Unit 06 “ Circular Motion, Gravitation and Black Holes” Gravitation Problem Solving

Gravitational Force Equation F G = G m 1 m 2 r 2 F G = Gravitation Force m 1 = mass of object 1 m 2 = mass of object 2 r = distance between the objects (“r” is measured center to center) G = constant of universal gravitation (6.673x Nm 2 /kg 2 ) Gravitational Force The force of attraction between any two masses.

Which picture below shows how the radius between two masses should be measured?

r What happens to the Gravitational Force between two objects if the mass of an object increases? r r If the MASS increases the Gravitational Force also increases!

What happens to the Gravitational Force between two objects if the distance between the objects increases? r r If the distance increases the Gravitational Force decreases!!! The Inverse Square Law says that…. If the radius increased by a factor of 2, the gravitational force decreases by 2 2 or 4. Example: above: r 1 = 10m F 1 = 200N r 2 = 20m F 2 = 50N

The force between two masses is F g = 9000N. If the radius … …increases by a factor of 3. What is the F g ? …decreases by a factor of 10. What is the F g ? The F g would decrease by 3 2, or N/9 is 1000N. Fg = 1000N The F g would increase by 10 2, or Nx100 is 900,000N. Fg = 900,000N

Answer the following questions: 1.Are astronauts weightless in outer space? 2.Is there a gravitational force between two people? 3.Is there gravity in outer space? 4.How does the gravitational force change if the mass of an object increases? 5.How does the gravitational force change if the radius of an object decreases? No, there is gravity. It is just that since the objects are so far away, the gravitational force is very small! Yes, there is gravity. It is just that since the objects are so small, the gravitational force is very small! YES! There is gravity. It is just that since the objects are so far away, the gravitational force is very small! More mass, more gravitational force! Less radius, more gravitational force!.

Easy: What is the gravitational force between two students who are standing 5m away from each other and have masses of 80.0kg and 65.0kg?

F G = G m 1 m 2 r 2 F G = (6.67x Nm 2 /kg 2 ) (80kg)(65kg) (5m) 2 F G = (6.67x Nm 2 /kg 2 ) (5200kg 2 ) (25m 2 ) F G = (6.67x Nm 2 /kg 2 ) (208kg 2 /m 2 ) F G = 1.39x10 -8 N F G = N F G = ? r= 5m m 1 = 80kg m 2 = 65kg

Easy: What is the gravitational force between the sun (m=1.99x10 30 kg) and the planet Juptier (m=1.89x10 27 kg) if they are 7.8x10 11 m apart? F G = G m 1 m 2 r 2 F G = (6.67x Nm 2 /kg 2 ) (1.99x10 30 kg)(1.89x10 27 kg) ( 7.8x10 11 m) 2 F G = (6.67x Nm 2 /kg 2 ) (3.76x10 57 kg 2 ) (6.08x10 23 m 2 ) F G = 4.12x10 23 N F G =412,000,000,000,000,000,000,000N F G = ? r= 7.8x10 11 m m 1 = 1.99x10 30 kg m 2 = 1.89x10 27 kg F G = (2.51x10 47 Nm 2 ) (6.08x10 23 m 2 )

Easy: What is the escape velocity on the planet Venus which as a mass of 4.90x10 24 kg and a radius of 6.06x10 6 m?

V esc = 2(6.67x Nm 2 /kg 2 )(4.90x10 24 kg) (6.06x10 6 m) V esc = 6.54x10 14 m 3 /s 2 (6.06x10 6 m) V esc = 1.08x10 8 m 2 /s 2 V esc = 10386m/s √ √ √ Vesc = ? M = 4.90x10 24 kg R = 6.06x10 6 m

What if the Sun were replaced with a Black Hole with the same mass? Would we be pulled into it!? Earth & Sun F G = ? m E = 5.97x10 24 kg m S = 1.99x10 30 kg r= 1.5x10 11 m Earth & Black Hole F G = ? m E = 5.97x10 24 kg m BH = 1.99x10 30 kg r= 1.5x10 11 m F G = 3.52x10 22 N F G = 352,000,000,000,000,000,000N F G = 3.52x10 22 N F G = 352,000,000,000,000,000,000N

No, we would not be pulled in! The gravitational force would be the same because the mass and distances are the same. We would orbit around the black hole – just like we orbit around the sun!!! We are beyond the event horizon, so we would not get pulled in. We would have to be closer, or the black hole would have to have a larger mass! What if the Sun were replaced with a Black Hole with the same mass? Would we be pulled into it!?