Law of Universal Gravitation

Newton's Universal Law of Gravitation

3.1

Newton's Universal Law of Gravitation

	Newton's Universal Law of Gravitation
	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.

	Gravitational force between two bodies, \(F\)
	\(F=\dfrac{Gm_1m_2}{r^2}\)

	\(m_1\) = mass of first body \(m_2\) = mass of second body \(r\) = distance between the centre of first and second body \(G\) = gravitational constant (\(6.67\times 10^{-11} N\,m^2\,kg^{-2}\))

The larger the mass of the body, the larger the gravitational force
The further the distance the bodies. the smaller the gravitational force

	Relating g and G
	\(g=\dfrac{GM}{r^2}\)
	On surface of a planet, gravitational force is equal to weight.

	Relating g and r
	When \(r < R\), \(g\) is directly proportional to \(r\) \(g \propto r\) R = Earth's radius

	When \(r \geq R\), \(g\) is inversely proportional to \(r^2\) \(g=\dfrac{GM}{r^2}\) \(g\propto \dfrac{1}{r^2}\)

	Formulas to calculate the gravitational acceleration

	Below the surface \(g=\dfrac{GM}{(R-h)^2}\)

	On the surface \(g=\dfrac{GM}{R^2}\)

	At a height \(g=\dfrac{GM}{(R+h)^2}\)

	Centripetal force, \(F\)
	\(F=\dfrac{mv^2}{r}\), where \(m\) = mass, \(v\) = linear speed, \(r\) = radius of circle

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

	Centripetal acceleration, \(a\)
	\(a=\dfrac{v^2}{r}\), where \(v\) = linear speed, \(r\) = radius of orbit

	Mass of earth, \(M\)
	\(M=\dfrac{4\pi^2r^3}{GT^2}\), where \(r\) = radius of orbit, \(G = 6.67\times10^{-11}\text{ Nm}^2\text{kg}^{-2}\), \(T\) = period of revolution

Newton's Universal Law of Gravitation

3.1

Newton's Universal Law of Gravitation

	Newton's Universal Law of Gravitation
	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.

	Gravitational force between two bodies, \(F\)
	\(F=\dfrac{Gm_1m_2}{r^2}\)

	\(m_1\) = mass of first body \(m_2\) = mass of second body \(r\) = distance between the centre of first and second body \(G\) = gravitational constant (\(6.67\times 10^{-11} N\,m^2\,kg^{-2}\))

The larger the mass of the body, the larger the gravitational force
The further the distance the bodies. the smaller the gravitational force

	Relating g and G
	\(g=\dfrac{GM}{r^2}\)
	On surface of a planet, gravitational force is equal to weight.

	Relating g and r
	When \(r < R\), \(g\) is directly proportional to \(r\) \(g \propto r\) R = Earth's radius

	When \(r \geq R\), \(g\) is inversely proportional to \(r^2\) \(g=\dfrac{GM}{r^2}\) \(g\propto \dfrac{1}{r^2}\)

	Formulas to calculate the gravitational acceleration

	Below the surface \(g=\dfrac{GM}{(R-h)^2}\)

	On the surface \(g=\dfrac{GM}{R^2}\)

	At a height \(g=\dfrac{GM}{(R+h)^2}\)

	Centripetal force, \(F\)
	\(F=\dfrac{mv^2}{r}\), where \(m\) = mass, \(v\) = linear speed, \(r\) = radius of circle

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

	Centripetal acceleration, \(a\)
	\(a=\dfrac{v^2}{r}\), where \(v\) = linear speed, \(r\) = radius of orbit

	Mass of earth, \(M\)
	\(M=\dfrac{4\pi^2r^3}{GT^2}\), where \(r\) = radius of orbit, \(G = 6.67\times10^{-11}\text{ Nm}^2\text{kg}^{-2}\), \(T\) = period of revolution