Week 8_ Gravity and Review

10/19/2015 Week 8: Gravity and Review

https://session.masteringphysics.com/myct/assignmentPrintView?displayMode=studentView&assignmentID=4002083 1/12

Week 8: Gravity and ReviewDue: 8:00am on Wednesday, October 21, 2015

You will receive no credit for items you complete after the assignment is due. Grading Policy

A message from your instructor...

At the end of this homework set is one extra credit problem, followed by 5 review/practice questions. The practicequestions will not count toward your grade.

Item 1

A typical human diet is "2000 Calories" per day, where the "Calorie" describing food energy is actually 1000 calories = 1kilocalorie. 1 calorie = 4.184 J.

Part A

Express 2000 in watts.

ANSWER:

Item 2

A 75- long-jumper takes 3.1 to reach a prejump speed of 10 .

Part A

What is his power output?

Express your answer using two significant figures.

ANSWER:

Item 3

Three identical very dense masses of 6100 each are placed on the x axis. One mass is at = -100 , one is atthe origin, and one is at = 360 .

Part A

What is the magnitude of the net gravitational force on the mass at the origin due to the other two masses?

Take the gravitational constant to be = 6.67×10−11 .

kcal/day

= E W

kg s m/s

= P W

kg x1 cmx2 cm

Fgrav

G N ⋅ /km2 g2



Express your answer in Newtons.

You did not open hints for this part.

ANSWER:

Part B

What is the direction of the net gravitational force on the mass at the origin due to the other two masses?

ANSWER:

Item 4

Learning Goal:

To understand Newton's law of gravitation and the distinction between inertial and gravitational masses.

In this problem, you will practice using Newton's law of gravitation. According to that law, the magnitude of thegravitational force between two small particles of masses and , separated by a distance , is given by

,

where is the universal gravitational constant, whose numerical value (in SI units) is .

This formula applies not only to small particles, but also to spherical objects. In fact, the gravitational force betweentwo uniform spheres is the same as if we concentrated all the mass of each sphere at its center. Thus, by modeling theEarth and the Moon as uniform spheres, you can use the particle approximation when calculating the force of gravitybetween them.

Be careful in using Newton's law to choose the correct value for . To calculate the force of gravitational attractionbetween two uniform spheres, the distance in the equation for Newton's law of gravitation is the distance between thecenters of the spheres. For instance, if a small object such as an elephant is located on the surface of the Earth, theradius of the Earth would be used in the equation. Note that the force of gravity acting on an object located nearthe surface of a planet is often called weight.

Also note that in situations involving satellites, you are often given the altitude of the satellite, that is, the distance fromthe satellite to the surface of the planet; this is not the distance to be used in the formula for the law of gravitation.

There is a potentially confusing issue involving mass. Mass is defined as a measure of an object's inertia, that is, itsability to resist acceleration. Newton's second law demonstrates the relationship between mass, acceleration, and thenet force acting on an object: . We can now refer to this measure of inertia more precisely as the inertialmass.

= Fgrav N

+x direction-x direction

Fg m1 m2 r

= GFgm1m2

r2

G 6.67× 10−11 N⋅m2

kg2

rr

rEarth

= mF ⃗ net a ⃗



On the other hand, the masses of the particles that appear in the expression for the law of gravity seem to have nothingto do with inertia: Rather, they serve as a measure of the strength of gravitational interactions. It would be reasonableto call such a property gravitational mass.

Does this mean that every object has two different masses? Generally speaking, yes. However, the good news is thataccording to the latest, highly precise, measurements, the inertial and the gravitational mass of an object are, in fact,equal to each other; it is an established consensus among physicists that there is only one mass after all, which is ameasure of both the object's inertia and its ability to engage in gravitational interactions. Note that this consensus, likeeverything else in science, is open to possible amendments in the future.In this problem, you will answer several questions that require the use of Newton's law of gravitation.

Part A

Two particles are separated by a certain distance. The force of gravitational interaction between them is . Nowthe separation between the particles is tripled. Find the new force of gravitational interaction .

Express your answer in terms of .

ANSWER:

Part B

A satellite revolves around a planet at an altitude equal to the radius of the planet. The force of gravitationalinteraction between the satellite and the planet is . Then the satellite moves to a different orbit, so that itsaltitude is tripled. Find the new force of gravitational interaction .



ANSWER:

Part C

A satellite revolves around a planet at an altitude equal to the radius of the planet. The force of gravitationalinteraction between the satellite and the planet is . Then the satellite is brought back to the surface of theplanet. Find the new force of gravitational interaction .


