Why Moon does not fall on Earth

8/13/2019 Why Moon does not fall on Earth

1/21

14-1

The Moon does not fall to Earth because

A: It generates a gravitational force equal and opposite the

earth's pull.B: The net force on it is zero.C: It is beyond the main pull of Earths gravity.D: It is being pulled by the Sun and planets as well as byEarth.E: none of the above

Answer: E. None of these is correct. The moon DOES fall to earth, all the time!There is a large force, F = GM(earth)M(moon)/R(earth to moon)^2 which pulls ittowards us. If this force was NOT there, the moon would travel in a straight line andrun away! A circular orbit means acceleration towards the center, that's what youmean by "falling"! The reason the moon doesn't HIT the earth is because it has anice big SIDEWAYS velocity which never goes away (because it's alwaysperpendicular to the force of gravity). That means that as the moon falls towards us,it also drifts sideways, and ends up further around the circle. No CLOSER to usradially... You have to really think about this and make sense of it for yourself. The

moon IS accelerating towards you right now, but it keeps missing...


2/21

14-2

Suppose Earth had no atmosphere and a ball were firedfrom the top of Mt. Everest in a direction tangent to the

ground.If the initial speed were highenough to cause the ball totravel in a circular trajectoryaround Earth, the ballsacceleration would

A: be much less than g (because the ball doesnt fall tothe ground).B: be approximately g.C: much greater than gD: depend on the balls speed.

Answer: It's g. The acceleration of ANY object that is freely falling near the surface

of the earth is g. Sideways speed is irrelevant. This goes way back to the verybeginning of the class- it's Galileo's observation.

Air resistance would change it a wee bit. Also, "g" at th top of Mt. Everest is aTINY TINY bit smaller than "g" here.

image)


3/21

14-3

At some instant in time, two asteroids in deep space are adistance r=20 km apart. Asteroid 2 has 10 times the massof asteroid 1. What is the ratio of their resultingacceleration (due to gravitational attraction to each other),

m1 2 = 10 m1

a1 / a2 =

A: 10:1 B: 1:1 C: 1:10D: Not enough information

Answer: F(1 on 2) = F(2 on 1) (in magnitude), that's N-III.(Or, use F = G M1 M2/r^2, it's the same if you reverse M1 and M2)

Since F=ma, and F is the same, the acceleration of the little one must be 10 timesbigger than the acceleration of the big one, 10:1.


4/21

14-4

Planet X has the same mass as earth, but its radius is onlyhalf as big. How does g x compare with our g?

A: g x = gB: g x = 2gC: g x = 4gD: g x = (1/2) gE: g x = (1/4) g

Answer: F(grav) = G M(planet) m(object) / R^2, = m(object) g,so g = G M(planet)/R^2. Thus, if M is the same, and R is half, g(x) must be 4 timesbigger. (Same mass but smaller means planet X is very dense!)


5/21

14-5

The space shuttle orbits about 200 km above the earthssurface. The shuttle's "pre-launch weight" is its weight

measured on the ground. (Neglecting the fuel) Themagnitude of the force of gravity on the shuttle while it isin orbit is

A: Slightly greater than pre-launch.B: Equal to pre-launchC: Slightly less than pre-launchD: Lots less than pre-launch, almost (but

not quite) zero.E: Precisely zero.

Answer: Slightly less than pre-launch. The force of gravity is nearly the same 200km up, because G M(earth) M(ship)/R^2 depends on the distance R to the CENTERof the earth (6000 km deep), not to the survace. Of course, 6200^2 is a bit biggerthan 6000^2, so there is a small difference. But it's not as much as you'd think.The shuttle is not truly "weightless", it's just APPARENT weight that vanishes upthere. (It's in free-fall!)


6/21

14-6

An astronaut is floating around in the space shuttle's cabin.Her acceleration, as measured from the earth's surface, is

A: zero - she's floatingB: very small, in some random directionC: quite large, nearly g, directed towards

the center of the earthD: quite large, nearly g, directed along the line of

travel of the shuttle

Answer: She's in orbit, so she's in uniform circular motion. She has a largeacceleration! We just said F(grav) is almost the same up there, so her acceleration isalmost the same as on earth in freefall, namely, nearly g, directed towards the centerof the earth. She's just like a ball that's been tossed up in the air, accelerating straightdown. The only difference is she has such a large sideways speed that when she hasfallen 200 km, she's gone so far to the side that the curved surface of the earth hasmoved 200 km away from her, so she's still 200 km above the ground!!!


