Question

Newton's Law of Gravitation states that two bodies with masses my and m2 attract each other with a force F, where r is the distance between the bodies and G is the gravitational constant. mim2 F=G 72 Use Newton's Law of Gravitation to compute the work W required to launch a 1900 kg satellite vertically to an orbit 800 km high. You may assume that the earth's mass is 5.98x1024 kg and is concentrated at its center. Take the radius of the earth to be 6.37x106 m and G = 6.67x10-11 N·m2/kg?. (Round your answer to three significant digits.) W = * Your answer cannot be understood or graded. More Information J Enter an exact number. Need Help? Read It Talk to a Tutor

Answer 1

Mass of the earth = M = 5.98 x 10²⁴ kg.

Mass of the satellite = m = 1900 kg.

Distance of the orbit from the center of the earth = R

= Radius of the earth R_o + Height of the orbit

= 6.37 x 10⁶ m + 800 km = 6.37 x 10⁶ m + 8 x 10⁵ m = 7.17 x 10⁶ m.

Hence, work done required to move the satellite from a distance r to ( r + dr ) is dW = F . dr,

where, gravitational force = F = GMm / r², at a distance r from the center of the earth.

So, total work done to move the satellite from the surface of the earth to R is :

W = $\int$ dW. This integration is calculated below :

R R w=f dw= pe su GMm Gmm dor سم o2 "Ro Ro Ro AT W - GMm [ kho GMm [ Itho-] WE

So, plugging the known values in the above equation :

W = 6.67 x 10^-11 x 5.98 x 10²⁴ x 1900 { 1 / ( 6.37 x 10⁶ ) - 1 / ( 7.17 x 10⁶ ) } J

or, W = 1.33 x 10¹⁰ J.