Chapter 11 and 13 Homework
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Chapter 11 and 13 Homewor k
Chapter 11 and 13 Homework Due: 10:00pm on Wednesday, April 23, 2014 You will receive no credit for for items you com plete after after the ass ignm ent is due . Grading Policy
Exercise 11.1 A 0.1 0.10 00- , 40. 40.2 2-lo lon ng un unif ifo orm ba bar has a sma small ll 0. 0.0 080 80-mass ma ss glu glue ed to to its its le left en end and and a sma small ll 0.1 .15 50mass ma ss glu glue ed to the other end. You want to balance this system horizontally on a fulcrum placed just under its center of gravity.
Part A How far from the left end should the fulcrum be placed? ANSWER: = 24.4
Correct
Introduction to Static Equilibrium Learning Goal: To understand understand t he conditions necess ary for static stati c equilibr equilibrium. ium. Look around you, and you see a world at rest. The monitor, desk, and chair—and the building that contains them—are in a state described as static equilibrium. equilibrium. Indeed, it is the fundamental objective of many branches of engineering to maintain this state in spite of the presence of obvious forces imposed by gravity and static loads or the more unpredictable forces from wind and earthquakes. The condition of static equilibrium is equivalent to the statement that the bodies involved have neither linear nor angular acceleration. Hence static mechanical equilibrium equilibrium (as opposed to t herma hermall or electrical electric al equilibrium) equilibrium) requires requires that the forces acting on a body simultaneously satisfy two conditions: and ; that is, both external forces and torques sum to zero. You have the freedom to choose any point as the origin about which to take torque torques. s. Each of these equations is a vector equation, so each represents three independent equations for a total of six. Thus to keep a table static requires not only that it neither slides across the floor nor lifts off from it, but also that it doesn't tilt about either the x the x or y or y axis, axis, nor can it rotate about its vertical axis.
Part A Frequently, attention in an equilibrium situation is confined to a plane. An example would be a ladder leaning against a wall, which is in danger of slipping only in the plane perpendicular to the ground and wall. By orienting a Cartesian coordinate system so that the x the x and y and y axes axes are in this plane, choose which of the following sets of quantities must be zero to maintain static equilibrium in this plane.
Hint 1. Simplifying the equations http://sessi on.master ing physi cs.com/myct/assi g nmentPr i ntView?assi g nmentID= 2854127
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The motion (or possible motion) is confined to a plane, the xy the xy plane plane in this case, when there are no forces acting out of that plane (e.g., all or all z all z-component -component forces occur in pairs that are applied at the same points). Recalling that torque is defined as a cross product, you can eliminate the need for two of the three equations for the components of torque since they will equal zero.
ANSWER: and
and
and
and
and
and
and
and
and
and
and
Correct
Part B As an example, consider consi der the case of a board of length
and negligible mass. Take Take the x the x axis axis to be the horizontal
axis along the board and the y the y axis axis to be the vertic ertical al axis perpendicular perpendicular to the board. A mass of weight strapped st rapped to the board a dist distance ance
is
from the left-hand end. This This is a st static atic equilibrium problem, and a good first step st ep
is to write down the equation for the sum of all the forces in the y the y direction direction since the only nonzero forces of that exist are in the y the y direction. direction. What is
? Your equation for the net force in the the y y
direction on the board should contain all the forces acting vertically on the board. Express your answe answerr in terms of the we ight
and
the tensions in the two vertical ropes at the left and right ends and . Recall Recal l that pos positive itive forces point upward.
ANSWER: =
Correct
The only relevant component of the torques is the z z component; component; however, you must choose your pivot point before writing wr iting the t he equations. equations. This point could c ould be anywher anywhere; e; in fact, the pivot pivot point does not even even have have to be at a point on the body. You should choose this point to your advantage. Generally, the best place to locate the pivot point is where some unknown force acts; this will eliminate that force from the resulting torque equation. http://sessi on.master ing physi cs.com/myct/assi g nmentPr i ntView?assi g nmentID= 2854127
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Part C What is the equation that results from choosing the pivot point to be the point from which the mass hangs (where acts)? Express your answe answerr in terms of the unknown quantitie s
and
and the known le lengths ngths
and
.
Recall that counterclockwise torque is positive. ANSWER: =
Correct This This gives gives us one equation involv involving ing two unknowns, solv sol ve for
and
and
. We can use this result and
to
.
Part D What is the equation that results from choosi choosing ng the piv pivot ot point point to be the left end end of the plank (where (where Express your answe answerr in terms of
,
,
, and the dime ns nsions ions
and
acts)? act s)?
. Not al alll of thes these e varia ble bles s ma may y
show up in the solution. ANSWER: =
Correct
Part E What is the equation that results from from choosi choosing ng the piv pivot ot point point to be the right end of the plank (wher (where e Express your answe answerr in terms of
,
,
, and the dime ns nsions ions
and
acts)? act s)?
