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Newton’s law of universal gravitation

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11) Newton’s predicted that the minimum orbital speed is 8km/s. Satellite must have a speed greater than 8 km/s. He drew a mountain with a canon at the top.

 

He knew that the ball will fall 4.9m in one second .

(extra credits if you show that. Easy use one of the equations of motion)

 

Some geometry can show that Earth’s surface curves away at a rate of 4.9 m for every 8km.

So the canon ball has to be thrown at a speed of 8km/s. That way, the ball keep falling toward the Earth without ever touching ground. The satellite falls toward the Earth ’s center but the Earth curves away ! Newton’s thought experiment didn’t take in account the resistance of the air and the mountain would have to be 150km above Earth surface to be above most of Earth’s atmosphere.

Picture from: Physics of every day Phenomena, Mcgraw-hill

A) What is the highest mountain on Earth ? How High ?

B) A satellite moves in circular motion.

So Fc  = _____________   (centripetal force, use m, V, r the total distance Earth's center-object, m is the mass of the object and V its orbital speed)

This centripetal force is the attractive gravitational force Fg = G mM/r² 

Or from the point of view of the object, the gravitational force has to balance the centrifugal force Fc

M is the mass of the Earth and G is a constant (6.67 E –11)

So Fg = Fc (they balance each other), can you solve for v ?  v = _____________ 

( hint: Use only G, M, r in your expression)

Conclusion: The orbital speed of a satellite does not depend on its ______________

 

C) Remember ?  V = 2 pi r/T   , T is the orbital period

substitute the expression found for V in B) in this expression and solve for T = _________

(hint: use only r, G, M ). T does not depend on the _________ of the object either !

 

D) Using F = G m M / r² , can you find  the unit for G ?

12) You now know that the orbital speed of an satellite is:

V² = (G M / r)    or V = √(G M / r)   with :

G = 6.67E-11, M = 5.97 E 24kg is the mass of the Earth and, r is the orbital distance satellite-center of Earth

A) the Earth’s radius is about R= 6378km

Use the formula to find the speed that a satellite shot from the cannon must have in order to orbit Earth h =150km above its surface. V = ________ km/s

 (hint: r = Earth’s radius + height = 6370km + 150km, CONVERT IN METERS)

 

B) In fact, the gravity decreases as the distance increases but by only a few % (maybe 3) if you move by about 100km. So,  as an approximation you can suppose that g = 9.8m/s/s at a distance of 150km above ground. Let’s see if this is a good approximation.

 

The force of gravity is now Fg= mg (g=9.8m/s/s,  m the mass of the satellite). Fc=mV²/r

If Fg =Fc solve for V². The orbital speed still does not depend on the _______ of the satellite.

And V² = _________ (use only g and r the total distance center-satellite)

Using the new formula for V², find the speed of the satellite using r = 6370km + 150km

V= ________km/s . compare with A. Do you get the same thing ? (convert to meters)

If r = 6400km, what is the velocity you find? V = _______km/s

This is the minimum orbital speed a satellite must have not to fall back on the ground.

 

C) How long , in hours, would it take for the satellite to complete one orbit and return to the cannon ? (that is find the orbital period T, should be very short)

You can use: V = 2 pi r / T and solve for T. Or you can make your life harder and use the formula from 11.

 

D) These kind of satellites are Low orbit satellites (LEO satellites). Spy satellites are LEO satellites. They need to fly close to Earth to spy on enemy activities.

Using B, can you guess what is the problem of these spy satellites ?

(the enemy knows the location of the spies and they may hide their activity during the time of their passage).

 

E) Let’s suppose a spy satellite orbits 150km above the ground. You know the orbital speed from A. (should be about 8km/s)

V = _____ km/s. Convert this speed to mph. V = ______ mph.

If the satellite wants to spy a area of 300 miles across. How many times does the satellite to take pictures? (use V = d /t solve for t)

 

 

13)  Find the mass of the Sun M in an encyclopedia Ms = _____________

Find the distance between Mercury and the Sun r_m = _________m  (convert to meters)

Find the distance between Saturn and the Sun rs = ___________m (convert to meters)

Using the formula V² = (G Ms / r)  Find the orbital speed of Mercury and Saturn.

Vm = ___________m/s = ___________ km/s = ___________ mph

and Vs = ________ m/s = ___________ km/s = ____________ mph

 

Does it make sense that Mercury is named after a speedy messenger of the gods, whereas Saturn is named after the father of Jupiter ?

