## 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