. Not al alll of thes these e varia ble bles s ma may y
show up in the solution. ANSWER: =
Correct
Part F Solve for
, the tensi tension on in the right rope.
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Express your answe answerr in terms of
and the dime ns nsions ions
and
. Not al alll of thes these e varia ble bles s ma may y show up
in the solution.
Hint 1. Choose the correct equation Whic h single equati equation on of the ones ones you'v you've e deriv derived ed can be be used to solve solve for
in terms of known quantit quantities? ies?
ANSWER:
ANSWER: =
Correct
Part G Solve for
, the tens tension ion in the left rope.
Express your answe answerr in terms of
and the dime ns nsions ions
and
. Not al alll of thes these e varia ble bles s ma may y show up
in the solution.
Hint 1. Choose the correct equation Whic h single equation of the ones ones you've you've deriv derived ed can be used to solve solve for
in terms of known quantit quantities? ies?
ANSWER:
ANSWER: =
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Correct
Part H Solve Solv e for the tension in the left rop rope, e,
, in the special case that
. Be sure the result checks with your
intuition. Express your answe answerr in terms of
and the dime ns nsions ions
and
. Not al alll of thes these e varia ble bles s ma may y show up
in the solution. ANSWER: =
Correct Only one set of forces, exactly balanced, produces static equilibrium. From this perspective it might seem puzzling that so much of the world is static. One must realize, however, that many forces—like those of the tensions in the ropes here or those between the floor and an object resting on it—increase very quickly as the object moves. If there is a slight imbalance of the forces, the object accelerates so that its position changes until the object has adjusted itself to restore the force balance. It then oscillates about this point until friction or some other dissipative mechanism causes it to become stationary at the exact equilibrium point.
Precarious Lunch A uniform st steel eel beam of length and mass is att attached ached via a hinge to the side of a building. Th The e beam is supported by a steel cable att ached to the end of the beam at an angle , as shown. Through Through the hinge, hinge, the wall exerts an unknown force, , on the beam. A workman of mass sits sit s eating lunch a dist distance ance from the building.
Part A http://sessi on.master ing physi cs.com/myct/assi g nmentPr i ntView?assi g nmentID= 2854127
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Find
, the tension tension in the cable. cable. Remember Remember to account for all the the forces in the problem.
Express Expres s your ans answe we r in terms of
,
,
,
,
, and
, the magnitude of the accelera tion due to
gravity.
Hint 1. Pick the best origin This is a statics problem so the sum of torques about any axis a will be zero. In order to solve solve for wantt t o pick the axis such that wan t hat
, you
will give give a torque, torque, but as few as possible other unk unknow nown n forces will enter
the equations. So where should you place the origin for the purpose of calculating torques? ANSWER: At the center of the bar At the hinge At the connection of the cable and the bar Where the man is eating lunch
Calculate ate the sum torques Hint 2. Calcul Now find the sum of the torques about center of the hinge. Remember that a positive torque will tend to rotate objects counterclockwise around around the t he origin. origin. Answe Ans we r in terms of
,
,
,
,
,
, and
.
ANSWER:
= 0 =
ANSWER:
=
Correct
Part B Find
, the
-component of the force exerted by the wall on the beam (
), using the axis shown. Remember to
pay attention to the direction that the wall exerts the force. Express your answe answerr in terms of
and other other given quantities.
Hint 1. Find the sign of the force http://sessi on.master ing physi cs.com/myct/assi g nmentPr i ntView?assi g nmentID= 2854127
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The beam is not accelerating accelerating in the given giv en coordinate sy syst stem, em, is
-direction, -dir ection, so the sum of the forces forces in the
-direction -dir ection is zero. zero. Using the
going to hav have e to be posit positiv ive e or negativ negative? e?
ANSWER: =
Correct
Part C Find
, the y-compone y-component nt of for force ce that the wall exerts on the beam beam (
), using the axis shown. Remember Remember to pay pay
attention to the direction that the wall exerts the force. Express your answe answerr in terms of
,
,
,
, and
.
ANSWER: =
Correct If you use your result from part (A) in your expression for part (C), you'll notice that the result simplifies somewhat. The simplified result should show that the further the luncher moves out on the beam, the lower the the magnitude of the upward force the wall exerts on the beam. Does this agree with your intuition?
Exercise 11.5 A ladder ladder of of leng length th 20.0 is carried carried by by a fire fire truck. truck. The ladd ladder er has has a weigh weightt of 3000 3000 center. The ladder is pivoted at one end ( A A)) about a pin ; you can ignore the friction torque at the pin. The ladder is raised into position by a force applied by a hydraulic piston at C .
and its cen c enter ter of of grav gravity ity is at its
Point C is is a dist ista ance 8. 8.0 0 from A A,, and the force exerted by the piston makes an an angle of = 40 40 with the ladder. ladder.
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Part A What magnitude must
have hav e to just lift the ladder off the support bracket at B?
