Work, Energy and Power

Hello and Greetings for the day! TODAY, we'll be looking at an important component in classical mechanics, or the 'currency' of classical mechanics: Work, Energy and Power.

Let's introduce you:
Work, energy and power are fundamental components in classical physics. Much of classical physics run with these laws, and a strong understanding of these concepts are crucial for benefitting you in your long journey of Physics(I guarantee this is true).


Let's put it simply, since we're under the premises of Physics Simplified.

Work, is defined as the force applied over a particular displacement. It is a mathematical scalar quantity that can be determined by the dot product of force and displacement. We may also define Work being the flow of energy, and we'll understand why later in the article, because, again, we're in the premises of Physics Simplified, and for the same reason, we'll get to the formula later(but we'll surely get there).


Now, Power, is defined as the rate of work done. Now we remember work being 'The flow of energy' and therefore, let's define Power, being the 'Rate of flow of energy'. Again, we'll understand why this is the case later, and you know exactly why.


Let's get to energy, the real currency of Physics. Remember work being the force over a particular displacement. What do you need when you use your useful muscular force when jogging? Energy! What do you need when you accelerate a car from rest to have a particular displacement(hopefully it has some displacement)? Energy! Therefore, we'll define energy being the Ability to do work.


Now that you're introduced in the simplest and purest and cleanest and neatest and excellentest manner possible, let's proceed a little bit deeper. 


"Look, Physics Simplified, my reader understood, ok?"

-me

Work is being done here, although...

Work

OK, we start with Work, because I did work in making this article but sadly it does not count, and you're doing work in reading this article and it also does not count.  So basically, as we looked above, work is the force applied F  to produce a displacement d. That is logical, because, for example if I push a car that shows off its resistance strength by staying on spot(either that or I do not have enough force, but we do not need to go there). This means, yes I'm applying force, but sadly I'm not doing anything except for wasting my energy. Thus, the work done is zero. So the formula of Work should be as such, that if any one quantity is zero, Work done is zero, and this can only be achieved by multiplying the two quantities. Here comes the best bit for me and the worst bit for Physics Simplified, work can be described as:

W=F*d, where 'W' is the work done, 'F' is the force applied and 'd' is the displacement.

Here, a curious question arises, can the above be the final formula for Work? Can we simply arrive at a conclusion being:

W=F*d?

Well, unfortunately(and fortunately) the answer is no! Remember what we discussed previously? Work is a scalar quantity, whereas Force and Displacement are vector quantities? Now you cannot all of the sudden take these poor quantities and torture them to transform into scalars. We need to smoothly transform them, and that's why we take a dot product. Now when we take the dot product, which is a type of multiplication, we're removing the vector qualities of these quantities. This means we're removing their directions, and therefore, the direction has to go somewhere! It obviously does not go out of the equation, because that wouldn't only make me and you and Physics simplified angry, but the whole world of Physics itself. So we extract this direction 'out' of these quantities and still keep them in the equation by 'cos'. Thus, we have:

W=F.d*cosθ 

Now don't worry about the 'cos' for now, it is a fantastic term used for receiving some useful values after giving the useful input value, theta, or 'θ'. If you're still interested in learning the 'cos' from our simplified series, just click on the 'E-mail' button below and you'll receive a link in your e-mail for the 'cos' function; of course, at the simplest it can be.

So what is this theta you might be wondering. Theta is basically the angle between the force and the displacement. Don't run away now, I'm going to simplify it further. So if I am pushing the car, theta would be 0 because there is simply put no angle. But let's say I'm pulling the box up with the useful help of a strong thread at an angle of 30 degrees and dragging it forwards, then theta is 30 degrees, because that's the angle between the force and the displacement.


Now that you've got the concept of Work, let's proceed to Power.

It is not uncommon to find us saying 'This task requires serious amount of power' or our computer declaring 'Power off', or perhaps hearing it in electrical circuits. Of course, these are all referring to power, but not all are referring to the classical physics power definition.


Power is, simply put, the work done over a particular time. So if I do some work in some time, then that is power. If I do some work in no time(perhaps impossible!) the power is undefined, and if I do nothing in some time, no power was used right? So the power is 0. To satisfy all these parameters above, which are, if there is no work being done in particular time, power is 0, or if there is work being done in no time, power is undefined (because it is closely impossible, right?) we have the equation:

P=W/t, where 'P' is the power, and 'W' is the work per unit time, 't'. Notice how if we substitute work being zero, the power is zero, and if time being zero, the power is undefined.


Now, some of us like to transform equations into other equations, like me, and to not make Physics Simplified furious, we'll do it slowly and logically.

So, we'll find another equation for power. Ready or not, here we start!

W=F.d agreed right? We looked at it previously.

P=W/t so far so good.

Now, if we just take the definition of work, and put it here, we have:

P=F.d/t, great isn't it?

Now, we'll just detach the 'F' from the fraction and multiply it to the whole fraction of d/t:

P=F(d/t), simple isn't it?

Now, if you went through 'Fundamentals of Classical Physics' you'll quickly realise that d/t is nothing but velocity. So, if we redefine that, we have:
P=F*v, simple, wasn't it?


