Gravity Is Solely an Attractive Topic, Nothing Repulsive Here!

2016 AP gravitation chapter 13

Gravity is solely an attractive topic, nothing repulsive here!

Gravity keeps us down to Earth!

1. What did Kepler discover about the motion of the planets?

First Law.

Second Law

1. The elliptical orbit of a comet is shown. Positions 1 and 2 are, respectively, the farthest and nearest positions to the Sun, and at position 1 the distance from the comet to the Sun is 10 times that at position 2. What is the ratio v1/v2 of the speed of the comet at position 1 to the speed at position 2?

1. 1/100
2. 1/10
3. 1
4. 10
5. 100

1. A satellite is in an elliptical orbit about a planet, as shown in the figure. At apogee, a distance R1 from the planet, the satellite’s angular speed is ω. What is the angular speed of the satellite at perigee, a distance R2=R1/2 from the planet?
2. (1/4)ω
3. (1/2)ω
4. (1/(2)1/2)ω
5. 2ω
6. 4ω

1. 
   When the satellite is at the desired altitude over the equator, its velocity vector has the correct direction, but the speed is slightly faster than the correct speed for a circular orbit of that radius. Draw the orbit below.

When the satellite is at the desired altitude over the equator, its velocity vector has the correct direction, but the speed is slightly slower than the correct speed for a circular orbit of that radius. Draw the orbit below.

1. Kepler’s 3rd Law

1. A space shuttle is initially in a circular orbit of radius r about Earth. At point P, the pilot briefly fires a forward-pointing thruster to decrease the shuttle’s kinetic energy K and mechanical energy E.

1. Which of the dashed elliptical orbits shown in the figure will the shuttle then take?
2. Is the orbital period T of the shuttle then greater than, less than, or the same as in the circular orbit?

1. An Earth satellite in a circular orbit of radius R has a period T. What is the period of an Earth satellite in a circular orbit of radius 4R?
2. T
3. 2T
4. 4T
5. 6T
6. 8T

1. Satellite X moves around Earth in a circular orbit of radius R, Satellite Y is also in a circular orbit around Earth, and it completes one orbit for every eight orbits completed by satellite X. What is the orbital radius of satellite Y?
2. ¼ R
3. ½ R
4. 2R
5. 4R
6. 8R

1. How did Newton expand on Kepler’s Laws? Derive Newton’s Law of Gravitation.

1. If F1 is the magnitude of the force exerted by the Earth on a satellite in orbit about the Earth and F2 is the magnitude of the force exerted by the satellite on the Earth, then which of the following is true?
2. F1 is much greater than F2
3. F1 is slightly greater than F2
4. F1 is equal to F2
5. F2 is slightly greater than F1
6. F2 is much greater than F1

1. Alice (50kg) and Arlo(100kg) are 20 meters apart. What is the gravitational force on each of them?

1. A spaceship is on a straight-line path between Earth and its moon. At what distance from Earth is the net gravitational force on the spaceship zero? (mass of earth=5.98x1024kg, mass of the moon =7.36x1022kg, distance between earth and moon is 3.28x108m)

1. Find the gravitational force on m due to M and M.

1. Force on a point mass located along the axis of a thin ring.

1. What was the centripetal force? How do I find velocity for a mass going in a circle? What if it is elliptical?

1. In March 1999 the Mars Global Surveyor (GS) entered its final orbit about Mars, sending data back to Earth. Assume a circular orbit with a period of 1.18 x 102 minutes = 7.08 x 103 s and orbital speed of 3.40 x 103 m/s .The mass of the GS is 930 kg and the radius of Mars is 3.43x106m .

(a)Calculate the radius of the GS orbit.

(b)Calculate the mass of Mars.

(c)If the GS was to be placed in a lower circular orbit (closer to the surface of Mars), would the new orbital periodof the GS be greater than or less than the given period?
______Greater than______Less than

Justify your answer.

(d) In fact, the orbit the GS entered was slightly elliptical with its closest approach to Mars at 3.71 x 105 m above the surface and its furthest distance at 4.36 x 105 m above the surface. If the speed of the GS at closest approach is 3.40 x 103 m/s, calculate the speed at the furthest point of the orbit.