Express your answer using two significant figures. ANSWER: = 5800
Correct
Tidal Forces Tidal forces are gravitational forces exerted on different parts of a body by a second body. Their effects are particularly visible on the earth's surface in the form of tides. To understand the origin of tidal forces, consider the earth-moon system sys tem to consist of two spherical bodie bodies, s, each with a spher spherical ical mass distribution. Let be the radius radius of the earth, be the mass of the moon, and be the grav gravitat itational ional const ant.
Part A Let
denote the dist distance ance between the center of the earth and the center of the moon. What is the magnitude of of the
acceleration acc eleration
of the earth due to the grav gravit itational ational pull of the moon?
Express your answe answerr in terms of
,
, and .
Hint 1. How to approach the problem Apply the law of gravitation to find the magnitude of the force exerted on the earth by the moon and use Newton's 2nd law to determine the magnitude of the acceleration of the earth. Recall that the gravitational interaction of two spherically symmetric bodies is calculated as if the mass of each body were concentrated in its center.
Hint 2. Find the gravitational force exerted on the ear th by the moon If
is the mass of the earth, what is the grav gravitat itational ional force
Express your answer in terms of
,
,
exerted by the moon on the earth?
, and .
ANSWER: =
ANSWER: =
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Correct Just as the earth accelerates toward the moon, the moon accelerates towards the earth. These are the centripetal accelerations that cause the earth and moon to follow circular orbits around their mutual center of rotation.
Part B Since Sinc e the grav gravitat itational ional force between two bodies decreases with dist distance, ance, the accel acceleration eration
experienced by a
unit mass located at the point on the earth's surface closest to the moon is slightly different from the acceleration experienced by a unit mass located at the point on the earth's surface farthest from the moon. Give a general expression for the qua quantity ntity Express your answe answerr in terms of
. ,
, , and
.
Hint 1. Find the acceleration at the point on earth nearest the moon What is
, the magnitude of the acc acceleration eleration due to the grav gravitat itational ional pull of the moon of a unit mass of
water located at the point on the earth's surface surface nearest nearest the moon? Express your answer in terms of
, ,
, and
.
Hint 1. How to approach the problem Repeat the same calculations as in Part A, taking into account, however, that a body located at the point on the earth's surface nearest the moon is closer to the moon than the center of the earth.
Hint 2. Find the distance between the center of the moon and the point on the earth's surface closest to the moon If you draw a line joining the center of the earth and the center of the moon, you will see that the point on the earth's surface nearest the moon is shifted toward the moon with respect to the center of the earth a distance equal to the earth's radius. How far is that point, then, from the center of the moon? Express your answer in terms of
and
.
ANSWER:
ANSWER:
=
moon Hint 2. Find the acceleration at the point on earth far thest from the moon What is
, the magnitude of the acc acceleration eleration of the same amount amount of of water at the point on on the earth's
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surface farthest from the moon? Express your answer in terms of
,
, ,
.
Hint 1. Find the distance between the center of the moon and the point on the earth's surface farthest fr om the moon moon If you draw a line joining the center of the earth and the center of the moon, you will see that the point on the earth's surface farthest from the moon is shifted away from the moon with respect to the center of the earth a distance equal to the earth's radius. How far is that point, then, from the center of the moon? Express your answer in terms of
and
.
ANSWER:
ANSWER:
=
ANSWER:
=
Correct If you simplified your answer, you found that .
Note that, that , since sinc e
, the difference difference is very well approximated by the expression express ion
.
On the side of the earth nearest the moon (near side), water has a 7% greater acceleration than on the farthest side (far side) and bulges out, causing a high tide. Water on the far side is less strongly attracted toward the moon and thus another tidal bulge occurs. In total then, the earth experiences two high tides. Note that these tidal bulges of water do not appear from nowhere; instead they are formed by water flowing away from other areas of the planet, where low tides are observed.
Part C The earth is subject not only to the gravitational force of the moon but also to the gravitational pull of the sun. However, the earth is much farther away from the sun than it is from the moon. In fact, the center of the earth is at an av average erage dist distance ance of from the center of the sun. Giv Given en that the mass of the sun is http://sessi on.master ing physi cs.com/myct/assi g nmentPr i ntView?assi g nmentID= 2854127
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, which of the following statements is correct?
Hint 1. How to approach the problem Since the force exerted on a body is proportional to its acceleration, to compare the gravitational effect of the sun with that of the moon simply compare the acceleration of the earth due to the moon's gravitational pull (which you calculated in Part A) with the same acceleration due to the sun's pull.
Hint 2. Find the acceleration of the earth due to the sun's gr avitational pull What is the magnitude of the acc acceleration eleration
of the earth earth due to the grav gravitat itational ional force of the sun?
Express your answer numerically in meters per second squared. ANSWER: = 5.90×10−3
ANSWER: The force exerted on the earth by the sun is weaker than the corresponding force exerted by the moon. The force exerted on the earth by the sun is stronger than the corresponding force exerted by the moon. The force exerted on the earth by the sun is of the same order of magnitude of the corresponding force exerted by the moon.
