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, \(F\)  

\(m_1\) = 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  
  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



On the surface



At a height


  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