7/21

14-7

A rock is released from rest, in deep space, far from theearth. At a certain point, as it falls towards us, it crosses the

moon's orbit.

moon earthrock

When the rock is the same distance from the earth as themoon, the acceleration of the rock is

A: Greater than B: Equal to C: Smaller thanD: (Need more information)the acceleration of the moonAnswer: The only force acting on either one is gravity,

F(grav) = G M(earth) m / R^2. The acceleration is then F/m, and that's the same forboth!!

As the rock falls toward the Earth, its acceleration is..A: constant. B: not constant.Answer: as it approaches, R is getting smaller so GM/R^2 is getting bigger. Theacceleration is thus getting larger. It's not constant.

(It's only approximately constant when you're NEAR the surface, because R isn'tchanging very much as you go from 6000km+0 to 6000 km + 20 feet, say. )


8/21

14--8

At time t=0, a satellite in circular orbit about the Earth isdirectly over Denver, 300 miles above the city, traveling

east at 16,000 mph.At the same time, a rock is released from rest 300 milesabove the city, very near the satellite.

A: true or B: FalseAt t=0+, the accelerations of the rock and the satellite areidentical in magnitude and direction.

Answer: We've seen in this in several recent concept tests. The acceleration of bothare identical. (nearly g, towards the center of the earth). Both are freely falling, andat the same radius.

Mass is irrelevant (it cancels from F=ma and F=G M m/r^2), and so is sidewaysspeed.

Denver

rock satellite


9/21

14-9

Two satellites A and B of the samemass are going around Earth in

concentric orbits.The distance of satellite B from Earthscenter is twice that of satellite A.What is the ratio of the magnitudes of the accelerations of A to B?

|a(A)| / |a(B)| =A: 8 B: 4 C: 2D: 1 E: Sqrt(2)

Answer: a = G M(earth)/R^2, so if you're half the distance, you have 4 times theacceleration.

A

B


10/21

14-10 A planet in elliptical orbit around a star moves fromthe point in its orbit furthest from the star (A) to the closestpoint (P). The work done by the force of gravity during

this movement is A : 0 B : + C: -

Answer: You go faster when you're closer to the sun. Since W(net) = Delta K,

if you go faster, something has done + work on you. That something can only begravity, there are no other forces.

Another way to see it:The force of gravity from the Sun is centripetal (toward the center). The force has a componentalong the direction of displacement. W = F . Delta r = F * Delta r * cos(theta) > 0

The planet executes one complete orbit starting from pointA and returning to A. The work done by the force of gravity during this orbit is:

A : 0 B : + C: -Answer: Zero. By the work energy theorem, or just conservation of energy.

AP

F Sun

! r


11/21

14-11

Two satellites A and B of the same massare orbiting Earth.

The distance of satellite B from Earthscenter is twice that of satellite A.What is the ratio of the speed of A tothat of B?

|v(A)| / |v(B)| =

A: 1/2 B: 1 C: 2D: Sqrt(2) E: Sqrt(1/2)

Answer: F= ma, so GM(earth)m /R^2 = m v^2/R,thus v = Sqrt[G M(earth)/R]. The farther you are, the slower you go.

So if A is 1/2 the radius, v is Sqrt[2] times bigger.

A

B


12/21

14-12

Two communications satellites are in orbit at the sameheight, but one weighs twice as much as the other. Thespeed of the heavier satellite isA: Less than B: Equal to C: Greater thanD: (need more information)

the speed of the lighter one.

Answer:See previous problem, mass cancels out. Speed depends only on radius, for

spherical orbits.


13/21

14-13

Kepler's 3rd law: T^2/R^3=constant for all planets.The period T for the earth is 1 year (!)The distance of the earth to the sun is called 1 A.U.Suppose an asteroid is in circular orbit around the sun at adistance of 2 A.U. How long does it take the asteroid toorbit the sun once?