Correct Although the sun is much farther away from the earth than the moon, it is much more massive. As a result, the gravitational force exerted on the earth by the sun is about 180 times stronger than the corresponding pull from the moon!
Part D The occurrence of tidal forces on the earth's surface is not limited to the gravitational effects of the moon. Tidal forces are produced every time different parts of a body are subject to different gravitational forces exerted by a second body. Therefore, tidal forces due to the gravitational effects of the sun are also present on the earth's surface. What can you conclude about the relative effects of these two tidal forces on the earth's surface?
Hint 1. How to approach the problem The strength of the tides is related to the difference between the acceleration of water on opposite sides of the earth due to the sun. Compute the difference and compare this to the result you obtained for the moon.
Hint 2. Find the difference in acceleration Find the difference between the acceleration on opposite sides of the earth due to the sun. Use the approximate formula for from the follow-up comm comment ent to Part B (but of course plug in the sun's mass rather than the moon's). http://sessi on.master ing physi cs.com/myct/assi g nmentPr i ntView?assi g nmentID= 2854127
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Express your answer numerically in meters per second squared. ANSWER: = 1.00×10−6
ANSWER: The moon exerts a stronger tidal force on the earth than the sun does. The sun exerts a stronger tidal force on the earth than the moon does. The moon and the sun cause tidal forces of equal magnitude.
Correct Even though the gravitational force exerted on the earth by the sun is about 180 times stronger than the corresponding pull from the moon, the differential pull is smaller. In fact, the difference in the sun's pull on opposite sides of the earth is about half the difference in the moon's pull. The effects of the tidal forces caused by the sun, however, become particularly evident when the sun, the moon, and the earth are aligned. On this occasion, the gravitational effects of the sun are added to the gravitational effects of the moon and the highest tides are observed (called spring tides, tides , although they have nothing to do with the spring season). When the sun is at an angle of with respect to the line joining the moon and the earth, inst ead, the gravitat gravitat ional effects of the sun partially cancel the effects of the moon and the least difference between high and low tide is observed (called neap tides). tides ).
Exercise 13.5 Two Tw o uniform spheres, each of mass 0.260
, are fixed at points
and
(the figure ).
Part A Find the magnitude magnitude of the initial acceleration of a uniform sphere with mass 0.010 0.010 http://sessi on.master ing physi cs.com/myct/assi g nmentPr i ntView?assi g nmentID= 2854127
if released from rest at point 12/31
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and act acted ed on only by forces of grav gravitat itational ional att attracti raction on of the spheres at
and
.
Express your answer using two significant figures. ANSWER: = 2.1×10−9
Correct
Part B Find the directi direction on of the init initial ial acc acceleration eleration of a uniform sphere with mass 0.010
.
ANSWER: upward to the right downward to the left
Correct
Gravitational Acceleration inside a Planet Consider a spherical spherical planet of uniform densit density y . The The dist distance ance from from the planet's planet's center to its surface (i.e., the t he planet's radius) rad ius) is . An object is located a distance from the center of the plane from planet, t, whe where re . (T (The he object is located inside of the planet.)
Part A Find an express expression ion for the magnitude of the accel acceleration eration due to grav gravity ity,, Express the accel accelera era tion due to gravity in terms of ,
,
, and
, insi inside de the planet. , the universal gravita gravitational tional cons constant. tant.
Hint 1. Force due to planet's mass outside radius From Newton's Principia Principia,, Proposition LXX, Theorem XXX: If to every point of a spherical surface there tend equal centripetal forces decreasing as the square of the distances from those points, I say, that a corpuscle placed within that surface will not be attracted by those forces any way. In other words, you don't have to worry about the portion of the planet's mass that is located outside of the radius . The The net net gravitat gravitational ional force force from from this "outer shell" of mass will equal zero. You only have have to worr worry y aboutt that portion abou portion of the planet's planet's mass that is located within a radius radius
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.
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Hint 2. Find the force on an object at distance Suppose the object has a mass it is located a distance
. Find the magnitude of the grav gravit itational ational force force act acting ing on this object when
from fr om the center of the plane planet. t.
Express the force in terms of
,
,
,
, and
, the universal gravita tional cons constant. tant.
Hint 1. Find the mass within a radius Find the net mass of the planet located within the radius
. Remember, the volume of a sphere sphere is
. Express your answer in terms of ,
, and
.
ANSWER: =
Magnitude tude of gravitational force Hint 2. Magni The Th e general equation for the magnitude of the grav gravitat itational ional force is
.
ANSWER: =
Hint 3. Finding
from
According Ac cording to Newton's 2nd law, the net force act acting ing on an object is giv given en by and
. In this problem,
since sinc e the only force act acting ing on the object is the grav gravitat itational ional force. Th Therefo erefore, re, , where
Note that in this usage, both
is the force you found in the prev previous ious hint. and
are magnitudes and hence are posit positiv ive. e. By conv convention, ention,
(or
in this case) is the magnitude of the gravitational field. This gravitational field is a vector, with direction downward.
