# Energy and work relationship definition

### Introduction to work and energy (video) | Khan Academy

This definition can be extended to rigid bodies by defining the work of the torque and rotational This relationship is generalized in the work-energy theorem. Kinetic energy can be defined as the capacity of an object to do work by is the Work-Energy theorem or the relation between Kinetic energy and Work done. To do joules of work, you must expend joules of energy. conservation of energy and the application of the relationships for work and energy, so it is.

Let's say I apply a force, F, for a distance of, I think, you can guess what the distance I'm going to apply it is, for a distance of d. So I'm pushing on this block with a force of F for a distance of d. And what I want to figure out is-- well, we know what the work is. I mean, by definition, work is equal to this force times this distance that I'm applying the block-- that I'm pushing the block.

But what is the velocity going to be of this block over here? It's going to be something somewhat faster. Because force isn't-- and I'm assuming that this is frictionless on here. So force isn't just moving the block with a constant velocity, force is equal to mass times acceleration. So I'm actually going to be accelerating the block.

So even though it's stationary here, by the time we get to this point over here, that block is going to have some velocity. We don't know what it is because we're using all variables, we're not using numbers. But let's figure out what it is in terms of v.

So if you remember your kinematics equations, and if you don't, you might want to go back. Or if you've never seen the videos, there's a whole set of videos on projectile motion and kinematics.

## Work (physics)

But we figured out that when we're accelerating an object over a distance, that the final velocity-- let me change colors just for variety-- the final velocity squared is equal to the initial velocity squared plus 2 times the acceleration times the distance. And we proved this back then, so I won't redo it now. But in this situation, what's the initial velocity? Well the initial velocity was 0.

So the equation becomes vf squared is equal to 2 times the acceleration times the distance. And then, we could rewrite the acceleration in terms of, what? The force and the mass, right? So what is the acceleration? Well F equals ma. Or, acceleration is equal to force divided by you mass. So we get vf squared is equal to 2 times the force divided by the mass times the distance. And then we could take the square root of both sides if we want, and we get the final velocity of this block, at this point, is going to be equal to the square root of 2 times force times distance divided by mass.

And so that's how we could figure it out. And there's something interesting going on here. There's something interesting in what we did just now. Do you see something that looks a little bit like work?

You have this force times distance expression right here.

Force times distance right here. So let's write another equation. If we know the given amount of velocity something has, if we can figure out how much work needed to be put into the system to get to that velocity. Well we can just replace force times distance with work. Because work is equal to force times distance. So let's go straight from this equation because we don't have to re-square it. So we get vf squared is equal to 2 times force times distance. Took that definition right here.

Let's multiply both sides of this equation times the mass. So you get mass times the velocity.

### Momentum, Work and Energy

And we don't have to write-- I'm going to get rid of this f because we know that we started at rest and that the velocity is going to be-- let's just call it v. So m times V squared is equal to 2 times the work. Divide both sides by 2. Or that the work is equal to mv squared over 2. Just divided both sides by 2. And of course, the unit here is joules. So this is interesting. Now if I know the velocity of an object, I can figure out, using this formula, which hopefully wasn't too complicated to derive.

I can figure out how much work was imputed into that object to get it to that velocity. And this, by definition, is called kinetic energy. This is kinetic energy. Therefore, the total momentum at the end must be what it was at the beginning. You may be thinking at this point: Nevertheless, we do know that momentum will be conserved anyway, so if, for example, the two objects stick together, and no bits fly off, we can find their final velocity just from momentum conservation, without knowing any details of the collision.

First, it only refers to physical work, of course, and second, something has to be accomplished. Consider lifting the box of books to a high shelf. If you lift the box at a steady speed, the force you are exerting is just balancing off gravity, the weight of the box, otherwise the box would be accelerating.

Putting these together, the definition of work is: To get a more quantitative idea of how much work is being done, we need to have some units to measure work.

This unit of force is called one newton as we discussed in an earlier lecture. Note that a one kilogram mass, when dropped, accelerates downwards at ten meters per second per second.

This means that its weight, its gravitational attraction towards the earth, must be equal to ten newtons. From this we can figure out that a one newton force equals the weight of grams, just less than a quarter of a pound, a stick of butter. The downward acceleration of a freely falling object, ten meters per second per second, is often written g for short. Now back to work. In other words approximately lifting a stick of butter three feet.

This unit of work is called one joule, in honor of an English brewer. To get some feeling for rate of work, consider walking upstairs. A typical step is eight inches, or one-fifth of a meter, so you will gain altitude at, say, two-fifths of a meter per second. Your weight is, say put in your own weight here! A common English unit of power is the horsepower, which is watts.

- Momentum, Work and Energy
- Introduction to work and energy
- Work, power and efficiency - AQA

Energy Energy is the ability to do work. For example, it takes work to drive a nail into a piece of wood—a force has to push the nail a certain distance, against the resistance of the wood. A moving hammer, hitting the nail, can drive it in. A stationary hammer placed on the nail does nothing. Another way to drive the nail in, if you have a good aim, might be to simply drop the hammer onto the nail from some suitable height.

By the time the hammer reaches the nail, it will have kinetic energy. It has this energy, of course, because the force of gravity its weight accelerated it as it came down. Work had to be done in the first place to lift the hammer to the height from which it was dropped onto the nail. In fact, the work done in the initial lifting, force x distance, is just the weight of the hammer multiplied by the distance it is raised, in joules.

But this is exactly the same amount of work as gravity does on the hammer in speeding it up during its fall onto the nail. Therefore, while the hammer is at the top, waiting to be dropped, it can be thought of as storing the work that was done in lifting it, which is ready to be released at any time. To give an example, suppose we have a hammer of mass 2 kg, and we lift it up through 5 meters.

This joules is now stored ready for use, that is, it is potential energy. We say that the potential energy is transformed into kinetic energy, which is then spent driving in the nail. We should emphasize that both energy and work are measured in the same units, joules.

### Work (physics) - Wikipedia

In the example above, doing work by lifting just adds energy to a body, so-called potential energy, equal to the amount of work done. From the above discussion, a mass of m kilograms has a weight of mg newtons. It follows that the work needed to raise it through a height h meters is force x distance, that is, weight x height, or mgh joules. This is the potential energy. Historically, this was the way energy was stored to drive clocks.

Large weights were raised once a week and as they gradually fell, the released energy turned the wheels and, by a sequence of ingenious devices, kept the pendulum swinging. The problem was that this necessitated rather large clocks to get a sufficient vertical drop to store enough energy, so spring-driven clocks became more popular when they were developed.