Evans SPH 4U1

Unit 4: Energy & Momentum

Note 7: More on Gravitational Potential Energy

Reference: Chapter 6.3

Recall:

Any two masses have a Gravitational Potential Energy of  at a separation distance of r.
As discussed earlier, it is negative because of the reference we use where earth is in a potential well.

Refer to Figure 4 (Text page 288)

If a rocket is at rest on the surface of the earth its kinetic energy is zero.  If the rocket is given a speed at the surface of the earth, it now has kinetic energy and begins to rise.  As it rises its potential energy increases (becomes less negative) and it will reach a point where its potential energy is at its maximum and its kinetic energy is zero.  At this point gravity will pull the rocket back to earth . When it hits the earth it has the same kinetic energy it left with (i.e. total energy is constant).

There is a speed the rocket can have to escape the potential well of Earth (i.e. escape gravitational force). This is when the rocket's kinetic energy equals the depth of the potential well at the Earth's surface (when the total energy is 0). Derive this speed.  11.2 km/s

If a rocket's total energy is negative then it will not be able to escape from earth's potential well.

Escape Speed: The minimum speed needed to project a mass to just escape the gravitational force.

Escape Energy: The minimum kinetic energy needed to project a mass to just escape the gravitational force.

Binding Energy: The amount of additional kinetic energy needed by a mass m to just escape from a mass M.

general formula for binding energy

Kinetic Energy of a satellite in circular orbit -derive:

Total energy of a satellite in circular orbit (it is negative and is equal to): - derive

Binding energy for a satellite bound to earth is equal to:

It is important for you to understand the difference between binding energies and total energies of a stationary object in space and an object which is in orbit.

Event Horizon: The surface of a black hole

Example 1:

A 60.0 kg space probe is in a circular orbit around Europa. If the orbital radius is 2.00 x 106m and the mass of Europa is 4.87 x 1022kg, determine the:

a) kinetic energy of the probe and it orbital speed (Answer: 4.87 x 107 J; 1.27 x 103 m/s)

b) gravitational potential energy of the probe (Answer: -9.74 x 107 J)

c) total orbital energy of the probe (Answer: -4.87 x107 J)

d) binding energy of the probe (Answer: 4.87 x 107 J)

e) additional speed that the probe must gain in order to break free of Europa (Answer: 528 m/s)

f) the energy needed to move the probe (from the 2.00 x 106m orbital radius) to an orbital radius of 4.00 x 106m above Europa (Answer: 2.43 x 107J)

Suggested Text Questions:

See sample problem #2 page 291

Page 294 # 6-8

Page 300 Review #'s 10-12, 17