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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