ANSWER:

F0F1

F0

= F1

F0F2

F0

= F2

F0F4

F0

= F4



Part D

Two satellites revolve around the Earth. Satellite A has mass and has an orbit of radius . Satellite B has mass and an orbit of unknown radius . The forces of gravitational attraction between each satellite and the Earth

is the same. Find .


ANSWER:

Item 5

A satellite that goes around the earth once every 24 hours iscalled a geosynchronous satellite. If a geosynchronoussatellite is in an equatorial orbit, its position appearsstationary with respect to a ground station, and it is known asa geostationary satellite.

Part A

Find the radius of the orbit of a geosynchronous satellite that circles the earth. (Note that is measured fromthe center of the earth, not the surface.) You may use the following constants:

The universal gravitational constant is .The mass of the earth is .The mass of the satellite is .The radius of the earth is .

Give the orbital radius in meters to three significant digits.


ANSWER:

m r6m rb

rb

r

= rb

R R

G 6.67 × N /k10−11 m2 g2

5.98 × kg1024

2.10 × kg102

6.38 × m106

= m R



Item 6

Two identical point masses, each of mass , always remain separated by a distance of . A third mass is thenplaced a distance along the perpendicular bisector of the original two masses, as shown in the figure.

Part A

What is the direction of the gravitational force on the third mass?

ANSWER:

Part B

What is the magnitude of the gravitational force on the third mass?

Express your answer in terms of the variables , , , , and appropriate constants.


ANSWER:

Item 7

M 2R mx

inward along the perpendicular bisectoroutward along the perpendicular bisectortoward the lower mass toward the upper mass

M

M

M m R x

= F



Part A

Two identical satellites orbit the earth in stable orbits. One satellite orbits with a speed at a distance from thecenter of the earth. The second satellite travels at a speed that is less than . At what distance from the center ofthe earth does the second satellite orbit?


ANSWER:

Part B

Now assume that a satellite of mass is orbiting the earth at a distance from the center of the earth with speed. An identical satellite is orbiting the moon at the same distance with a speed . How does the time it

takes the satellite circling the moon to make one revolution compare to the time it takes the satellite orbitingthe earth to make one revolution?


ANSWER:

Item 8

Very far from earth (at ), a spacecraft has run out of fuel and its kinetic energy is zero. If only the gravitationalforce of the earth were to act on it (i.e., neglect the forces from the sun and other solar system objects), the spacecraftwould eventually crash into the earth. The mass of the earth is and its radius is . Neglect air resistancethroughout this problem, since the spacecraft is primarily moving through the near vacuum of space.

Part A

Find the speed of the spacecraft when it crashes into the earth.

Express the speed in terms of , , and the universal gravitational constant .

v rv

At a distance that is less than .At a distance equal to .At a distance greater than .

r

r

r

m rve vm Tm

Te

is less than . is equal to . is greater than .

Tm Te

Tm Te

Tm Te

R = ∞

Me Re

se

Me Re G




ANSWER:

Part B

Now find the spacecraft's speed when its distance from the center of the earth is , where the coefficient .

Express the speed in terms of and .


ANSWER:

Item 9

A satellite of mass is in a circular orbit of radius around a spherical planet of radius made of a materialwith density . ( is measured from the center of theplanet, not its surface.) Use for the universal gravitationalconstant.

Part A

Find the kinetic energy of this satellite, .

Express the satellite's kinetic energy in terms of , , , , , and .

= se

R = αReα ≥ 1

se α

= sα

m R2R1

ρ R2G

K

G m π R1 R2 ρ




ANSWER:

Part B

Find , the gravitational potential energy of the satellite. Take the gravitational potential energy to be zero for anobject infinitely far away from the planet.

Express the satellite's gravitational potential energy in terms of , , , , , and .


ANSWER:

Part C

What is the ratio of the kinetic energy of this satellite to its potential energy?

Express in terms of parameters given in the introduction.

ANSWER:

A message from your instructor...

The next question is for extra credit.

Item 10

A projectile is launched vertically upward from an airless planet of mass and radius ; its initial speed is timesthe escape speed.

Part A

Derive an expression for its speed as a function of the distance from the planet's center.

Express your answer in terms of the variables , , and the constant of universal gravitation .

= K

U

G m π R1 R2 ρ

= U

KU

= KU

M R 2√

r

M R G

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Week 8_ Gravity and Review