Video: Finding the Gravitational Potential Energy of Water behind a Tidal Barrage

Water is allowed to pass through a tidal barrage until the tide is at its highest. The barrage is then closed, and the tide beyond the barrage lowers, while the water behind it stays at the same height. The water behind the barrage is then allowed to flow back through the turbines into the sea. If the mass of the extra water held back by the barrage is 2.4 × 10¹⁰ kg, and it has an average height of 1.5 m above the water in the sea beyond the barrage, what is the gravitational potential energy of the water held back by the barrage? Use a value of 9.8 m/s² for the acceleration due to gravity.

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Video Transcript

Water is allowed to pass through a tidal barrage until the tide is at its highest. The barrage is then closed and the tide beyond the barrage lowers, while the water behind it stays at the same height. The water behind the barrage is then allowed to flow back through the turbines into the sea. If the mass of the extra water held back by the barrage is 2.4 times 10 to the power of 10 kilograms and it has an average height of 1.5 meters above the water in the sea beyond the barrage, what is the gravitational potential energy of the water held back by the barrage? Use a value of 9.8 meters per second squared for the acceleration due to gravity.

Okay, so this question is really, really long and there’s a lot of information that we’ve been given. So let’s start by underlining all the important bits so we don’t miss any information out. Firstly, we’re told that water is allowed to pass through a tidal barrage until the tide is at its highest. We’re told that the barrage is then closed and the tide beyond the barrage lowers, while the water behind the barrage stays at the same height.

Next, we’re told that the water behind the barrage is allowed to flow back through the turbines into the sea. We’re also told that the mass of the extra water held back by the barrage is 2.4 times 10 to the power of 10 kilograms and that it has an average height of 1.5 meters above the water in the sea beyond the barrage.

What we’ve been asked to do is to find the gravitational potential energy of the water held back by the barrage. And we’re told to use a value of 9.8 metres per second squared for the acceleration due to gravity. okay, so let’s draw a diagram to help us visualise what’s going on.

First of all, let’s say that this is our tidal barrage and this is the turbine through which the water will flow later. Let’s say that to the left of the barrage we have the sea and to the right of the barrage we have some sort of reservoir. Now, this reservoir is where the extra water gets held back by the barrage.

So firstly, we’ve been told that water is allowed to pass through the barrage until the tide is at its highest. So let’s say that the sea is at its highest tide around this level and this water is allowed to pass through the barrage until the reservoir is also at the same height. Now this level that we’ve drawn here is high tide. And once the sea reaches high tide, we’re told that the barrage closes. So now there is no movement of water from one side of the barrage to the other. Now, the sea is gonna do its own thing and the tide is going to change again. So the tide will lower and that’s exactly what we’ve been told will happen. The tide lowers. So the sea level lowers. But the barrage is still holding back the water in the reservoir. So that stays at the same height.

Now, once the sea level is lowered, what we do is start to allow the water from the reservoir to flow through the turbines back into the sea. Now, the turning turbines are connected to a generator so they will generate electricity. And in essence, what we’re doing is using the gravitational potential energy of the water in the reservoir and harnessing this energy to turn the turbines. So we’re converting gravitational potential energy from the water into mechanical energy — the turning of the turbines — which is then converted to electrical energy. And this electricity can be then passed on to consumers.

So what we need to work out is the gravitational potential energy of the water — more specifically the gravitational potential energy of the extra water held back by the reservoir. That’s all of this water here. Now, what we can do is to set the gravitational potential energy to be zero at this level — at the level of the sea at low tide. This way we can work out the gravitational potential energy of just the excess water.

Now, obviously, the water up here is going to have more gravitational potential energy than the water down here. But that’s why we’ve been given the average height of the water above sea level. The average height is of course going to be half the height of the excess water because some water is down here, some water is higher up. And the average height comes out to be this midpoint here.

Now, we can use just this average height to work out the gravitational potential energy because the excess gravitational potential energy of water in this region is cancelled out by the lack of gravitational potential energy in this region. And for this reason, we can just use the average height when we’re calculating gravitational potential energy.

So how do we calculate gravitational potential energy? Well, we can recall that the gravitational potential energy 𝐸 sub grav of an object is given by the mass of that object multiplied by the gravitational field strength of the Earth multiplied by the height of the object. And this height can be above whatever we defined to be the surface, where gravitational potential energy is zero.

So in this case, we’re looking at the height above this level here. So in other words, gravitational potential energy is zero here because that’s how we’ve defined it. And so the gravitational potential energy of the excess water is going to be the mass of the excess water 𝑚 multiplied by the gravitational field strength of the Earth multiplied by the height — the average height of the water. And we can label this distance ℎ.

Now, at this point, we can sub in all our values because we’ve been told that the mass of the excess water is 2.4 times 10 to the power of 10 kilograms. We’ve also been told the gravitational field strength of the Earth or the acceleration due to gravity is 9.8 meters per second squared. And the average height of the water is 1.5 metres. So we can plug all these values into our calculator to give us an answer of 3.528 times 10 to the power of 11 joules.

However, we can write this in a more convenient fashion. We can write the same number as 352.8 times 10 to the power of nine joules. Now, essentially, what we’ve done is multiplying this part of the number by 100 and divide this part of the number by 100. And when we multiply by 100 and divide by 100, we’re essentially just multiplying by one. So we’ve got the same number again. And therefore, 3.528 times 10 to the power of 11 is the same as 352.8 times 10 to power of nine.

But why would we want to do that? Well, like we said earlier, this is more of a convenient way to write it because 10 to the power of nine joules is the same as one gigajoule. So we can replace 10 to the power of nine with capital 𝐺 which stands for giga. And so if we write it in gigajoules, then we don’t have to worry about standard form. We don’t need to write times 10 to the power of something.

But anyway, so both answers are equally valid. And hence, we’ve got our final answer now. The gravitational potential energy of the extra water held back by the barrage is 3.528 times 10 to the power of 11 joules or 352.8 gigajoules.

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