Newton’s Law of Gravity

Starter

Newton’s Law of Gravity

Starter If the mass of the Earth stayed the same but the radius of the Earth doubles, what happens to your weight?

a. it doubles also. b. it stays the same. c. it drops to 1/2 of the original. d. it drops to ¼ of the original.

g = MG/R2

Newton’s Law of Gravity

StarterTo increase the speed of an Earth satellite, should it be placed in an orbit closer to the Earth or further away from the Earth?

Practice

F = m1m2G/r2

Newton’s Law of Gravity : Every two objects attract each other with a gravitational force given by:

m1 = mass of the first object in kg m2 = mass of the second object in kg r = distance between the two masses in meters G = 6.67 x 10-11

Practice / Example

Find the gravitational force between the Earth and a 50,000kg asteroid that is 12 x 106 m away.masteroid = 5.00 x104 kg mEarth = 6.00 x 1024 kg

G = 6.67 x 10-11 r = 12 x 106 m

F = m1m2G/r2

= (5.00 x104 )( 6.00 x 1024 )(6.67 x 10-11 ) / (12 x 106 )2

= 1.39 x 105 Newtons = 139,000 Newtons

Practice / ExampleFind the gravitational force between Saturn and the sun.

msun = 1.99 x1030 msaturn = 5.71 x 1026

G = 6.67 x 10-11 r = 1430 x 109

F = m1m2G/r2 =

(1.99 x1030 )(5.71 x 1026 )(6.67 x 10-11 ) / (1430 x 109)2

= 3.71 x 1022 Newtons

Where does g come from?

Consider a person, mass m, standing on the Earth.

His weight is: F = mMG/R2 = mg Cancel the person’s mass m: g = MG/R2

=(6x1024)(6.67x10-11)/(6.4x106)2 = 9.8 m/s2

Circular OrbitsF = mv2/R = mMG/R2 and v = 2pR / T

Orbit Velocity

Period

R = Orbit Radius M = mass of planet or sun being orbited

(The mass of the satellite is not in these 2 equations.)

ExampleThe moon is 385 x 106 m from the Earth’s center.

R = 385 x 106 MEarth = 6 x 1024

1. What is its orbit velocity? v = (MG/R)1/2 = ( ( 6 x 1024)(6.67 x 10-11)/(385 x106))1/2

= 1019 m/s

2. What is the period?

T = 2pR/v = (2 p ) (385 x 106)/ 1019 = 2.37 x 106 seconds

2.37 x 106 seconds ( 1hr/3600s)(1 day/24 hr) = 27.5 days

Useful Data

Quantity Mercury Venus Earth Mars Jupiter Saturn Uranus Neptune Pluto

Mean distance from sun

57.9 108 150 228 778 1,430 2,870 4,500 5,900(109 m)

Mass (kg)

3.331E+23

4.866E+24 5.97E+24

6.388E+23

1.898E+27

5.677E+26

8.657E+25

1.027E+26 5.97E+22

Period of revolution, years 0.241 0.615 1 1.88 11.9 29.5 84 165 248

Orbital speed,

47.9 35 29.8 24.1 13.1 9.64 6.81 5.43 4.74(103 m/s)

Mass (Earth=1) 0.0558 0.815 1 0.107 318 95.1 14.5 17.2 0.01

                   

Moon mass

Distance to Earth    

Mass of Sun  

Mass of Earth

Radius of Earth    

7.36E22 kg 3.84E8 m    

1.99E30 kg  

5.97 E24 kg 6.4E6m