ANSWER: =
Correct
Part B http://sessi on.master ing physi cs.com/myct/assi g nmentPr i ntView?assi g nmentID= 2854127
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Rewrite Rewr ite your result for
in terms of
, the grav gravitat itational ional accel acceleration eration at the surface of the planet, tim times es a
function of R. Express your answe answerr in terms of
,
, and
.
Acceleration eration at the surface Hint 1. Accel Note that the acc acceleration eleration at the surface should be equal to the value of the function evaluated ev aluated at the radius of the planet:
from Part A
.
ANSWER:
=
Correct Notice that
increases linearly linearly with
, rather than being propor proportional tional to
. This assures that it is zero at
the center of the planet, as required by symmetry.
Part C Find a numerical value for
, the av average erage densit density y of the earth in kil kilograms ograms per cubic meter. Use
the radius of the earth,
, and a value of
at the surface of
for .
Express your answer to three significant figures.
Hint 1. How to approach the problem You already derived the relation needed to solve this problem in Part A: . At wha whatt distance
is
known so that you could use this relation to find ?
ANSWER: = 5500
Correct
Exercise Exercise 13.11
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Part A At what what distance distance abov above e the surf surface ace of of the the earth earth is the accele accelera ration tion due due to the ear earth' th's s grav gravity ity 0.620 0.620
if the the
acceleration due to gravity at the surface has magnitude ANSWER: 1.90×107
Correct
Energy of a Spacecraft Very far from earth (at ), a spacec spacecraft raft has run out of fuel and its kinetic energy is zero. If only the grav gravitat itational ional force of the earth were to act on it (i.e., neglect the forces from the sun and other solar system objects), the spacecraft would wou ld eventually eventually crash into t he earth. earth. Th The e mass of the earth earth is and its radius is . Neglect air resist ance throughout this problem, since the spacecraft is primarily moving through the near vacuum of space.
Part A Find the speed
of the spacec spacecraft raft when it crashes into the earth.
Express the spee d in terms of
,
, and the universal gravita gravitational tional cons constant tant
.
Hint 1. How to approach the problem Use a conservation-law approach. Specifically, consider the mechanical energy of the spacecraft when it is (a) very far from the earth and (b) at the surface of the earth.
Hint 2. Total energy What is the total mechanical ene energy rgy of the spacecraf spacecraftt whe when n it is far fr from om ear earth, th, at a distance
?
ANSWER: = 0
Hint 3. Potential energy If the spacecraf spacecraftt has mass
, wha whatt is its potential potential ene energy rgy
surface of the earth, the spacec spacecraft raft is a dist distance ance Express your answer in terms of
,
,
at the surfa surface ce of the ear earth? th? Note that, at at the
from the cent center er of the earth. , and the universal gravita tional cons constant tant
.
Hint 1. Formula for the potential energy The Th e grav gravit itational ational potenti potential al energy
of a sy syst stem em of 2 mass masses es
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and
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is .
ANSWER: =
ANSWER:
=
Correct
Part B Now find the spacec spacecraft's raft's speed when it its s dist distance ance from the center of the earth is
, where the coefficient
. Express the spee d in terms of
and
.
Hint 1. General approach This problem is very similar to the problem that you've just done. Note that the potential energy of the spacecraftt at a distance spacecraf is diff differe erent nt fr from om its potential ener energy gy at the ear earth's th's surfa surface. ce.
Hint 2. First step in finding the speed Find the spacecraf spacecraft's t's speed at
in terms of
,
,
, and
.
ANSWER:
=
ANSWER: =
Correct http://sessi on.master ing physi cs.com/myct/assi g nmentPr i ntView?assi g nmentID= 2854127
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Properties of Circular Orbits Learning Goal: To teach you how to find the parameters characterizing an object in a circular orbit around a much heavier body like the earth. The motivation for Isaac Newton to discover his laws of motion was to explain the properties of planetary orbits that were observed by Tycho Brahe and analyzed by Johannes Kepler. A good starting point for understanding this (as well as the speed of the space shuttle and the height of geostationary satellites) is the simplest orbit--a circular one. This problem concerns the pro proper perties ties of circular orb orbits its for a satellit satellite e orb orbiting iting a plane planett of mass . For all parts parts of this problem, where where appropr appropriate, iate, use
for the univ universal ersal grav gravitat itational ional const constant. ant.
Part A Find the orbital speed
for a sat satellit ellite e in a circ circular ular orbit of radius
Express the orbital spee d in terms of
,
, and
.
.
Hint 1. Find the force Find the radial force
on the sat satellit ellite e of mass
Express Expres s your ans answe we r in terms of
,
,
. (Note that
, and
will cancel out of your final answer for .)
. Indicate outward radia radiall direction wi with th a
positive sign and inward radial direction with a negative sign. ANSWER: =
Hint 2. A basic kinematic relation Find an expre expression ssion for for the rad radial ial acceleration Express your answer in terms of
and
for the satellit satellite e in its circular orb orbit. it.