1. A student is given the set of orbital data for some of the moons of Saturn shown below and is asked to use the data to determine the mass of Saturn. Assume the orbits of these moons are circular.

1. Use the assumption of circular orbits to derive an equation for the orbital period T of a moon as a function of its orbital radius R.

1. Which quantities should be graphed to yield a straight line whose slope could be used to determine Saturn’s mass?

1. A satellite moves in a stable circular orbit with speed vo at a distance R from the center of a planet. For this satellite to move in a stable circular orbit a distance 2R from the center of the planet, the speed of the satellite must be
2. vo
3. 2vo

1. Two artificial satellites, 1 and 2, orbit the Earth in circular orbits having radii R1 and R2, respectively, as shown above. If R2 = 2R1, the accelerations a2 and a1 of the two satellites are related by which of the following?

(A) a2 = 4a1 (B) a2 = 2a1 (C) a2 = a1 (D) a2 = a1/2 (E) a2 = a1/4

1. Two stars, A and B. are in circular orbits of radii ra and rb, respectively, about their common center of mass at point P, as shown above. Each star has the same period of revolution T. Determine expressions for the following three quantities in terms of ra, rb, T, and fundamental constants.

1. The centripetal acceleration of star A

1. The mass Mb of star B

1. The mass Ma of star A

Determine expressions for the following two quantities in terms of Ma, Mb, ra, rb, T, and fundamental constants.

1. The moment of inertia of the twostar system about its center of mass.

1. The angular momentum of the system about the center of mass.

1. Two identical stars, a fixed distance D apart, revolve in a circle about their mutual center of mass as shown above. Each star has mass M and speed v. G is the universal gravitational constant. Which of the following is a correct relationship among these quantities?
2. v2=GM/D
3. v2=GM/2D
4. v2=GM/D2
5. v2=MGD
6. v2=2GM2/D

2016 AP gravitation chapter 13

1. What is the force due to gravity between a mass m and an extended mass?

What is the center of mass

of an extended mass?

2016 AP gravitation chapter 13

1. A spherical, nonrotating planet has a radius R and a uniform density ρ throughout its volume. Suppose a narrow tunnel were drilled through the planet along one of its diameters, as shown in the figure above, in which a small ball of mass m could move freely under the influence of gravity. Let r be the distance of the ball from the center of the planet.

1. Find the force due to gravity inside the planet.

1. On the axes below, sketch the force F on the ball as a function of distance r from the center of the planet.

1. What physics quantity is the area of the above graph?

1. What if Earth was a shell, what would the force of gravity be inside the shell?

1. What if the Earth was a shell with thickness. Draw the Force due to gravity on a mass m as a function of r in the range 0r

F

r

1. A uniform solid sphere of radius R produces a gravitational acceleration of ag on its surface. At what two distances from the center of the sphere is the gravitational acceleration ag/3?

Inside the planetOutside the planet

1. The mass of Planet X is one-tenth that of the Earth, and its diameter is one-half that of the Earth. The acceleration due to gravity at the surface of Planet X is most nearly
2. 2m/s2
3. 4m/s2
4. 5m/s2
5. 7 m/s2
6. 10 m/s2

1. A newly discovered planet has twice the mass of the Earth, but the acceleration due to gravity on the new planet’s surface is exactly the same as the acceleration due to gravity on the Earth’s surface. The radius of the new planet in terms of the radius R of Earth is
2. ½ R
3. 2R
4. 4R

1. The acceleration due to gravity on the surface of the Moon is about 1/6 of that on the surface of the Moon is about 1/6 of that on the surface of Earth. An astronaut weighs 600 N on Earth. The astronaut’s mass on the Moon is most nearly
2. 600kg
3. 360 kg
4. 100 kg
5. 60 kg
6. 10 kg

1. At the surface of the Earth is the only acceleration we have that due to gravity? Why or why not?

1. Three uniform spherical planets that are identical in size and mass. The periods of rotation T for the planets are given, and six lettered points are indicated- three points are on the equators of the planets and three points are on the north poles. Rank the points according to the value of the free fall acceleration g at them, greatest first.

