GRAVITATIONAL FIELDS

• A field created around any object with mass

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• Extends all the way to infinity

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• Diminishes as the distance from the centre of mass of the object increases

The gravitational force exerted per unit mass at a point within a gravitational field

• The gravitational force on an object

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• An attractive force towards the centre of mass of the object producing the gravitational field

g = F / m

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F = ma and F = mg

∴ gravitational field strength at a point = acceleration of free fall of an object at that point

9.81 Nkg^-1

• Lines of force used to map the gravitational field pattern around an object having mass

• Do not cross

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• The arrows show the direction of the force on an object at that point in the field
  • Gravitational force is always attractive
    •  The direction of the field is always towards the centre of mass of the object producing the field

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• Closer together = stronger field

• A symmetrical field

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• Diminishes with distance from the centre of mass of the object producing the field

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• Shown by the field lines getting further apart

A mass with negligible volume

• The field lines are parallel

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• The value for gravitational field strength remains constant

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• The gravitational field close to the surface of a planet is approximately uniform

The force between two point masses is:

• Directly proportional to the product of the masses, F ∝ Mm

• Inverseley proportional to the square of their separation, F ∝ 1 / r²

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F = - GMm / r² 

G = 6.67e-11 Nm²kg^-2

g = F / m and F = - GMm / r²

g = - GMm / mr²

g = - GM / r²

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The negative sign shows that g at that point is in the opposite direction to r from the centre of mass of the object producing the gravitational field- a gravitational field is an attractive field.

The orbit of a planet is an ellipse with the Sun at one of the two foci

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Ellipse = an elongated 'circle' with two foci

Eccentricity = a measure of the elongation of an ellipse

Aphelion = the furthest point from the Sun in an orbit

Perihelion = the closest point to the Sun in an orbit

• A line segment connecting a planet to the Sun sweeps out equal areas during equal intervals of time

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• As a planet moves on its elliptical orbit around the Sun, its speed is not constant

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• When it is closer to the Sun it moves faster

mv² / r = GMm / r²

v² = GM / r

v = 2πr / T so 4π²r² / T² = GM / r

T² = (4π² / GM) r³

∴ T² ∝ r³ or T² / r³ = k

k is a constant for the planets orbiting the Sun approximately equal to 1 y²AU^-3

F =  mv² / r = centripetal force on the satellite

F = GMm / r² = gravitational force between the satellite and the Earth

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Only force acting on a satellite is the gravitational attraction between it and the Earth

∴ always falling towards the Earth

↑ speed and ↑ distance

∴ as it falls the Earth curves away beneath it

∴ its height above the Earth's surface remains constant

• v = √ GM / r

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• All satellites placed in a given orbit at a given height will be travelling at the same speed, even if their mass varies

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• Once launched they are normally above the atmosphere

There is no air resistance to slow them down

Their speed remains constant

• Communications (e.g. satellite phones)

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• Military uses (e.g. reconnaissance)

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• Scientific research (e.g. looking down onto the Earth to monitor pollution)

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• Weather and climate (e.g. monitoring long-term changes in climate)

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• Global positioning

Polar orbit

• Circles the poles
• Offers a complete view of the Earth over a given time period as the Earth rotates beneath the path of the satellite
• Useful for reconnaissance

Low Earth orbit

• In orbit close to the Earth
• T² ∝ r ³ so takes less than 2 hours for the satellite to orbit the Earth

Equatorial orbit

• In orbit above the equator

• Are in a geostationary orbit

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• Are in orbit above the equator

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• Rotate in the same direction as the Earth's rotation

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• Have an orbital period of 24 hours

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• ∴ Remain above the same point of the Earth whilst the Earth rotates

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• Are used for satellite television

• The work done per unit mass to bring an object from infinity to a point in the gravitational field

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• All masses attract each other

External work must be done to move masses apart 

Gravitational potential is a maximum at infinity

= 0 Jkg^-1

∴ all values of gravitational potential are negative

A distance so far from the object producing the gravitational field that the gravitational field strength is zero

V_g = - (GM / r)

• All values of V_g within the region of the gravitational field are negative
• When r = ∞ then V_g = 0

• Moving towards a point mass results in a decrease in gravitational potential

∴ the change in gravitational potential would be negative

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• Moving away from a point mass (towards infinity) results in an increase in gravitational potential

∴ the change in gravitational potential would be positive

The capacity for doing work as a result of an object's position in a gravitational field

E = mV_g

↑ r 

= ↑ V_g

= ↑ E

E = mV_g and V_g = - (GM / r)

∴ E = - (GMm / r)