What does it mean when total mechanical energy is negative?

[goonking]

Homework Statement

Homework Equations

The Attempt at a Solution

So for part a, i just plugged 3 into the equation, got the U(3) = -5.36

to get mechanical energy, (KE + PE):

5 J + (-5.36 J)= -0.36 J

is this correct? what does it mean when the total energy is a negative value? (I have never seen this before)

and part b) I have no clue how to do this.

 

[AlephNumbers]

part b) I have no clue how to do this.

Think about this. If you have an object at a height h, it has gravitational potential energy equal to mgh. If you then drop the object, the force of gravity acts on the object, reducing its potential energy.

 

[Orodruin]

b) I have no clue how to do this.

How does the force relate to the potential?

 

[goonking]

.

 

[goonking]

Think about this. If you have an object at a height h, it has gravitational potential energy equal to mgh. If you then drop the object, the force of gravity acts on the object, reducing its potential energy.

yes, I sort of understand what you said , but I always thought the decrease in height lowered the potential energy.

 

[goonking]

How does the force relate to the potential?

W_grav = mgh
W = F d

mgh = F d

?

 

[AlephNumbers]

yes, I understand what you said, but how does that relate to the x axis?

And why would the height change, if not for the force acting on the object? Since U_g = mgh, how would the gravitational potential energy of an object change if not for the force acting on it?

 

[goonking]

And why would the height change, if not for the force acting on the object? Since U_g = mgh, how would the gravitational potential energy of an object change if not for the force acting on it?

yes. that is true. makes complete sense now

but isn't height on the y axis? how can I find where the particle is located on the x-axis at a given time?

 

[AlephNumbers]

yes. that is true. makes complete sense now

So how is the conservative force related to the potential energy in the OP?

 

[goonking]

So how is the conservative force related to the potential energy in the OP?

-dU / dx = F(x) = g

?

 

[AlephNumbers]

-dU / dx = F(x)

Exactly! Now apply that to the original problem statement. The thing about gravity was just an analogy. Find the opposite of the derivative of the potential energy function listed in the problem statement, and you will have the conservative force as a function of x.

 

[goonking]

Exactly! Now apply that to the original problem statement. The thing about gravity was just an analogy. Find the opposite of the derivative of the potential energy function listed in the problem statement, and you will have the conservative force as a function of x.

dU/dx = 4e^-2/x ( x - 2) = F = 0

?

 

[goonking]

also, can you give me a real life example where the total mechanical energy of a system is negative?

 

[Orodruin]

also, can you give me a real life example where the total mechanical energy of a system is negative?

Energy is only defined up to an additive constant (at least for the purposes of classical mechanics). By redefining the zero level of potential energy, any system can have a negative total energy. The prime example would be anything on Earth bound by gravity (the zero level of gravitational potential energy is usually taken to be at infinite separation).

 

[gneill]

also, can you give me a real life example where the total mechanical energy of a system is negative?

A planet orbiting a star is an example. Objects in bound orbits (circles, ellipses) have negative total mechanical energy associated with them. A body on a hyperbolic trajectory that passes by the Sun only once: approaching, swinging around the Sun, then heading out again never to return, has a positive total mechanical energy. The borderline case where the total mechanical energy is zero corresponds to a parabolic orbit (such as some comets are thought to have) and are also "one time guests" to the solar system.

 

[Orodruin]

A planet orbiting a star is an example. Objects in bound orbits (circles, ellipses) have negative total mechanical energy associated with them. A body on a hyperbolic trajectory that passes by the Sun only once: approaching, swinging around the Sun, then heading out again never to return, has a positive total mechanical energy. The borderline case where the total mechanical energy is zero corresponds to a parabolic orbit (such as some comets are thought to have) and are also "one time guests" to the solar system.

Just to be clear here: This is true for the convention of having zero potential at infinite separation as I indicated above. You can always add a constant to the potential.