So let us have a look at the principle of work and energy.
Now, before we start to look at the definition of work, I want to state that the principle of work
and energy for our case is a tool that we can use to solve some kinetics problems so we can use the
principle of work and energy, for instance, to solve for the velocity of a particle.
And this is very handy.
As you'll see, we'll do an example of applying the principle of work and energy.
So we're going to use it as a tool in our toolbox to solve kinetics problems.
OK, so in physics, work is defined as a measure of energy transfer that occurs to an object that is
moved over a distance while an external force is applied.
So a force will do work on a particle only if the particle is undergoing a displacement in the direction
of the force.
So if if the force is constant, you can calculate the work done on a certain body by multiplying the
length of the path that the body has traveled by the force acting along this path.
Now, as you'll see, sometimes a force will not act in the same direction as the path traveled by the
body or the particle.
But then we need to take the component of the force in the direction of travel and we need to multiply
that component by the distance travelled to get the work done.
So then we also have the concepts of positive and negative work.
So if the force or component of the force and the displacement have the same direction, then the work
done is positive.
Well, if the force or the component of the force and the displacement caused by this force of opposite
directions, then the work done is negative.
So let's look at an example to illustrate when his work positive and when his work negative.
So what I've drawn here is a person doing a bungee jump.
So there's a cliff and the person stands on the edge of the cliff and is attached with a bungee rope
and will jump off the cliff and then.
The bungee rope will slow down the person's velocity and it will pull him back up and so forth, like
someone doing a bungee jump.
So on the left hand side, in the first case, the person has just just jumped over the cliff and the
bungee rope has not yet stretched.
So you can assume that the person is kind of in a freefall.
In this case, gravity does positive work, so the force of gravity pulls the person down and the person
moves in the direction in which the force of gravity acts, that is downwards.
And so the force and the direction of movement is in the same direction.
So the work done is positive.
If we go to the chaos in the middle, this is where the bungee rope has now tightened and started to
stretch, so now the bungee rope applies a force to the person in the upward direction, but the person
is still moving downward.
Remember, this bungee jump is still stretching out.
So gravity still does positive work on the person because the person is moving downward still.
But the spring force, that's the force and the bungee rope works upwards.
So that would be a negative work because it's in the opposite direction of movement.
Now we go to the case of the right person has now moved down up until a point where the velocity became
zero and is now starting to move up again.
So the Bunshiro pulls the person up.
Now, the work done by gravity is negative because the person moves in the opposite direction.
Then the direction in which the force of gravity acts, but now the work of the spring force is positive
because the spring force works up and the person moves up.
So these three cases illustrates to you when a work will be negative and when it will be positive.
It has to do with the direction of the force and the direction of the movement.
Now, it's very important to have the correct sign for the work because you will see when we apply the
principle of work and energy, the work either adds kinetic energy or it subtracts takes away kinetic
energy, and that determines the final velocity of a particle.
And so it's important that we realize where the energy is added to the system or taken away from the
system.
It's added when work is positive and it's taken away when work is negative.
We will look at an example in the end.
So now that we know how work is defined in the in terms of physics and we know when work is positive
and when work is negative, we can state the principle of work on energy.
Now, the principle of work and energy says.
T one plus the some of the work.
Then on the particle is equal to T to now D one and two, those are kinetic energies.
So it says the kinetic energy you start off with of a certain particle.
Plus, the work done on that particle will give you this kinetic energy after the work has been done
on the particle, now that work can be positive or it can be negative if it's positive in the kinetic
energy after the work has been done on, the particle will be larger and the kinetic energy before the
work has been done.
If the work is negative, then the kinetic energy will decrease.
So one, that's kinetic energy, one, we know kinetic energy is given by half in the square, so that
be halftimes, the mass times, the initial velocity squared plus the work done will give you the kinetic
energy in the end, which is one of in the two squared.
So the velocity of the particle has either gone up with positive work or it has decreased with negative
work.
Now the work part can be done.
In our case, we will consider a constant force the force by linear spring and also the force of gravitation.
And so for each one of these forces, there's a slightly different formula that you use to calculate
the work.
And then if we have the velocity at the start and we have the mass and we can calculate the work, then
we can calculate the velocity of the particle after the work has been applied.
And that will be the example that we'll look at in the end as well.
So let us look.
Our work can be calculated for different types of force applied to a body.
So first of all, let's look at a constant force applied to a body.
And what I've drawn here in red is that let's say it's great and it has a certain mass and is a force,
if applied to it at a certain angle to the horizontal, this red great moves only horizontally.
So there's only work done in the horizontal direction.
It's not lifted off the ground.
It's only moved in a horizontal direction.
So we will calculate that work by taking the force component in a horizontal direction and multiplying
it by the distance at this great travels and that distance here is indicated by us.
So how do we get the component of the force in the horizontal direction that would be if cosine of theta?
And if we multiply that by the distance, we get the work done on this that we can also write down as
if DOT is the DOT product indicates that we take cosine of the angle between the force and the direction
of movement.
And if we then write down the principle of work and energy equation, it would be one of M.V. squared,
the kinetic energy plus the work done if cost theta is is equal to half M.V. two squared.
So that's in the case of a constant force applied at an angle.
If we look at a spring.
I have drawn a crate that is pressed against a spring that is attached to a wall, so the spring will
exert a force to the right, and if we push the crate to the left, it means the force of the spring
is in the opposite direction of movement.
If we push the crate to the left.
So are we calculate spring work is to say you caused by this spring is minus of K, which is the spring
constant multiplied by X2 squared minus X one squared.
Now the Xs, those are the positions, the start and the end position of this movement.
Now it is important that we use the correct points for the start.
In the end, you'll notice that in the standard definition of work by spring it is given with a negative
sign.
So that means that X two must be larger than X one if we push the grade to the left in this picture,
because the amount in brackets need to be positive so that we can have a negative work.
OK, I will explain that again in the example, but if we then write down the principle of work and
energy with a spring force, we'll see it's half M.V. one squared, minus one half K X2 squared minus
X, one squared is equal to a half in the two squared.
And then the third force that we consider is that of gravity.
So if a great falls down, gravity does work on this crate and that work is given by minus M g y to
minus Y one y y two is the end position and one is the start position now.
Why two is lower than why one?
Because we want that distance in the brackets to be negative, because we multiply that with a negative
sign in front of the M, we get positive work when it falls down.
In the case of the bungee jumper, when the bungee jump, it moves up, then work.
Done is negative.
OK, so there's a standard way to define the work done by gravity, and that is, what, a minus sign
in front of the IMG?
But you should just be sure that it makes sense if the object moves in the same direction as the force,
then the work that you get out of that equation has to be positive.
So just double check that always.
So if we write down the principle of work and energy with gravity as a force and is one half of one
squared minus N g y two minus one is equal to one half in the squared.
Now we can have different forces doing work on a body at the same time, like we saw in the bungee jumping
example, we've got the spring force of the bungee robe and you have gravity.
So in the next video, we're going to look at an example where a spring falls and the force of gravity
works in on a body.
And we want to calculate the final velocity of the body.