- #1
Bashyboy
- 1,421
- 5
Hello,
I am currently reading about the conception of a general expression for gravitational potential energy. I understand that we have to use methods of calculus to generate a general expression of gravitational potential energy, because when considering alterations to the configuration of an object in a gravitational field that are large, the force varies. The force is varying as it is being applied over a distance to counter the gravitational force it is experiencing from another object. What I am getting confused over is this part of the derivation:
[itex]U_f-U_i = -GM_Em(\frac{1}{r_f}-\frac{1}{r_i})[/itex]
The paragraph accompanying this fuction:
"As always, the choice of a reference configuration for the potential energy is completely arbitrary. It is customary to choose the reference configuration for zero potential energy to be the same as that for which the force is zero. Taking [itex]U_i=0[/itex] at [itex]r_i= \infty[/itex], we obtain the important result,
[itex]U(r)=-\frac{GM_Em}{r}[/itex]"
Th portion in red I don't quite understand. Is that saying when the two bodies in question are separated by an infinite distance, the initial potential energy is zero? What is that suppose to mean, and why does it allow us to do away with the [itex]U_i[/itex] term?
Also, in my textbook it says, "When two particles are at rest and separated by a distance r, an external agent has to supply an energy at least equal to [itex]+Gm1m2/r[/itex]to separate the particles to an infinite distance." I know that if I wanted to separate, say, the moon and earth, which as a system possesses gravitational energy, that I would need to apply a force on the moon. Moreover, I would have to be applying this force over a distance, meaning I am putting energy into the moon. But how am I suppose to know that the equal that I provide by applying a force over a distance to free the moon from Earth's gravitational pull is exactly [itex]+Gm1m2/r[/itex]? Honestly, if someone was in the enterprise of moving the moon, they'd be more concerned with the force that Earth pulls on the moon, so that they could counter it and move the moon, and not with the gravitational potential energy of the moon. Again, this infinite distance business comes up, what exactly do they mean by this?
Finally, I am also reading about Energy Considerations in Planetary and Satellite Motion. The mechanical energy equation the derive, of which I understand, is [itex]E = 1/mv^2 - \frac{GMm}{r}[/itex], where we don't consider the kinetic energy of the object the satellite is rotating around. One thing the author says in my textbook is, "...[itex]E[/itex] may be positive, negative, or zero, depending on the value of [itex]v[/itex]." Could someone give scenarios that would correspond to those possibilities, that [itex]E[/itex] is positive, negative, zero?
I would appreciate the help. Thank you in advance!
I am currently reading about the conception of a general expression for gravitational potential energy. I understand that we have to use methods of calculus to generate a general expression of gravitational potential energy, because when considering alterations to the configuration of an object in a gravitational field that are large, the force varies. The force is varying as it is being applied over a distance to counter the gravitational force it is experiencing from another object. What I am getting confused over is this part of the derivation:
[itex]U_f-U_i = -GM_Em(\frac{1}{r_f}-\frac{1}{r_i})[/itex]
The paragraph accompanying this fuction:
"As always, the choice of a reference configuration for the potential energy is completely arbitrary. It is customary to choose the reference configuration for zero potential energy to be the same as that for which the force is zero. Taking [itex]U_i=0[/itex] at [itex]r_i= \infty[/itex], we obtain the important result,
[itex]U(r)=-\frac{GM_Em}{r}[/itex]"
Th portion in red I don't quite understand. Is that saying when the two bodies in question are separated by an infinite distance, the initial potential energy is zero? What is that suppose to mean, and why does it allow us to do away with the [itex]U_i[/itex] term?
Also, in my textbook it says, "When two particles are at rest and separated by a distance r, an external agent has to supply an energy at least equal to [itex]+Gm1m2/r[/itex]to separate the particles to an infinite distance." I know that if I wanted to separate, say, the moon and earth, which as a system possesses gravitational energy, that I would need to apply a force on the moon. Moreover, I would have to be applying this force over a distance, meaning I am putting energy into the moon. But how am I suppose to know that the equal that I provide by applying a force over a distance to free the moon from Earth's gravitational pull is exactly [itex]+Gm1m2/r[/itex]? Honestly, if someone was in the enterprise of moving the moon, they'd be more concerned with the force that Earth pulls on the moon, so that they could counter it and move the moon, and not with the gravitational potential energy of the moon. Again, this infinite distance business comes up, what exactly do they mean by this?
Finally, I am also reading about Energy Considerations in Planetary and Satellite Motion. The mechanical energy equation the derive, of which I understand, is [itex]E = 1/mv^2 - \frac{GMm}{r}[/itex], where we don't consider the kinetic energy of the object the satellite is rotating around. One thing the author says in my textbook is, "...[itex]E[/itex] may be positive, negative, or zero, depending on the value of [itex]v[/itex]." Could someone give scenarios that would correspond to those possibilities, that [itex]E[/itex] is positive, negative, zero?
I would appreciate the help. Thank you in advance!