Gravity

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This article is a topic within the subject Higher Physics 1A.

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Newton's law of Universal Gravitation

[1]Newton's law of universal gravitation states that every particle in the universe experiences an attractive force (Fgrav) from every other particle. This force is proportional to the product of their masses (m1 and m2), and inversely proportional to the square of the distance between them (r). The constant of proportionality is G, the universal gravitation constant. The gravitational force (Fgrav) will pull the two masses together, and so is in the opposite direction to the distance (r), hence the negative sign (-).

Fgrav = -G(m1m2)/r2

G = 6.674*10-11Nm2/kg2

Note that r is measured from the center of each mass (although this is only important for large objects like planets).


Superposition Principle

[2]If there are more than two masses involved, then there will be multiple gravitational force vectors. These vectors may be combined using vector addition to find the net gravitational force on each mass.


Gravity near a planet's surface

The force of gravity on an object at a planet's surface may be found using Newton's Law of Universal Gravitation. The distance (r) will be the planet's radius, and the masses will be those of the planet and the object.

[3]Gravitational acceleration (g) may also be found anywhere that a gravitational force is known by using Fgrav=mg. Combining this equation with Fgrav=-G(m1m2)/r2 gives:

g = (Gmplanet)/(r2)

r = radiusplanet + altitudefalling mass


Complications

Gravity on a planet's surface is rarely uniform.

  • The planet's rotation means that objects near the equator will experience a greater centripetal force, decreasing the effect of gravity.
    • This centripetal force also causes a bulge on the planet's surface near the equator, decreasing the force of gravity at the surface (since it is further away).
    • For this reason, gravity is greatest at the poles.
  • The density of a planet is rarely uniform, so gravity will be stronger in regions with more dense material and won't point exactly to the planet's centre.


Gravitational Potential Energy

As shown previously, gravitational potential energy near a planet's surface is given by the following formula:

U = mgh

[4]Since gravitational force is not constant, a new formula is needed for the general case. For two masses (m1 and m2), the gravitational potential energy (U) is proportional to the product of the masses and inversely proportional to the distance between them (r). The constant of proportionality is once again the universal gravitation constant (G).

U = -(Gm1m2)/r

Usually, U=0 is chosen to be at r=infinity, since the gravitational force will also be zero here.


Escape Velocity

[5]Escape velocity is the minimum velocity required for an object to escape to an infinite distance from a gravitational field. On a planet with mass mplanet and radius rplanet the escape velocity of any object is given by:

vesc = sqrt((2Gmplanet)/rplanet)

Note that vesc is independent of the escaping objects's mass.


Planetary Motion

Kepler's Laws

[6]Kepler's laws are given below. These particular descriptions describe the laws in our solar system, but they work in the same way for any system of masses orbiting a central body.

1 All planets move in elliptical orbits with the sun at one focus.

  • The orbits of the planets are mostly circular. Pluto and Halley's comet have more elliptical orbits.
  • There are no masses of importance located at the center of an orbit, or the second focus.
  • Kepler's first law can be proved using Newton's second law and gravitational theory.

2 The radius vector drawn from a sun to a planet moves across equal areas in equal time periods.

  • This law may be proved using conservation of angular momentum.
  • The effect of this law is that planets will move faster when their orbit takes them close to the sun, and slower when far away.
  • Kepler's second law may be expressed mathematically:

dA/dt = L/(2Mp) dA/dt is the rate at which the radius vector sweeps across the orbit area, L is the length of the radius vector and Mp is the planet mass.

3 Orbital period squared is proportional to the cube of the orbit's semi-major axis.

  • The effect of this law is that masses with smaller orbital radii will orbit faster than those with larger orbits.
  • Kepler's first law can be proved using Newton's second law and gravitational theory.
  • Kepler's third law may be expressed mathematically:

For a body with orbital period T and orbital radius r:

T2 = ((4π2)/(Gmcentral)) r3

G is the universal gravitation constant and mcentral is the mass of the central body.

For two bodies orbiting the same central mass (eg. planets orbiting the sun):

(T1/T2)2 = (r1/r2)3 = K

K is a constant.


Orbits and Energy

[7]Non-conservative forces do no work in an orbit, so mechanical energy is conserved. The energy of an orbit is the sum of kinetic and potential energy, and is given by the following formula (for circular orbits):

E = -(Gmplanetmsatellite)/(2r)

mplanet is the planet mass, msatellite is the orbiting object's mass, and r if the orbit height.

For an elliptical orbit, replace "r" with "a", the length of the orbit's semi-major axis.

Low orbits have a large negative energy and so are faster.


End

This is the end of this topic. Click here to go back to the main subject page for Higher Physics 1A.

References

Textbook refers to Serway & Jewett, Physics for Scientists and Engineers (Brooks/Cole , 8th ed, 2010)
(Slides) refers to those distributed by Wolfe, J (2012) on his First Year Physics site

  1. Textbook, pp375-376
  2. (Slides) Gravitation
  3. Textbook, p377
  4. Textbook, pp385-386
  5. Textbook, p389
  6. Textbook, pp379-382
  7. Textbook, pp387-388
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