Energy of an Object in a Gravitational Field
The kinetic energy of an object is given by the formula:
Ek =
The gravitational potential energy is given by:
Eg =
The totalenergy is the sum of the kinetic and potential energies.
ET =
Note that:
- gravitationalpotential energy is always______.
- kineticenergy is always ______.
- the total energy can be ______, ______, or ______.
- Zero total energy:
- Positive total energy:
- Negative total energy
Binding Energy
If an object has a negative total energy, it is bound (trapped) in the gravitational well. It is in gravitational debt, so to speak. The earth - moon system for example, has a negative total energy; the moon is bound by the earth’s gravitational field (and the earth is bound by the moon’s gravitational field).
The magnitude of this debt is called the binding energy. The binding energy is the
energy ______.
Note:
- objects are bound to each other ONLY when the total energy is negative
- binding energy is always a positive amount.
Ex.An asteroid of mass 6.5 x 1013kg passes within 4.5 x 109 m of the Earth. When it does, it reaches a speed of 1300m/s.
a)Find the gravitational potential energy of the asteroid
b)Find the kinetic energy of the asteroid
c)Is the asteroid bound in the Earth-asteroid system? If so, find the binding energy. If not, find the surplus energy.
d)What will happen to this asteroid? What does this picture not take into account?
Energy of an Orbiting Satellite
A satellite is any object that is kept in orbit by the force of gravity. The moon is a natural satellite. The earth is a natural satellite. The hubble telescope is an artificial satellite. In each case, the centripetal force that pulls the satellite to the center is actually the gravitational pull:
The left side of the equation is very close to the equation for kinetic energy. We can multiply both sides by ½ (or divide by 2) and get:
This gives us an equation for the kinetic energy of an orbiting satellite:
notice that this is similar to another formula:
By coincidence, a satellite in orbit must have kinetic energy that is exactly half the gravitational potential energy debt. A satellite is always “halfway out of the hole”.
The total energy of the satellite is then Ek + Eg = ½ Eg
The total energy of a satellite is half the gravitational potential energy. In other words, the satellite is still “halfway into the hole”.
Summary:
- A satellite is always bound in a gravitational field (it is not free).
- A satellite has a negative total energy. This is the satellite’s energy debt.
- A satellite’s binding energy is the same as the total energy, except the binding energy is positive. This is the amount of energy needed to “pay” to get out of debt.
- A satellite’s gravitational potential energy is the reason why it is in debt. It is double the binding energy and kinetic energy and total energy.
A satellite example with meaningless numbers:
Eg = -100 J
Ek =
ET =
Ebinding =
Converting Eg to Ek
Imagine an apple being held a certain distance away from the earth. Suppose at that distance it has an Eg of -100J. Now suppose that we let that apple fall from rest to a point closer to the earth. What happens to the apples value of Eg?
As the apple falls, it falls further into the gravitational potential well – it is further in debt.
Suppose the new value of Eg is ______J. This is ______energy than it had previously. The law of conservation of energy tells us that energy cannot be created or destroyed. The apple’s total energy must remain constant, so what happened?
Calculate the speed of the apple at its new position if it has a mass of 1kg.
Problem:
A book of mass 5.0kg is dropped from rest from a height of 10.0m above the ground. Use these formulas to calculate:
a)Eg before it is dropped.
b)Eg after it is dropped.
c)Ek just before hitting the ground.
d)The speed just before hitting the ground.