doPhysics · Gravitation
Gravitational Energy and Orbits
The energy of masses that attract, and the potential of a point, a ring, a shell and a sphere. The speed that escapes the Earth, satellites and the geostationary orbit, the energy of an orbit, Kepler's three laws and binary stars.
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Why this chapter
Why this chapter exists
The last chapter found the pull of gravity. This one follows its energy, and energy answers the big questions quickly. How much does it cost to lift a satellite hundreds of kilometres, where \(mgh\) no longer holds? How fast must a body leave the Earth never to return? Why does a satellite never fall, and why does the space station need no engine to stay up? Where must a television satellite sit to hang still in the sky? And why does a comet race past the Sun and then crawl for decades? Every answer comes from one expression, \(U = -GMm/r\), used with the conservation of energy and of angular momentum. Here \(g = 9.8\,\text{m/s}^2\), \(G = 6.67 \times 10^{-11}\,\text{N m}^2/\text{kg}^2\) and the Earth's radius is \(6.4 \times 10^6\,\text{m}\).
You will learn
By the end of this chapter you will
- Find the work done against gravity between two distances, and explain why \(U = -GMm/r\) is negative and how \(mgh\) follows from it.
- Add the potential energy of many masses pair by pair, and find the self-energy of a shell and of a solid sphere.
- Find the gravitational potential of a point mass, a ring, a shell and a solid sphere, and get the field from its slope.
- Find the work of an outside agent from a difference in potential.
- Derive the escape speed, 11.2 km/s for the Earth, and use the gravitational well to decide whether a body is bound.
- Find the speed and period of a satellite, explain weightlessness in orbit, and find the height of the geostationary orbit.
- Use \(K = -E\) and \(U = 2E\) on a circular orbit to find binding energies, orbit changes and escape from an orbit.
- State and use Kepler's three laws, and find the orbits and the period of a binary star.
Before you begin
Before you begin
From the chapter on Gravitational Force and Field, you should know the law \(GMm/r^2\), the field of a point mass and \(g = GM/R^2\). You should know that a uniform sphere pulls an outside body as if its mass sat at its centre. From the chapter on Potential Energy and Conservation of Energy, you should know \(\Delta U = -W\) and the conservation of mechanical energy. You should be able to read an energy diagram for turning points and for bound and unbound motion. You should know the centripetal requirement \(mv^2/r\), the centre of mass of two particles, and the conservation of angular momentum. You should be able to integrate \(1/r^2\).
Lessons
1
Gravitational Potential Energy
Far from the ground the Earth's pull weakens, so \(mgh\) no longer holds. The work of the pull \(GMm/r^2\) gives \(U = -GMm/r\), with its zero at infinity. The value is negative because the pair is bound, and \(mgh\) returns as its near-surface form.
Work done by gravity far from the earthPotential energy with its zero at infinityLifting near the surface
2
Energy of Many Masses
Assemble masses one at a time, and each new one adds a term for every mass already there. So a system has one term \(-Gm_im_j/r_{ij}\) for every pair. The same count gives the energy released when masses fall together, and the self-energy of a shell and of a solid sphere.
One term for every pairEnergy is shared when many masses moveSelf energy of a shell and a sphere
3
Gravitational Potential
The potential \(V = U/m\) describes a place, not a body. It adds as a plain number, the field is minus its slope, and an outside agent does \(m\) times the change in \(V\). The lesson finds \(V\) for a point mass, a ring, a shell and a solid sphere.
Potential is energy per unit massThe field is minus the slope of potentialPotential of a ring a shell and a solid sphereWork done by an external agent
4
Escape Velocity
A body escapes a planet when its total energy is zero or more. That gives \(v_e = \sqrt{2GM/R} = \sqrt{2gR}\), 11.2 km/s for the Earth. The speed needs neither the mass nor the direction. The well \(-GMm/r\) shows why a negative energy means bound, and a small \(v_e\) explains the Moon's missing air.
Escape needs a total energy of zeroThe gravitational wellEscape speed ignores mass and directionWhy the moon has no air
5
Satellites in Orbit
A satellite is a body that falls all the time and never lands. Gravity supplies its centripetal force, which fixes its speed \(\sqrt{GM/r}\) and its period \(2\pi\sqrt{r^3/GM}\). The lesson explains weightlessness on a space station and finds the one orbit that keeps a satellite above the same place on the Earth.
An orbit is a fall that never landsOrbital speed and periodWeightlessness in orbitThe geostationary orbit
6
The Energy of a Satellite
On a circular orbit \(K = GMm/2r\), \(U = -GMm/r\) and \(E = -GMm/2r\), so \(K = -E\) and \(U = 2E\). The binding energy is \(-E\). A higher orbit holds more energy but moves more slowly. A satellite slowed by air speeds up, and escape from an orbit needs \(\sqrt{2}\) times the orbital speed.