. Indi Indicate cate outwa outward rd radi radial al direction wi th a pos positive itive sign
and inward radial direction with a negative sign. ANSWER: =
ewton's ton's 2nd law Hint 3. New Apply
to the radia radiall coord coordinate. inate.
ANSWER:
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Chapter 11 and 13 Homewor k
=
Correct
Part B Find the kinetic ener energy gy
of a satellit satellite e with mass
Express your answe answerr in terms of
,
,
, and
in a circular orb orbit it with rad radius ius
.
.
ANSWER: =
Correct
Part C Express the kinetic ene energy rgy
in terms of the potential ener energy gy
.
Hint 1. Potential energy What is the potential ener energy gy
of the satellit satellite e in this orbit? orbit?
Express your answer in terms of
,
,
, and
.
ANSWER: =
ANSWER: =
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Chapter 11 and 13 Homewor k
Correct This is an example of a powerful theorem, known as the Virial Theorem. Theorem. For any system whose motion is periodic or remains forever bounded, and whose potential behaves as , Rudolf Clausius proved that , where where the brackets brackets denote denote the t he temporal temporal (time) avera average. ge.
Part D Find the orbital period
.
Express your answe answerr in terms of
,
,
, and
.
Hint 1. How to get star ted Use the fact that the period is the tim time e to make one orbit. Th Then en
.
ANSWER:
=
Correct
Part E Find an expression for the square of the orbital period. Express your answe answerr in terms of
,
,
, and
.
ANSWER: =
Correct This shows that the square of the period is proportional to the cube of the semi-major axis . This is Kepler's Third Third Law, in the case of a circular circ ular orbit where where the semi-major axis is equal equal to the radius, radius, .
Part F http://sessi on.master ing physi cs.com/myct/assi g nmentPr i ntView?assi g nmentID= 2854127
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Find
, the magnitude magnitude of the angular angular momentum of the satellite satelli te with respect to the center of the planet.
Express your answe answerr in terms of
,
,
, and
.
Hint 1. Definition of angular momentum Recall that
, where
is the momentum of the object and
is the vect ector or from the piv pivot ot point.
Here the pivot pivot point is the center of the planet, and sinc since e the object is moving moving in a circ circular ular orbit, perpendicul perpend icular ar to
is
.
ANSWER: =
Correct
Part G The quan quantit tities ies
,
,
, and
all rep represent resent physical qua quantities ntities characterizing the orb orbit it that dep depend end on rad radius ius
.
Indicate the exponent (power) of the radial dependence of the absolute value of each. Express your answe answerr as a comma -s -sepa epa rated list of ex ponents corres corresponding ponding to , order. For ex exampl ample, e, -1 -1,-1/2 ,-1/2,-0.5 ,-0.5,-3/2 ,-3/2 would mea n
,
,
, and
, in that
, and so for forth. th.
Example ample of a power law Hint 1. Ex The Th e potenti potential al energy behav behaves es as power pow er for this is
(i.e.,
, so
depends inv inversely ersely on
. Th Therefo erefore, re, the appropr appropriate iate
).
ANSWER: -0.500,-1,-1,0.500
Correct
Geosynchronous Satellite A satellite that goes around the earth once every 24 hours is called a geosynchronous geosynchronous sat satellite. ellite. If a geosynchronou geosynchronous s satellite is in an equatorial orbit, its position appears stationary with respect to a ground station, and it is known as a geostationary satellite. satellite.
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Part A Find the rad radius ius
of the orb orbit it of of a geosynchro geosynchronous nous satellit satellite e that circles the ear earth. th. (Note (Note that
is measured measured fr from om
the center of the earth, not the surface.) You may use the following constants: The Th e univ universal ersal grav gravitat itational ional const constant ant
is
.
The Th e mass of the earth is
.
The Th e radius of the earth is
.
Give the orbital radius in meters to three significant digits.
Hint 1. Find the force on the satellite If we just consider the earth-satellite system, then there is only one force acting on the satellite. Suppose the mass of the satellit satellite e is , the mass of the earth is , and the rad radius ius of the satellit satellite's e's orb orbit it is . What is the magnitude of the force that acts on the satellite? Answe Ans we r in terms of
,
,
, and the universal gravita gravitational tional cons constant tant
. (Us (Use e varia ble bles s, not
numerical values.) ANSWER: =
Hint 2. Angular frequency of satellite The gravitational force on the satellite provides a centripetal acceleration that pulls the satellite inward, holding it in a circ circular ular orbit. A generic formula for the magnitude of the cent centripetal ripetal acc acceleration eleration is where whe re
,
is the angular angular frequ frequency ency of the satellite's sat ellite's orbit.
What is the numerical value value for
in radians per sec second ond for a geosync geosynchronou hronous s satellite? satelli te?
