Lesson Video: Mechanical Energy | Nagwa Lesson Video: Mechanical Energy | Nagwa

# Lesson Video: Mechanical Energy Science • First Year of Preparatory School

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In this video, we will learn how to relate the mechanical, kinetic, and gravitational potential energies for an object that work is done to.

09:40

### Video Transcript

In this video, we will learn how to relate the mechanical, kinetic, and gravitational potential energies for an object that work is done to. Along the way, we’ll come to understand what mechanical energy is.

But before we talk about categories of energy, let’s consider the term work. This is an everyday familiar word, but in physics, work has a very specific meaning. It is defined as the energy transferred when a force moves an object some distance. So work in physics is energy, and it’s the energy transferred by a force moving an object over a distance. For example, say a box is at rest on the ground. We come along and lift the box. Raising it to some height requires that we exert a force on the box. So there’s a force involved, and this force moves the box through a distance equal to its height above ground. We say then that we’ve done work on the box. And notice that by doing this work, we have indeed transferred energy to the box. Its gravitational potential energy has increased.

Gravitational potential energy — we can call it potential energy here for short — is one energy category. Another one is kinetic energy, which is energy due to motion. Imagine that the box is back at ground level. Now, instead of lifting it, we push it along the ground. Our push makes the box move, so we are once again doing work on the box. Now, though, the energy transferred is kinetic energy; the box has started moving after being stationary. So doing work on an object, exerting a force on it to make it move some distance, can transfer both potential and kinetic energy to the object. If we combine these two categories of energy, we get a third category called mechanical energy. The mechanical energy of an object equals the object’s kinetic and potential energies added together.

Let’s think about the mechanical energy of the box when it’s in different positions. At rest on the ground, the kinetic and potential energy of the box are both zero, so its mechanical energy is zero. We’ve seen we can lift the box straight up. When we bring it to rest again, the box’s kinetic energy is zero, but its potential energy has increased to a positive value. So the box’s mechanical energy in this position is positive. We’ve also started with the box on the ground and given it horizontal motion. By pushing on the box, we made it slide along the ground. Though its potential energy remains zero all during this time, its kinetic energy increases from zero to a positive value. So overall, its mechanical energy is positive.

Lastly, imagine we pick up the box and moved it diagonally. This way, the box has positive potential and kinetic energy. Therefore, the box’s mechanical energy is positive.

We can use a formula with symbols to express the fact that mechanical energy is the sum of kinetic and potential energy for an object. Let’s say that 𝐾 stands for kinetic energy. Then we’ll say 𝑃 stands for potential energy. Lastly, if we say that 𝑇 stands for mechanical energy, then we can write 𝑇 equals 𝐾 plus 𝑃. Let’s look now at a few examples to help us better understand mechanical energy.

In which of the following cases is the mechanical energy of an object zero? (A) The object is at rest in any position. (B) The object is at rest on the surface of Earth. (C) The object is on the surface of Earth with any motion.

Let’s remember that the mechanical energy of an object is defined as the sum of that object’s kinetic and potential energies. So in order for mechanical energy to be zero, the object’s potential energy and its kinetic energy must add up to zero. One way for this to happen is if both of those energies are zero.

If we picture this object in relation to Earth’s surface, we can see that the object’s gravitational potential energy is zero when it is on the surface. An object that has zero gravitational potential energy must also have no kinetic energy for its mechanical energy to be zero. This means the object on Earth’s surface must be at rest. We see this agrees with answer choice (B). If an object is at rest on the surface of Earth, its mechanical energy is zero.

Let’s look now at another example.

Which of the following is correct about the difference between the initial and final mechanical energies of a ball that is thrown upward and then caught and held at the same height that it was thrown from? (A) The difference is zero. (B) The difference is nonzero.

