters, that value is the amplitude of the
wave, ๐ด. As we look farther along the
equation, the value multiplying ๐ฅ, which in our case is 0.40, is equal to the wave
number of our wave represented by ๐. In further along, the value
multiplying the time ๐ก, which in our case is 1170.12, that is equal to the waveโs
angular frequency ๐. So weโve now greatly increased are
known information about this problem by analyzing this position and time
function.

Now letโs refer to a relationship
between the average power of a wave on a string and the properties of that
string. The average power on the string,
that is the power over one wavelength, is equal to one-half times the square root of
๐, the mass of the string per unit length, multiplied by ๐, the tension in the
string, times ๐ squared multiplied by the amplitude, ๐ด, squared.

When we apply this relationship to
our scenario, we can see that for each of the variables on the right side of this
equation, we have a value either given in the problem statement or derived from the
wave position equation, except for the variable ๐, the mass per unit length of the
string. But, given that ๐ is defined as
the mass of the string divided by its length, we can use the given mass and length
of our string to solve for that variable. Weโre now ready to begin inserting
the values for our variables to the right side of our equation for average
power.

Before we multiply these values
together, letโs convert the mass of 90 grams to a mass in kilograms. 90 grams is equal to 0.090
kilograms. Now all the values in this
expression are in SI base units and ready to be multiplied.

When we enter these values on our
calculator, to two significant figures, we get an average power of 92 watts. That is how much power the wave has
over one wavelength.