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Newton's Universal Law of Gravitation |
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Newton's Universal Law of Gravitation |
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Gravitational force is directly proportional to the product of masses of two bodies and inversely proportional to the square of the distance between the centers of two bodies.
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Gravitational force between two bodies, \(F\) |
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\(F=\dfrac{Gm_1m_2}{r^2}\) |
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\(m_1\) = mass of first body
\(m_2\) = mass of second body
\(r\) = distance between the centre of first and second body
\(G\) = gravitational constant (\(6.67\times 10^{-11} N\,m^2\,kg^{-2}\))
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- The larger the mass of the body, the larger the gravitational force
- The further the distance the bodies. the smaller the gravitational force
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Relating g and G |
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\(g=\dfrac{GM}{r^2}\) |
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On surface of a planet, gravitational force is equal to weight. |
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Relating g and r |
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- When \(r < R\), \(g\) is directly proportional to \(r\)
- \(g \propto r\)
- R = Earth's radius
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- When \(r \geq R\), \(g\) is inversely proportional to \(r^2\)
- \(g=\dfrac{GM}{r^2}\)
- \(g\propto \dfrac{1}{r^2}\)
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Formulas to calculate the gravitational acceleration |
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Below the surface
\(g=\dfrac{GM}{(R-h)^2}\)
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On the surface
\(g=\dfrac{GM}{R^2}\)
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At a height
\(g=\dfrac{GM}{(R+h)^2}\)
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Centripetal force, \(F\) |
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\(F=\dfrac{mv^2}{r}\), where \(m\) = mass, \(v\) = linear speed, \(r\) = radius of circle |
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Centripetal force in the motions of Satellites and Planets
- In the absence of force, object move in the same direction.
- \(0\text{ N}\) force \(\rightarrow\) object move in the same direction
- In an orbit, object is always changing its direction
- There is force acting on the object in direction towards the center of the circle (orbit) \(\rightarrow\) centripetal force
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Centripetal acceleration, \(a\) |
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\(a=\dfrac{v^2}{r}\), where \(v\) = linear speed, \(r\) = radius of orbit |
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Mass of earth, \(M\) |
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\(M=\dfrac{4\pi^2r^3}{GT^2}\), where \(r\) = radius of orbit, \(G = 6.67\times10^{-11}\text{ Nm}^2\text{kg}^{-2}\), \(T\) = period of revolution |
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