A 2 yearsB: 3 years

C: 2^(3/2) = 2.83 yearsD: 2^(2/3) = 1.59 yearsE: Not enough information (An asteroid is not a planet -can't figure it out from Kepler's third law.)

Answer: T (period) goes like R^(3/2), so if you're 2 times as far, your year is 2^3/2times longer.


14/21

14-14

Kepler's 3rd law: T^2/R^3=constant for all planets. Theperiod T for the earth is 1 year.The distance of the earth to the sun is called 1 A.U.Suppose a satellite orbits the earth at a distance of about1/1000 AU. How long does it take the satellite to orbit theearth once?

A: Not enough information given.B: 1/1000 year.

C: 1000 years.D: (1/1000)^(3/2) yearsE: (1/1000)^(2/3) years

Answer: Not enough information given. Kepler's law only works if the objects orbita common center. We need to know the earth's mass (or the distance to the moon, orsomething) to figure this one out.


15/21

14-15

A rock, initially at rest withrespect to Earth and locatedfar away is released andaccelerates toward Earth.An observation tower isbuilt 3 Earth-radii high to observe the rock as it plummetsto Earth.

Neglecting friction, the rocks speed when it hits the

ground isA: twice B: three timesC: four times D: eight timesE: sixteen times

its speed at the top of the tower.

Answer: Conservation of energy says 1/2 mv^2 - G M(e) m/R = constant.If it starts infinitely far away with v=0, then at "4R" it has1/2 mv^2 = constant/(4R).And at "R" it has1/2 mv^2 = same constant/R, i.e. 4 times BIGGER KE.So 2 times BIGGER v.

3 R R


16/21


17/21

14-17

A projectile is fired straight upward from the surface of anairless planet (radius R) with escape velocity. What is theprojectile's speed when it is a distance 4R from the planet'scenter (3R from the surface)?

A: 1/2 v escB: 1/4 v escC: 1/9 v esc

D: 1/3 v escE: None of these is correct.

(Ignore the gravity of the Sun and other astronomicalbodies.)

Answer: 1/2 m vesc 2 - G M m / R = 0 (that's the condition for escape velocity)So later, 1/2 m v^2 - G M m / (4R) must still = 0. (energy is conserved)

So 1/2 m v^2 = (1/4) * 1/2 m vesc^2 is.Therefore v^2 = (1/4) vesc^2.or v = (1/2) vesc


18/21


19/21

14-19

A planet is in elliptical orbit around the Sun. The zero of potential energy U is chosen at r = ! , so U r GMm

r ( ) = ! .

How does the magnitude of U, i.e. |U| compare to theKE? A: |U| > KE B: |U| < KEC: |U| =KE D: depends on the position in the orbit.

Answer:The earth is bound, so K + U


20/21

14-20 A satellite is in circular orbit around a planet that has avery tenuous atmosphere extending up to the altitude of thesatellite. Due to atmospheric drag, the speed of the

satelliteA: increasesB: decreasesC: remains constant

Answer: Very tricky. Drag will make the satellite's orbital radius drop, and since v =Sqrt[ G M/r], the speed will INCREASE. How can this be? Well, K+U = totalenergy, so K - GM m/R is the total energy. K is the smaller term (because you're inorbit), the potential "wins". If R decreases, the negative term gets MORE negative,the positive term gets MORE positive, but the total gets MORE negative (becausethe negative one is winning). Your total energy is decreasing, your potential energyis decreasing, your Kinetic energy is increasing (but not as fast). So they spiral infaster and faster, burning up more and more energy and ultimately burning upentirely!


21/21

14-21

Astronaut Dave Bowman is standing in the centrifuge of the spaceship Discovery. He drops his pen and observes itfall to the floor.Which statement below is most accurate?

A: After he releases the pen, the net force on the pen iszero.B: The pen falls because the centrifugal force pulls ittoward the floor.

C: The pen falls because the artificial gravity pulls ittoward the floor.

Answer: A is correct. The pen was moving in a circule, and if he lets go, it willhenceforth follow a straight line path... which APPEARS to make it fall to theground in his reference from, but in fact there is no net force on it....

Why Moon does not fall on Earth

Documents

Transcript of Why Moon does not fall on Earth