Hint 1. How to calculate http://sessi on.master ing physi cs.com/myct/assi g nmentPr i ntView?assi g nmentID= 2854127
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Calculate Calculat e
using the definition of of a geosync geosynchronous hronous orbit; the angular velocit y should be suc such h that
the satellite satelli te makes one orbit per per day. The The equation relati relating ng the angular angular velocit y
and the time period period
is .
Hint 2. What is
?
Here the time period
is one day, but you are ask asked ed for the angular angular velocit y in radians per per sec second. ond.
ANSWER: = 7.27×10−5 radians/s
expression pression for Hint 3. Find an ex Type an express expression ion for the radius radius Express your answer in terms of
of the circular orbit of of a sat satellit ellite e orbiting the earth. ,
(the ma mas ss of the ea rth), and
, the angul ar veloci velocity ty of the
satellite.
Hint 1. Putting it all together Using Newton's 2nd law,
, giv gives es us .
Find
from this equation.
ANSWER: =
ANSWER: = 4.23×107 m
Correct
A Satellite in a Circular Orbit Consider a satellit Consider satellite e of mass that orb orbits its a plane planett of mass in a circle a distance fr from om the center of the plane planet. t. The satellite's mass is negligible compared with that of the planet. Indicate whether each of the statements in this http://sessi on.master ing physi cs.com/myct/assi g nmentPr i ntView?assi g nmentID= 2854127
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problem is true or false.
Part A The information given is sufficient to uniquely specify the speed, potential energy, and angular momentum of the satellite.
Hint 1. What constitutes sufficient initial conditions? An initial position and velocity plus a knowledge of the forces at all points will suffice to specify the subsequent subseque nt motion mot ion in Newtonian Newtonian mechanics. Are you able to deter determine mine the s atellite's velocity velocity fro from m the t he information given?
ANSWER: true false
Correct
Part B The total mechanical energy of the satellite is conserved.
Hint 1. When is mechanical energy conserved? A system's total mechanical energy is conserved if no nonconservative forces act on the system. Is gravity a conservative force?
ANSWER: true false
Correct
Part C The linear momentum vector of the satellite is conserved.
Hint 1. When is linear momentum conserved? The linear momentum vector of a system is conserved if the net external force acting on it is zero.
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ANSWER: true false
Correct
Part D The angular momentum of the satellite about the center of the planet is conserved.
Hint 1. When is angular momentum conserved? Angular momentum about a particular axis is conserved if there is no net torque about that axis.
ANSWER: true false
Correct
Part E The equations that express the conservation laws of total mechanical energy and linear momentum are sufficient to solve for the speed necess necessary ary to maintai maintain n a circ circular ular orbit at
without using
.
Hint 1. How are conservation laws used? Conservation laws are generally applied to a system that has some "initial" and "final" conditions that are different. Motion in a circular orbit, however, possesses no obvious initial and final points that are different.
ANSWER: true false
Correct
Kepler's 3rd Law http://sessi on.master ing physi cs.com/myct/assi g nmentPr i ntView?assi g nmentID= 2854127
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A planet planet moves moves in an ellipt elliptical ical orbit around the sun. Th The e mass of the sun is of the planet from the sun are and , respect respectiv ively. ely.
. The The minim minimum um and maxi maximum mum distances
Part A Using Kepler's 3rd law and and Newton's Newton's law of univ universal ersal gravitat gravitation, ion, find the period of rev revolution olution
of the planet as it
moves around the sun. Assume that the mass of the planet is much smaller than the mass of the sun. Use
for the grav gravitat itational ional const ant.
Express the peri period od in terms of
,
,
, and
.
Hint 1. Kepler's 3rd law Kepler's 3rd law states that the square of the period of revolution of a planet around the sun is proportional to the cube of the semi-major axis of its orbit. Try finding the period of a circular orbit and then using Kepler's 3rd law (which applies equally to circular and elliptical orbits) to extend your result to an elliptical orbit.
Hint 2. Find the semi-major axis Find the semi-major axis
.
Express the sem emi-ma i-major jor ax is in terms of
and
.
Hint 1. Definition of semi-major axis The semi-major axis of an ellipse is half of its major axis. The sun is at the focus of the elliptical orbit and the focus lies on the major axis.
ANSWER: =
Hint 3. Find the period of a circular orbit Find the period
of a planet in a circ circular ular orbit of semi semi-major -major axis
Express the peri period od in terms of ,
, and
.
.
Hint 1. Formula for the period The Th e period is
, where
is the radius of the orbit and
is the speed of the object object.. Note that this
is the distance traveled in one orbit divided by the speed.
Hint 2. Find the velocity Find the velocit elocity y acceleration accel eration
of an object in an orbit of radius
by set setting ting the magnitude of the centri centripetal petal
equal to the magnitude of the acc acceleration eleration due to grav gravit ity. y.
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Express your answer in terms of ,
, and
.
ANSWER: =
Hint 3. Radius of the orbit For a circle, the semi-major axis is just the radius.
