I'm going to show you some examples of how
to solve problems involving work.
Imagine a 4 kilogram trashcan.
The trashcan is disgusting.
So someone ties a string to it and pulls
on the string with a force of 50 newtons.
The force of kinetic friction on the trashcan while it slides
is 30 newtons.
The trash can slides across the ground
for a distance of 10 meters.
Let's try to find the work done by each force on the trash can
as it slides across the ground.
To find the work done by each force,
we should recall the formula definition of work.
Work equals Fd cosine theta, where
theta is the angle between the force doing
the work and the direction the trashcan is moving.
There are four forces involved here-- tension,
the normal force, the gravitational force,
and the force of kinetic friction.
In finding the work done for all of these forces,
the displacement is going to be 10 meters.
But the value of the force and the angle
between that force and the displacement
is going to differ for each of the forces.
For instance, to find the work done by the force of tension,
we'll plug in the size of the tension, which is 50 newtons.
The displacement is 10 meters.
And since the tension force is pointed
in the same direction as the displacement,
the angle between the force of tension and the displacement
is 0 degrees.
And since cosine of 0 is 1, the work done by the tension force
is 500 joules.
To find the work done by friction,
we'll plug in the size of the force of friction,
which is 30 newtons.
The displacement is still 10 meters.
And since the force of friction points
in the opposite direction as the displacement,
the angle between the force of friction and the displacement
is 180 degrees.
Since cosine of 180 is negative 1,
the work done by the force of friction
is negative 300 joules.
Now let's figure out the work done
by the gravitational force.
The force of gravity is mg.
So the force of gravity is 4 kilograms
times 9.8 meters per second squared, which is 39.2 newtons.
The displacement is again 10 meters.
But the angle between the gravitational force
and the direction of the displacement
is 90 degrees in this case.
And since cosine of 90 is 0, the gravitational force
does no work on this trashcan.
Similarly, if we were to find the work done
by the normal force, the angle between the direction
of the displacement and the normal force is 90 degrees.
So the normal force also does no work on the trashcan.
This makes sense because forces that
are perpendicular to the motion can never
do any work on that object.
So that's how you can find the work done by individual forces.
And if we wanted to know the net work done on this trashcan,
we could just add up the work done by each individual force.
So the net work is going to be 200 joules.
Now that we know the net work done on the trashcan,
we can use the work-energy principle
to figure out the speed of the trashcan
after it's slid the 10 meters.
The work-energy principle says that the net work done
on an object is equal to the change
in kinetic energy of that object.
So 200 joules is going to equal the difference
in kinetic energy.
If we assume the trashcan started
at rest, which seems reasonable, the initial velocity is 0.
So we can solve for the final speed of the trashcan, which
comes out to be 10 meters per second.
This time, let's say you take the trashcan
and lift it upwards with a constant velocity
for a distance of 2 meters.
In order to lift the trashcan up with a constant velocity,
you need to push with a force equal to the weight
of the trashcan, which means you have to push upwards
with a force of 39.2 newtons.
So to find the work done by the force that you exert,
the force is going to be 39.2 newtons.
The displacement is going to be 2 meters.
And the angle between the force and the displacement
is going to be 0 degrees because the direction of the force
that you exert is in the same direction as the displacement
of the trashcan.
So the work that you've done in lifting up this 4 kilogram
trashcan is 78.4 joules.
To find the work done by the force of gravity,
we can use the force of gravity, which is again 39.2 newtons.
The displacement is again 2 meters.
But the angle between the direction
of the displacement and the gravitational force
is 180 degrees because the displacement points up
and the gravitational force points down.
So the work done by the gravitational force
is negative 78.4 joules, which means
the net work done on the trashcan is 0.
And that makes sense.
Because since the trashcan moved upwards with constant velocity,
there was no change in the kinetic energy of this object.