Kinetic potential and total energy in orbitBinding energy of a satelliteMoving between orbitsEscape from an orbit
7
Kepler's Laws and Binary Stars
Planets move on ellipses with the Sun at a focus. They sweep equal areas in equal times, because gravity exerts no torque about the Sun, and \(T^2\) is proportional to \(a^3\). Two stars of similar mass both circle their common centre of mass, and the reduced mass turns the pair into one body.
The law of orbitsThe law of areasThe law of periodsBinary stars and the reduced mass
Read in order. The small labels are the topics of each lesson. They take your colour as you answer the checks.
0%The Workbook
Questions to solve on paper, marked by you.
Glossary
The key words of this chapter, with the meaning this chapter gives them.
- aphelion
- The point of a planet's orbit farthest from the Sun, at a distance \(a(1 + e)\). The planet moves slowest there.
- apparent weight
- The reading of a weighing scale, which is the normal force that the floor exerts. In orbit it is zero, because the body and its surroundings fall together.
- binary star
- A pair of stars bound by their mutual gravity, both circling their common centre of mass with the same period.
- binding energy
- The energy that must be supplied to free a body from gravity, taking it to infinity. It is minus the total energy. For a satellite on a circular orbit of radius \(r\) it is \(GMm/2r\); for a body at rest on the surface, \(GMm/R\).
- bound
- Held together by gravity, with a negative total energy. A bound body cannot travel to infinity; it has a greatest distance, where it stops and returns.
- eccentricity
- A number \(e\) from 0 to 1 that measures how stretched an ellipse is. A circle has \(e = 0\); the foci lie a distance \(ae\) from the centre.
- ellipse
- A closed curve shaped like a stretched circle. For every point on it, the distances to two fixed points, the foci, add up to the same length \(2a\).
- escape speed
- The least speed with which a body must leave the surface of a planet to reach infinity and never return, \(v_e = \sqrt{2GM/R} = \sqrt{2gR}\). It is \(11.2\,\text{km/s}\) for the Earth and does not depend on the mass or the direction of the body.
- focus
- One of the two special points inside an ellipse. The Sun sits at one focus of every planet's orbit; the other focus is empty.
- geostationary
- Describes an orbit over the equator, travelled west to east with a period of one day. A satellite on it stays above one point of the Earth. Its height is about \(3.6 \times 10^7\,\text{m}\).
- gravitational field
- The gravitational force per unit mass at a point, in N/kg. It equals minus the slope of the potential, \(E = -dV/dr\), and points towards lower potential.
- gravitational potential
- The potential energy per unit mass of a small body at a point, \(V = U/m\), in J/kg. It equals the work per kilogram needed to bring a body slowly from infinity to that point. For a point mass it is \(-GM/r\).
- gravitational potential energy
- The potential energy of two masses due to their mutual pull. For masses \(M\) and \(m\) a distance \(r\) apart it is \(U = -GMm/r\), with \(U = 0\) at infinite separation. It is negative and belongs to the pair.
- gravitational well
- The curve \(U = -GMm/r\) drawn against \(r\): deepest at the planet's surface and rising towards zero far away without ever reaching it. A body with negative total energy is trapped in it.
- perihelion
- The point of a planet's orbit nearest the Sun, at a distance \(a(1 - e)\). The planet moves fastest there.
- reduced mass
- \(\mu = m_1m_2/(m_1 + m_2)\) for two bodies. Their separation changes like the orbit of a single body of mass \(\mu\) about a fixed centre. When one mass is far larger, \(\mu\) is almost the smaller mass.
- satellite
- A body that orbits a planet, falling towards it all the time but always missing it. The Moon is a natural satellite; the space station is an artificial one.
- self-energy
- The gravitational potential energy of all the parts of one body, taken in pairs. It is the energy released in assembling the body from matter spread out at infinity. It is \(-GM^2/2R\) for a thin shell and \(-3GM^2/5R\) for a uniform solid sphere.
- semi-major axis
- Half the longest diameter of an ellipse, written \(a\). It is the mean of the nearest and farthest distances of an orbit from its focus.
- spherical shell
- A thin hollow sphere, with all its mass at one distance \(R\) from its centre, like the skin of a football.
- superposition
- The rule that the gravitational effects of several masses simply add: forces and fields as vectors, potential energies and potentials as plain numbers.
- unbound
- Free to travel to infinity, with a total energy of zero or more. The energy line never meets the curve of \(U\), so nothing turns the body back.