Question

Newton's Law of Gravitation states that two bodies with masses m1 and m2 attract each other with a force F, where r is the distance between the bodies and G is the gravitational constant.
F=G m1m2/r2

Use Newton's Law of Gravitation to compute the work W required to propel a 1400 kg satellite out of the earth's gravitational field. You may assume that the earth's mass is 5.98✕1024 kg and is concentrated at its center. Take the radius of the earth to be 6.37✕106 m and G = 6.67✕10-11 Nm2/kg2. (Round your answer to three significant digits.)

Answer 1

Answer:

The workdone is  $W = 3.14*10^{10} \ J$

Explanation:

From the question we are told that

    The equation for the force is   $F= \frac{G * m_1 * m_2}{r^2}$

     The gravitational constant is $G = 6.67*10^{-11} \ Nm^2/kg^2$

     The mass of satellite is  $m_1 = 1400 \ kg$

      The radius of the earth is  $r = 6.37*10^6 m$

        The mass of earth is  $m_2 = 5.98* 10^ 24 \ kg$

Generally the work required is mathematically represented as

           $W = \int\limits^a_b {F} \, dr$

Where  b =  G this is because the satellite has to overcome the earth gravitational pull

 And  a  =  $\infty$  this because at infinity the there is not effect of gravitational force

        So

                $W = \int\limits^{\infty}_{6*10^{-11}} {F} \, dr$

                $W = \int\limits^{\infty}_{6*10^{-11}} { \frac{G * m_1 * m_2}{r^2}} \, dr$

               $W = G * m_1 * m_2 \int\limits^{\infty}_{6*10^{-11}} { \frac{1}{r^2}} \, dr$

  substituting values

             $W = 6.67*10^{-11} * 1400 *5.98*10^{24} \int\limits^{\infty}_{6*10^{-11}} { \frac{1}{r^2}} \, dr$

              $W = 2 *10^{17}\int\limits^{\infty}_{6*10^{-11}} { \frac{1}{r^2}} \, dr$

               $W = 2 *10^{17}} [{ - \frac{1}{r}} ] \left | \infty \atop {6*10^{-11}}} \right.$

                $W = 2 *10^{17}} [{ - \frac{1}{\infty }} + \frac{1}{6*10^{-11} }}]$

                  $W = 2 *10^{17}} [{ - 0 + \frac{1}{6*10^{-11} }}]$

                   $W = 2 *10^{17}} * 1.57*10^{-7}$

                   $W = 3.14*10^{10} \ J$