Question

In: Physics

Ocean waves carry a lot of energy – according to the Bureau of Ocean Energy Management,...

Ocean waves carry a lot of energy – according to the Bureau of Ocean Energy Management, about 2,640 TWh/yr (terawatt-hours per year) off the immediate coast of the US. 1 TWh/yr is enough to power 93,850 average U.S. homes. If some of this energy could be harnessed, it could be a great source of clean, renewable energy. Assume that an ocean wave is described by a sinusoidal transverse wave along the surface of the ocean and that has an amplitude of 2.0 m, a frequency of 0.25 Hz, and a wavelength of 130 m. Make a model (i.e. write down an equation and calculate) for the energy that a wave-energy harnessing buoy shown below will generate from those waves per cycle and in one whole day. How many homes could this amount of energy power per year? Assume that the buoy always floats at the surface and that the generator shown on the right will convert all of the gravitational energy of the buoy into usable electric energy. The combined mass of the buoy and the translator is 50.0 kg. Deeper thinking question (write a sentence or two in response): Where does the wave energy come from? What happens to the wave after it has passed the buoy? Why?

(i) What is the total vertical movement of the buoy in one wave cycle?

(ii) How much gravitational potential energy does the buoy gain from the upward movement in each wave cycle?

(iii) How many wave cycles pass the buoy in one hour and in one day?

Solutions

Expert Solution

Answer:-

Here, we have some data,

Let's begin with hints to start answering point to point;

(1) Total vertical movement of buoy in one wave cycle is 4 meter because it has an amplitude of 2m.

(2) Gravitational potential energy buoy gain from upward movement in each wave cycle is,

GPE=Mgh=50kg*9.8m/sec2*4m

=1960J/cycle [ Therefore g=9.8m/s2 , m =50kg, h=4m ]

(3) Buoy having 0.25Hz Frequancy (f):-

So,  

--> It takes 4s for one cycle

So, we can calculate that

in 60sec (1 hour) thare are 15 cycle/min

in 36005(1 hour) thare are 900 cycle/hr

in 3600*245(1 day) thare are 21600 cycle/day

Thus, 21600 cycle passes in one day:-

* The wave equation for given data is,

[ initial phase = 0 ]

(Where, t=time, A= amplitude, w= = angular frequancy, = wave vector, x= distance)

* Buoy having a mass (m) =50 kg if it converts all gravitational energy in to electrical energy.

than,

--> As we know GPE is per cycle is 1960J and 1 cycle contain 4s.

thus power per 1 cycle is 1960/4 J/s

Therefore, Power = 490 w

And we having 21600 cycle/day

So, power generated per day is (490*21600)w

p=1.0584*107w

* Power generation per year is,

--> As we know

is enough for 93,850  ( homes)

2 s

  (approx)

--> Approximatly 362 homes get this amount of energy per year.

* This energy comes from ocean waves which is created by moon's (Luna) and earth's gravitations.

-->It converts in gravitational energy and in this case we convert it into electrical energy------


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