So we conclude power with 2 equations:
1. P=W/t

2. P=F*v

Both mean the same thing, like we saw above. Now for everything we need energy, which is the real currency of Physics. It does not go anywhere, it's just a payment from one body to another, and this is what we call 'Work', as we saw previously, the 'Flow of energy'

Energy in Physics(we can also relate it to real-life) is defined as the ability to do work. As you see, without energy, I cannot push the car(although I have energy, it is not sufficient), Everytime work is done, there is a flow of energy. This simply means that there is a change in form of energy. There are many different types of energy, but we'll be looking at two important ones(if you want to learn more, email physics@palrishabh.com). We'll be looking at Kinetic energy and Gravitational Potential energy. Kinetic energy is basically the energy of movement. So if I manage to somehow push the resilient car, The car(and I) have some Kinetic energy. Now imagine throwing a ball from a building roof(I used the word imagine here, so please don't do it in real). Well, if I release the ball, it has some Kinetic energy, because it is moving; but while I hold it at a particular height, it has Potential energy, or it has a 'Potential' to have Kinetic energy. In other words, it has stored energy which can be converted into Kinetic Energy, in this case, upon the release of the ball. Of course, since we're studying Physics, no numbers, no fun!

So let's start with only 2 formulas. Physics Simplified, close your eyes!


 The formula for Potential energy, is:

mgh, where 'm' is the mass of the body(not weight), 'g' is the acceleration due to gravity(not the gravity) and 'h' is the height. Notice how all of them multiplied and all of them are key in the formulas:

If the mass of the body is 0, there's no body so no Potential energy(We'll look at waves like light in terms of classical physics in another article)

If the acceleration due to gravity is 0, then there is also no potential energy, because nothing is pulling it down. For example, if the same ball was released in empty space, it would relax on-spot because its lazy on its own, and there's nothing to move it!

If the height of the body is 0, then there is no Potential energy, since there is no stored energy. If I have an object on the ground and I release it, it is not going to roll anywhere, because there is no height and thus no potential energy. In other words, the potential energy is directly proportional to the height(and of course, mass and acceleration due to gravity as well).


Now we have the formula for Kinetic energy:

1/2mv^2, yes, it's that simple! Here 'm' is the mass, and 'v' is the velocity.


It makes sense, because if the mass is 0, there's again, nothing(unlike something such as waves) and if the velocity is 0, then it means the object is not moving, and thus the kinetic energy must be 0. 

Connect: Work-energy theorem

We're in Physics and building connections is an important part in Physics. In this connection sector, we shall connect 2 concepts we learnt, Work and Energy, which is connected by Physicists with the 'Work-Energy theorem'. If we remember a definition of work, 'The flow of energy', then we know that Work is basically the change in energy, fancified by physicists with the word 'flow'.


Now, work there also has to be some displacement in work, so there has to be some velocity. Thus, work should be related to Kinetic energy right? YES! Work is defined as 'delta kinetic energy', or the change in Kinetic energy. Here's how one can define work in mathematical formula:

W=ΔKE

Now, if you see in the glossary below, delta simply means change. The mass cannot change, because we have the same body, so its the velocity that changes:
W=KEf-KEi, or work is the change in Kinetic energy.

Now if we substitute the values, we have:
W=1/2mv^2-1/2mu^2, where 'W' is the work done, 'm' is the mass of body with initial velocity 'u' and final velocity 'v'.


If you'd like to learn how to derive it, click on the sector below!

Derive Work-Energy theorem

Let's detive the Work-Energy theorem together shall we?


So, if you remember the Kinematic Equations(we have a dedicated video right here), one of them states:

v^2=u^2+2ad


Now if we take u^2 to the left hand side, we have:

v^2-u^2=2ad


We want to get rid of the 2, so we'll divide both sides by 2:
1/2v^2-1/2u^2=ad


What's missing in both sides of the equation is 'm', so let's multiply both sides by 'm':

1/2mv^2-1/2mu^2=mad


Notice 'ma' is force from Newton's second law, so we have:

1/2mv^2-1/2mu^2=F.d

And now notice 'F.d' is work, so:

W=1/2mv^2-1/2mu^2!


Well done, you learnt to derive a formula you use!

OK, so with your permission, let's conclude our fun journey of Work, Energy and Power. If you'd like access to our specially crafted notes, do not hesitate in reaching out to us at help@palrishabh.com. Finally, this article requires a serious amount of work(and unfortunately it does not count in Physics) and if you found it useful by any means, please contribute to the well-being of the Physics community.


Of course, that's not the end! Let's journey through other Physics trips shall we? How about we change our mood and perspective of journeying to the citysides and try something more complex, Quantum Mechanics? Don't hesitate by the name, we've made it at the simplest form it can be, just for you!

Signing off,

The Phyics Simplified Foundation by palrishabh.com.

I beg pardon, Physics Simplified, if there were any vocabulary my dear reader couldn't understand. As a command from Physics Simplified, here are some key vocabulary:

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This article is an element of palrishabh.com and may not be reproduced, modified or copied. 

Author: Rishabh Pal 

Published: 16-11-2023

Publisher: Physics Simplified 

Organisation: Palrishabh.com