1. Find the acceleration at the equator of the Earth.

1. What is the work done by gravity?

1. The graph above shows the force of gravity on a small mass as a function of its distance R from the center of the Earth of radius Re, if the Earth is assumed to have a uniform density. The work done by the force of gravity when the small mass approaches Earth from far away and is placed into a circular orbit of radius R2 is best represented by the area under the curve between

a. R=0 and R=Re

b. R=0 and R=R2

c. R=Re and R=R2

d. R=Re and R=

e. R=R2 and R=

1. When an object is moved from rest at point A to rest at point B in a gravitational field, the net work done by the field depends on the mass of the object and

1. the positions of A and B only

b. the path taken between A and B only

c. both the positions of A and B and the path taken between them

d. the velocity of the object as it moves between A and B

e. the nature of the external force moving the object from A to B

1. What is the work done on the moon as it orbits the Earth?

1. Why does the gravitational force do work in an elliptical orbit, but not in a circular orbit?

1. What is the net work done for one complete elliptical orbit?

1. Find the gravitational potential associated with Newton’s Law of Universal outside a planet.

1. How does this compare to other potential energy (mgh).

Close to planet surface object fallingFar from planet surface object falling

UU

t t

Why the difference?

1. Planet X has a mass M, radius R, and no atmosphere. An object of mass m is located a distance 2R above the surface of planet X, as shown. The object is released from rest and falls to the surface of the planet. What is the speed of the object just before it reaches the surface of planet X?

1. As your spacecraft travels along an x axis through an asteroid belt, the gravitational potential energy U(x) of the spacecraft-asteroid system is given by the curve below. Rank the magnitude of F at points A, B, C, D, E , greatest first. What is the direction of the force at each of the points?

U

x

1. The potential energy U as a function of the position x of an object is given by U(x)=-400/x, where U is in joules and x is in meters. The object is released from rest at position x=30 m and is free to move along the x-axis. When the object has moved a distance of 10 m from its initial position, the force associated with this potential energy function is
2. 20 N
3. 10 N
4. 4 N
5. 1 N
6. 0.25 N

1. Consider Earth to be stationary, and the Moon as orbiting Earth in a circle of radius R. If the masses of Earth and the Moon are Mg and Mm, respectively, which of the following best represents the total mechanical energy of the Earth-Moon system?
2. GMgMM/(2R)
3. GMEMM/ (R)
4. –GMgMM / (2R)
5. –GMg MM/ (R)
6. -2GMgMM/(R)

1. Rank the positions of a satellite in elliptical orbit from greatest to least.

a. gravitational force

b. speed

c. momentum

d. Kinetic Energy

e. Potential energy

f. total energy

1. A satellite of mass m is in an elliptical orbit around the Earth, which has mass Me and radius Re. The orbit varies from closest approach of a at point A to maximum distance of b from the center of the Earth at point B. At point A, the speed of the satellite is vo Assume that the gravitational potential energy Ug = 0 when masses are an infinite distance apart. Express your answers in terms of a, b, m, Me, Re, vo, and G.

a. Write the appropriate definite integral, including limits, that can be evaluated to show that the potential energy of the satellite when it is a distance r from the center of the Earth is given by

Ug = -GMem

r

b. Determine the total energy of the satellite when it is at A.

c. What is the magnitude of the angular momentum of the satellite about the center of the Earth when it is at A ?

1. Determine the velocity of the satellite as it passes point B in its orbit.

As the satellite passes point A, a rocket engine on the satellite is fired so that its orbit is changed to a circular orbit of radius a about the center of the Earth.

e. Determine the speed of the satellite for this circular orbit.

f. Determine the work done by the rocket engine to effect this change.

1. Find the speed an object would have to attain to escape the gravitational pull of an astronomical body of mass M and radius R.

1. The escape speed for a rocket at Earth’s surface is ve. What would be the rocket’s escape speed from the surface of a planet with twice Earth’s mass and the same radius as Earth?
2. 2ve
3. ve
4. ve/2

1. A small rock is launched straight upward from the surface of a planet with no atmosphere. The initial speed of the rock is twice the escape speed ve of the rock from the planet. If gravitational effects from other objects are negligible, the speed of the rock at a very great distance from the planet will approach a value of
2. Zero
3. ve/2
4. ve
5. (2)1/2ve
6. (3)1/2ve