Question about law of conservation of mechanical energy

[Sinistar9]

Hello, i was wondering if someone can help me out.
I finished law of momentum, and learned that for the conservation of momentum to be applicable, the system must be isolated.

An isolated system means no external forces, correct?

if i am correct, then i don't understand the law of conservation of mechanical energy.
it says that conservation of mechanical energy must be in an isolated system.

this is K + U (initals) = K + U (finals)

but the U term is mgh, which has gravity in it. isn't gravity an external force?
so how is this an isolated system?

 

[bossman27]

For energy, Earth is considered part of the system. Since the change in energy imparted to the Earth as a result of for instance, "falling toward a ball," is negligible, we can ignore the change in the Earth's energy in this equation as it essentially cancels itself out. That is essentially the basis of the way we define gravitational potential energy for these common situations.

 

[AJ Bentley]

That's a good question. Let's take a simple example.

If a 1 kg stone falls under gravity, from a height of 10 metres it begins with a U of 1*9.81*10 = 98.1 Joules.

According to the conservation of energy, it ends with a velocity given by 98.1 = 1/2 * mv²
so v= sqrt(2*98.1) = 14 m/sec

Is momentum conserved? Initial momentum is zero. final momentum is 14.

So an external force (gravity) messes with the momentum conservation but not the energy conservation.
It's a bit subtle but with practice you'll learn to recognise situations where momentum and/or energy are being 'injected' into a system (or taken out) and how to handle it. Talk to your teacher about it.

 

[mfb]

For energy conservation, it is sufficient that all external forces are conservative - that is equivalent to "you can introduce a potential energy". Gravity is conservative.
"No external forces" is required for momentum conservation only.

 

[jtbell]

An isolated system means no external forces, correct?

if i am correct, then i don't understand the law of conservation of mechanical energy.
it says that conservation of mechanical energy must be in an isolated system.

this is K + U (initals) = K + U (finals)

but the U term is mgh, which has gravity in it. isn't gravity an external force?
so how is this an isolated system?

You've already taken gravity into account by way of the potential energy, so it doesn't count as an "external force" in your definition.

A better statement of the law of conservation of mechanical energy is that the sum of K + U is constant, provided that there are no non-conservative forces (e.g. friction), and that you've taken all the conservative forces into account in the potential energy U.