Let’s picture this situation. Here we’re holding a ball, and we imagine throwing it straight up from a certain height. The released ball rises, slows down, and comes to a stop. Then it falls back down to our waiting hand. We’re told that we catch the ball at the same height from which it was released. The question is, in this process, does the mechanical energy of the ball change or not? Let’s recall that an object’s mechanical energy is the sum of the object’s kinetic and potential energies. So to find our answer, we’ll want to look at the kinetic and potential energy of the ball at the start and end of its journey.

At the instant the ball is released upward, it has some amount of gravitational potential energy. This is because the ball is some height above Earth’s surface. Let’s label that energy GPE one. Also at that time, the ball has kinetic energy because it’s in motion. We don’t know exactly what the kinetic energy of the ball is, but we do know that it is some specific amount, and we can call that quantity KE one. If we add GPE one and KE one together, we have the mechanical energy of the ball when it is thrown upward. Let’s call this energy ME one.

We want to compare ME one with the ball’s mechanical energy at the instant we catch it when it’s falling downward. That energy, which we can call ME two, equals the ball’s gravitational potential energy plus its kinetic energy at this instant. Notice that we catch the ball at the same height it was released from. So if GPE two is the ball’s gravitational potential energy when it lands in our hand, then GPE two equals GPE one.

Now how about the ball’s kinetic energy? In general, an object’s kinetic energy is one-half its mass times its speed squared. If we think about the ball’s motion as it rises and falls, we know that as it rises, it slows down, and as it falls, it speeds up. If we ignore the effect of the air slowing the ball down, then the amount the ball slows down while ascending and the amount it speeds up while descending are the same. All this means that the ball’s speed upward when we first release it will be the same as its speed downward when we catch it. The speeds are the same. And since the mass of the ball doesn’t change, the ball’s kinetic energy at both of these two instants must be equal as well.

We’re finding then that ME two, the mechanical energy of the ball when we catch it, is equal to ME one, the mechanical energy of the ball at the instant it’s released. For our answer, we’ll choose option (A). The difference between the mechanical energy of the ball at these two instants is zero.

Let’s now work an example where we calculate a specific energy quantity.

The mechanical energy of an object that is at Earth’s surface is 36 joules. What is the kinetic energy of the object?

These two energy categories, kinetic energy and mechanical energy, are related. An object’s mechanical energy equals its kinetic energy plus its potential energy. We can write this as an equation using symbols: 𝑇, mechanical energy, equals 𝐾, kinetic energy, plus 𝑃, potential energy. In this example, we’re given 𝑇, the object’s mechanical energy, as 36 joules. We also know that the object is at Earth’s surface. This means its gravitational potential energy is zero. That’s true for every object at the surface of the Earth.

So in our equation, 𝑇 is 36 joules and 𝑃 is zero joules. 𝑃 plus 𝐾 is 36 joules. Therefore, it must be the case that 𝐾 is 36 joules. Our answer is that the kinetic energy of the object is 36 joules.

Let’s look now at one last example.

A car drives up a hill, slowing down as it climbs the hill. The kinetic energy of the car decreases by 12 joules and the gravitational potential energy of the car increases by 12 joules. What is the change in the mechanical energy of the car?

As this car drives uphill, it slows down. This change in speed impacts its kinetic energy. The change in the car’s elevation changes its gravitational potential energy. We want to find the change in the car’s mechanical energy. This is equal to the change in the sum of its potential and kinetic energies.

We are told that as the car ascends, its kinetic energy decreases by 12 joules. This means that the change in the car’s kinetic energy is negative 12 joules. The reason this change is negative is because the car is slowing down, so its kinetic energy is decreasing. We also know that the change in the car’s potential energy equals positive 12 joules. This change is positive because the car gets higher up as it climbs, which increases its potential energy. Adding these changes in energies together gives zero joules.

Therefore, the change in the car’s mechanical energy is zero joules.

Let’s now summarize what we’ve learned in this video. Doing work on an object will cause its kinetic energy or potential energy or both to change. An object’s mechanical energy is the sum of its kinetic energy and potential energy. Written as an equation with symbols, mechanical energy, 𝑇, equals kinetic energy, 𝐾, plus potential energy, 𝑃.

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