 

 

 

13) advanced

The Sun is considered to be a satellite of our galaxy, the milky way. The Sun revolves around the center of the galaxy with a radius of 2.2 E20 m. The period of a revolution is 2.5 E8 years.

A) find the mass of the galaxy

B) Assuming that the average star in the galaxy has the same mass as the sun, find the number of stars.

C) Find the speed with which the sun moves around the center of the galaxy.

 

14) extra credits: show work

A satellite orbits Earth 225km above its surface. What is its speed in orbit and its period.

 

15) extra credits: take that quiz:

http://highered.mcgraw-hill.com/sites/0072509775/student_view0/chapter5/mastery_quiz.html

 

Write down the questions and answers. Extra credits assignment.

 

16) conservation of energy shows that, to escape the gravity of the Earth and fly free to space, an object, of any mass, needs to be given a speed of at least 11km/s

Convert 11km/s to mph = _________________

 

17) The GPS satellite are medium orbit satellites (MEO). Their speed is about 12,000mph.

A) convert that speed to km/s V= __________

B) Using the formula derived in 12) can you find the distance from the center of the Earth and the distance above the ground? In km ? Rearth = 6370km

Hint: use V² = (G M / r)   , r is the total distance Earth-satellite, M is the mass of the Earth. First solve for r then find h, the distance from ground, using  r = h + R

C) How long does it take them to  orbit the Earth ?

Use v = 2 pi r / T, r is the total distance Earth-satellite, solve for T.

 

18) The geosynchronous satellites are stationary. They stay above the same point. They are used for TV.

A) Why do they are to be above the equator?

B) If the are stationary, what is their orbital period ? (hint: they turn with the Earth)

C) Advanced only (extra credits) : Find their height above the ground.

Hint: Use V² = (G M / r) but V = 2 pi r / T , use T = 24h = 24 x 60 x 60 seconds,  solve for r in meters. Then find h with r = h + 6378km. G = 6.67E-11, M = 5.97 E 24kg is the mass of the Earth and, r is the orbital distance satellite-center of Earth

 

19) A satellite is orbiting the Earth. There is no friction. It could orbit, with the same speed, at the same distance forever. How can you can it back (you can expel gas, but in which direction) ? Draw to illustrate the situation. Nice drawing will get extra credits.

 

20) The escape velocity for Earth is 11km/s

 A) This is derived from the conservation of energy. To escape the Earth, the ship needs enough kinetic energy (1/2 m V² ) to over come the gravitational force . The gravitational force gives the ship a gravitational potential energy (G mM /R) as long as the ship does not escape the gravity of Earth.

If the kinetic energy balances the potential energy, can you derive V, the escape velocity?

Use : Kinetic energy = potential energy at ground level or:

1/2 m V²  = GmM/R

A) First solve for V (don’t use any number yet)  V= __________

m is the mass of the ship, M is the mass of Earth, R is the radius of Earth and V the speed of the ship.

B) Find the numerical value of V

V = ___________ km/s

G = 6.67E-11, M = 5.97 E 24kg , R = 6400,000 m

 

C) Using the same idea, find the escape speed for the Sun.

V = √ (2GMs/R)

With Ms the mass of the Sun and R the radius of the Sun.

V = ________________ km/s (to do the computations convert to meters and kg)

 

D) that’s a pretty big number ! Isn’t it ?

Convert to mph to be convinced: V = _____________mph

(1 mile = 1.6 km and 1 hour = 3600s)

Is there an object in space whose escape speed is the speed of light ?

(300,000 km/s ?) If you take the Earth, make it the size of a golf ball, that would be a black home. (same mass but smaller radius, that is huge density). Do the computation for fun !

 

21) puzzle:

1)   __________ due to gravity is less the farther you are from Earth.

2) The space shuttle in _________________ around the earth is in free-fall.

3) Newton invented a _______________ called calculus

4) Newton was one of the first scientist to understand projectile motion.

5) It Earth’s ____________ matches a projectile fall, the projectile’s will not hit Earth.

6) _____________ predicted projectile motion

7) Relative to a person on Earth a satellite that orbits once a day would appear to not be in

____________________

8) Newton was born on ___________Day in 1642

9) A satellite that is appears to be motionless is called a ____________________ satellite

10) Satellites orbiting the Earth never escape its __________

11) A stationary satellite orbits Earth ________ each day.

12) …………. Was the first artificial satellite and it is Russian.

 

 

Some of the questions were inspired by the book “Physics for future presidents ”by Richard A. Muller. He is a professor at Berkeley College, California. I highly recommend his book and webcasts available from the Web.  (no Math !)