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### Course: AP®︎/College Physics 1 > Unit 5

Lesson 4: Gravitational potential energy and conservative forces# Conservative forces

A conservative force exists when the work done by that force on an object is independent of the object's path. Instead, the work done by a conservative force depends only on the end points of the motion. An example of a conservative force is gravity. Created by David SantoPietro.

## Want to join the conversation?

- when can the mechanical energy be negative(real life example)?(8 votes)
- Mech. E = KE + PE

KE is never negative, and it can only be 0 if the object is not moving.

PE, however can be negative. In fact, PE is very often negative, but an extremely important fact about PE is that it is always up to YOU to decide what position is the PE = 0 position. For example, we usually say that gravitational potential energy PEg = mgy, where y is the height of the object. But the y = 0 height can be whatever you want. It can be the height where the object starts, where it finishes, the height of the ground level, or any other height. We just make the choice that we think will make calculations easiest. So to give you an example, if we are holding an object at y = 0 (say, ground level), and our friend is down in a hole (say, y = -10m), and we drop the object and he catches it and holds it at rest, then it has KE = 0 and PE = mgy < 0, so ME < 0.

Here is another very real-life example:**the earth has negative mechanical energy**, if you ask almost any astronomer! The reason is that the earth is "down in a hole" relative to the sun's gravitational pull. It has some KE, since it is moving in its orbit, but not enough to get free from the sun. Most astronomers choose to say that if an object has enough KE to get free from a big gravitational object, then it has positive ME, but if it is trapped by the big object, then the negative PE is greater in magnitude than the positive KE, so the ME is negative.(34 votes)

- Right near the end of the video it is stated that the thermal energy is dissipated into the ground and into the block, but doesn't it also radiate into the surrounding air?(8 votes)
- Yes, you are correct, the heat is dissipated all around the block.(12 votes)

- A problem in my physics book involves a mass hanging from a rope. It is pushed to the right by an applied variable force. It begins at rest and ends at rest. One of the parts of the question asks what is the work done by the tension in the rope. I reasoned that the work done by tension was a constant force and took the dot product of tension with the displacement of the mass to come up with an answer that it did some negative work on the mass. The answer in the book says that the work done was zero. Is this because it starts and ends at rest, or is it because the applied force is doing the work?(3 votes)
- If the force is always perpendicular to the displacement, as is the case with rope tension and a hanging mass, then the work will be zero since work is the dot product of force and displacement. The work that was done is by you when you lift the mass before releasing it.(10 votes)

- can an object have PE and KE at the same time?

like a bird flying -does it possess PE or KE?(4 votes)- I think so. A good example of this is when something is in the middle of falling due to gravity. Some of its energy is kinetic because gravity has moved it down from its original position. At the same time, if the object hasn't hit the ground yet, it still has some gravitational potential energy yet to be transferred into kinetic energy.(2 votes)

- As explained above, we came to know that mass spring system is a conservative system, but wait, when the block attatched to the spring moves to and fro, it is also subjected to frictional force. So in this case it should be a non-conservative system rather than the conservative.

Why is it so that the mass spring system is a conservative system?(2 votes)- In a real world situation there will be frictions forces and cause the system to be non-conservative. In most example systems you work with while learning physics you simplify them so that you deal with one aspect at a time.(5 votes)

- Why when you have friction, there will be thermal energy?(2 votes)
- what is meant by Torque?(2 votes)
- torque means a force that tends to cause rotation(4 votes)

- how can we know that a force is path dependent?(2 votes)
- Having the force's equation, we can find it, but it is required to know how to find a function's curl. For example, if you calculate the curl of any force of the type F = f(r) --- (function of radius only), like gravity (k/r²), it will result in 0. Having curl equal to 0 will immediately imply that force field is conservative. If curl is not 0, the force field will not be conservative.(3 votes)

- What are more examples of conservative forces?(2 votes)
- If i go for A to B in a straight line and in a semicircular path then the work done by friction will be same?

it should be same because work done = force*

displacement. hence if it is same then how is non conservative force path independent(2 votes)- intuitively, I do not think the work done by friction will be the same. The longer the path, the more work will be done against fristion(1 vote)

## Video transcript

What's a conservative force? Conservative forces
are any force wherein the work done by
that force on an object only depends on the initial and
final positions of the object. In other words, the work done by
a conservative force on a mass does not depend on the
path taken by that mass. If the work done by a
force follows this rule, then we call it a
conservative force. For instance, the gravitational
force on a 5 kilogram mass is 49 newtons. If the mass moves downwards
by an amount of 6 meters, the work done by gravity
is going to be 294 joules. Now let's start over. Say the mass again
moves down 6 meters. But then it moves up 6 meters,
then down again 6 meters. The work done by gravity
for the first downwards trip was 294 joules. Then for the upwards trip,
since the gravitational force is pointing in the opposite
direction of the motion of the mass, the
work done by gravity is going to be
negative 294 joules. Then for the last
trip downwards, the work again is
positive 294 joules. That means that the total work
done on the mass from gravity is still 294
joules, just like it was when the mass was
lowered only once. In other words, the work done
by the gravitational force doesn't depend on the specifics
of the path taken by the mass. The work done by
gravity only depends on the initial and final
position of the mass. In fact, you could
allow the mass to take any path from
this initial point to the final point, and
the work done by gravity is still just going
to be 294 joules. Because the work done
by gravity doesn't depend on the path
taken, we call gravity a conservative force. The force exerted by a
spring is another example of a conservative force. The total work done
on a mass by a spring does not depend on the
path taken by the mass. It only depends on the
initial and final positions of the mass. The term conservative
comes from the fact that conservative forces
conserve mechanical energy, whereas non-conservative
forces do not conserve mechanical energy. Mechanical energy is kinetic
energy and potential energy. An example of a non-conservative
force is friction. If I move a mass along a
table from point A to point B, friction does a certain
amount of negative work on the mass, which creates
some thermal energy. If instead of going
straight from A to B, I make the block go from A to B
back to A over and over again, the work done by friction
will become larger and larger. And it'll generate more
and more thermal energy. Because the work
done by friction depends on the path
taken, friction is not a conservative force. Similarly, air resistance
is not a conservative force since the work done
by air resistance depends on the specifics
of the path taken. It's useful to note that
if a force is conservative, you could define a potential
energy for that force. That's why conservative forces
like gravity and spring forces have potential energies
associated with them. And non-conservative
forces like friction do not have potential
energy associated with them. This makes sense
because if you do work against the
gravitational force by lifting a mass
in the air, you can get that energy back out
by letting the mass fall down, turning potential energy
into kinetic energy. Similarly, if you do work
against the spring force by compressing a spring,
you can get that energy back out by letting the
spring decompress, which turns the stored potential
energy into kinetic energy. But if you do work against
the force of friction, you'll have a hard time trying
to get that energy back out. The energy's been dissipated
into the form of thermal energy and is now randomly
distributed along the ground and into the block.