Gravitational fields Topic 10

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Review

• A field is said to exit when one object exerts a force on another object at a distance

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Gravitational fields • A gravitational force exists

between two objects that have both mass

• 𝐹 = −𝐺𝑀𝑚

𝑅2

• A gravitational field is associated with each mass

• A force is acting on each mass

• Gravity is always attractive

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Field lines

• Similar to negative point charge towards a mass b/c field is only attractive

• Field is radial

• Gravitational field is uniform

• Directed downward at 900 normal to the surface

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Equipotential surface • Always perpendicular to gravitational field lines

• Closer together where field is stronger

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Gravitational field strength

• Gravitational field strength = 𝑓𝑜𝑟𝑐𝑒 𝑎𝑐𝑡𝑖𝑛𝑔 𝑜𝑛 𝑡𝑒𝑠𝑡 𝑚𝑎𝑠𝑠

𝑚𝑎𝑔𝑛𝑖𝑡𝑢𝑑𝑒 𝑜𝑓 𝑡𝑒𝑠𝑡 𝑚𝑎𝑠𝑠

• 𝑔 = 𝐹

𝑚 (N 𝑘𝑔−1) on surface of Earth

• Since F = 𝐺𝑀𝑚

𝑅2 at a distance R from the surface of Earth

𝑔 = 𝐺𝑀𝑚

𝑅2

𝑚 at distance R

g = 𝐺𝑀

𝑅2

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Inside a solid sphere • g falls as

1

𝑅2 as moving away from the surface

• At the center g = 0

• Assume a mass is placed inside a sphere at a distance x, where x < R

Find an expression for the field strength at distance x from the center

x 𝑔𝑥= 𝐺𝑀𝑥

𝑥2 and 𝜌 = 𝑚𝑥

𝑉 𝑚𝑥= 𝜌𝑉 = 𝜌

4

3𝜋𝑥3

𝑔𝑥= 𝐺𝜌

4

3𝜋𝑥3

𝑥2 = 4

3𝜋𝐺𝜌x

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The gravitational field…

• Increases linearly inside a solid sphere from zero to its surface value

• This is inaccurate for Earth because the density of Earth varies

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Earth’s gravitational field

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