ANSWER:
=
ANSWER:
=
Correct
Exercise Exercise 13.26 In March 2006, two smal smalll sat satellit ellites es were disc discov overed ered orbiting Pluto, one at a dist distance ance of 48,000 and the other at 64,000 . Plut Pluto o already was known to hav have e a large sat satellit ellite e Charon Charon,, orbiting at 19,600 with an orbital period of 6.39 days.
Part A Assuming that the satellites do not affect each other, find the orbital periods of the two small satellites without using using the mass of Pluto. Enter En ter your answe answe rs numericall y sepa separated rated by a comma. ANSWER: = 24.5,37.7 days
Correct
Exercise Exercise 11.10 http://sessi on.master ing physi cs.com/myct/assi g nmentPr i ntView?assi g nmentID= 2854127
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A uniform ladder 5.0 long rests against against a fricti frictionless onless,, vertic ertical al wall with it its s lower end 3.0 from the wall. The ladder weighs weig hs 160 . The coefficient coefficient of static stati c friction between the foot foot of the ladder and and the ground ground is 0.40. A man weighing weighing 740 climbs slowly up the ladde ladder. r.
Part A What is the maximum frictional force that the ground can exert on the ladder at its lower end? Express your answer using two significant figures. ANSWER:
=
Part B What is the actual frictional force when the man has climbed 1.0 m along the ladder? Express your answer using two significant figures. ANSWER:
=
Part C How far along the ladder can the man climb before the ladder starts to slip? Express your answer using two significant figures. ANSWER:
=
Exercise 13.4 Two Tw o uniform spheres, each with mass
and radius
, touch one another.
Part A What is the magnitude of their gravitational force of attraction? Express your answe answerr in terms of the varia ble bles s
,
, and appropri appropriate ate cons constants tants..
ANSWER: http://sessi on.master ing physi cs.com/myct/assi g nmentPr i ntView?assi g nmentID= 2854127
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Orbiting Satellite A satellit satellite e of mass is in a circular orb orbit it of rad radius ius around a spherical planet of radius made of a material with density . ( is measure m easured d from from the t he center of the planet, not its surface. surface.)) Use for the univ universal ersal grav gravitat itational ional constant.
Part A Find the kinetic ener energy gy of this satellit satellite, e,
.
Express the sate atelli lli te's kinetic ene rgy in terms of
,
,
,
,
, and
.
Hint 1. Kinematics of circular motion The circular trajectory of the t he satellite satellit e implies that t her here e is a certain inward radial radial acceleration , which must be be due to the grav gravitat itational ional force .
Hint 2. Mass of the planet Find the mass of the planet in terms of its size and density. Express
in terms of
and
.
ANSWER: =
Hint 3. Gravitational force
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Chapter 11 and 13 Homewor k
Find
, the magnitude of the grav gravit itational ational force on the sat satellit ellite. e.
Express your answer in terms of
,
,
,
,
, and
.
,
,
,
,
, and
.
ANSWER:
=
Hint 4. Speed of the satellite Find the speed
of the satellit satellite. e.
Express your answer in terms of ANSWER:
=
ANSWER:
=
Correct
Part B Find
, the grav gravitat itational ional potent potential ial energy energy of the satellite. satelli te. Take the grav gravit itational ational potential energy to be zero for for an
object infinitely far away from the planet. Express Expres s the satell atellite's ite's gravitational potent potential ial energy in terms of
,
,
,
,
, and
.
Hint 1. What physical principle to use The Th e grav gravitat itational ional potenti potential al energy ass associat ociated ed with a
force is best remembered. If you hav have e to work it out,
remember that the potenti potential al energy of an object found at height
, relative to height
the negativ negative e of the work done by the grav gravitat itational ional force when the object is brought from
, is equal to to
:
, where whe re . In this case,
is a fun function ction of the rad radius ius of the satellit satellite e trajectory,
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, and and the zero point of the ener energy gy is 30/31
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Chapter 11 and 13 Homewor k
conventiona conv entionally lly taken at
, so you wan wantt to use .
Hint 2. How to handle the math You will need to solve an integral of the form
, where
is a const ant. This ev evaluates aluates to
, and you will need to evaluate this between the limits of integration.
ANSWER:
=
Correct
Part C What is the ratio of the kinetic energy of this satellite to its potential energy? Express
in terms of para parame meters ters given in the introduction.
ANSWER: = -0.500
Correct The The result of this problem may be expressed express ed as (i.e.
where
is the exponent of the force law
). This This is a specical speci cal case of a general general and powerful powerful theroem of advanced advanced
classical mechanics known as the Virial Theorem. Theorem. The theorem applies to the average of the kinetic and potential energies of of any one or multiple objects moving over any closed (or almost closed) path that returns very close to itself provided that all objects interact via potentials with the same power law dependence on their separation. separation. Thus it applies applies to stars in a galaxy, or masses tied together with springs (where (where the force law is
since
).
Score Summary: Your score on this assignment is 108%. You received 14.19 out of a possible total of 14 points, plus 0.99 points of extra credit.
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