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Assignment MasteringPhysics: Print View
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PHCC 141: Physics for Scientists and Engineers I - Fall 2007
6a. Work, Energy, and Power
Due at 11:59pm on Thursday, September 27, 2007
Hide Grading Details
Number of answer attempts per question is: 5 You gain credit for: correctly answering a question in a Part, or correctly answering a question in a Hint. You lose credit for: exhausting all attempts or requesting the answer to a question in a Part or Hint, or incorrectly answering a question in a Part. Late submissions: reduce your score by 100% over each day late. Hints are helpful clues or simpler questions that guide you to the answer. Hints are not available for all questions. There is no penalty for leaving questions in Hints unanswered. Grading of Incorrect Answers
For Multiple-Choice or True/False questions, you lose 100%/(# of options - 1) credit per incorrect answer. For any other question, you lose 3% credit per incorrect answer. Work Energy Theorem and Exercises
The Work-Energy Theorem
Learning Goal: To understand the meaning and possible applications of the work-energy theorem. In this problem, you will use your prior knowledge to derive one of the most important relationships in mechanics: the work-energy theorem. We will start with a special case: a particle of mass moving in the x direction at constant acceleration . During a certain interval of time, the particle accelerates from to , undergoing displacement given by . Part A Find the acceleration of the particle. Hint A.1 Some helpful relationships from kinematics
By definition, . Furthermore, the average speed is , and the displacement is . Combine these relationships to eliminate . Express the acceleration in terms of ANSWER: = , , and .
Part B Find the net force Hint B.1 acting on the particle.
Using Newton's laws Hint not displayed and .
Express your answer in terms of ANSWER: Part C =
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Find the net work
done on the particle by the external forces during the particle's motion. and .
Express your answer in terms of ANSWER: Part D =
Substitute for from Part B in the expression for work from Part C. Then substitute for from the relation in Part A. This will yield an expression for the net work done on the particle by the external forces during the particle's motion in terms of mass and the initial and final velocities. Give an expression for the work in terms of those quantities. Express your answer in terms of , ANSWER: = , and .
The expression that you obtained can be rearranged as
The quantity
has the same units as work. It is called the kinetic energy of the moving particle and is denoted by .
Therefore, we can write and .
Note that like momentum, kinetic energy depends on both the mass and the velocity of the moving object. However, the mathematical expressions for momentum and kinetic energy are different. Also, unlike momentum, kinetic energy is a scalar. That is, it does not depend on the sign (therefore direction) of the velocities. Part E Find the net work kinetic energies. done on the particle by the external forces during the motion of the particle in terms of the initial and final and .
Express your answer in terms of ANSWER: =
This result is called the work-energy theorem. It states that the net work done on a particle equals the change in kinetic energy of that particle. Also notice that if is zero, then the work-energy theorem reduces to . In other words, kinetic energy can be understood as the amount of work that is done to accelerate the particle from rest to its final velocity. The work-energy theorem can be most easily used if the object is moving in one dimension and is being acted upon by a constant net force directed along the direction of motion. However, the theorem is valid for more general cases as well. Let us now consider a situation in which the particle is still moving along the x axis, but the net force, which is still directed along the x axis, is no longer constant. Let's see how our earlier definition of work,
needs to be modified by being replaced by an integral. If the path of the particle is divided into very small displacements , we can assume that over each of these small displacement intervals, the net force remains essentially constant and the work
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MasteringPhysics: Assignment Print View
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done to move the particle from to
is ,
where is the x component of the net force (which remains virtually constant for the small displacement from to net work done on the particle is then given by . Now, using
). The
and , it can be shown that .
Part F Evaluate the integral Hint F.1 .
An integration formula is .
The formula for
Express your answer in terms of , ANSWER: =
, and
.
The expression that you havejust obtained is equivalent to . Not surprisingly, we are back to the same expression of the work-energy theorem! Let us see how the theorem can be applied to problem solving. Part G A particle moving in the x direction is being acted upon by a net force , for some constant . The particle moves from to . What is , the change in kinetic energy of the particle during that time? Hint G.1 Integrate Hint G.2 Finding the work to calculate the work done on the particle. An integration formula is . Express your answer in terms of ANSWER: = and .
The formula for
It can also be shown that the work-energy theorem is valid for two- and three-dimensional motion and for a varying net
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MasteringPhysics: Assignment Print View
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force that is not necessarily directed along the instantaneous direction of motion of the particle. In that case, the work done by the net force is given by the line integral
where and
and
are the initial and the final positions of the particle,
is the vector representing a small displacement,
is the net force acting on the particle.
When Push Comes to Shove
Two forces, of magnitudes and , act in opposite directions on a block, which sits atop a frictionless surface. Initially, the center of the block is at position . At some later time, the block has moved to the right, so that its center is at position , where .
Part A Find the work Hint A.1 done on the block by the force of magnitude as the block moves from to .
Formula for the work done by a force Hint not displayed
Express your answer in terms of some or all of the variables given in the problem introduction. ANSWER: Part B Find the work Hint B.1 done by the force of magnitude as the block moves from to . =
Is the work positive or negative? Hint not displayed
Express your answer in terms of some or all of the variables given in the problem introduction. ANSWER: Part C What is the net work ANSWER: Part D Imagine that the two forces are equal in magnitude, , and that there are no other horizontal forces acting on the block. Determine the change in the kinetic energy of the block as it moves from to . Hint D.1 If the forces are equal, how can the block be moving? Hint not displayed ANSWER: =0 = done on the block by the two forces? =
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Work from a Constant Force
Learning Goal: To understand how to compute the work done by a constant force acting on a particle that moves in a straight line. In this problem, you will calculate the work done by a constant force. A force is considered constant if This is the most frequently encountered situation in elementary Newtonian mechanics. Part A Consider a particle moving in a straight line from initial point B to final point A, acted upon by a constant force . The force (think of it as a field, having a magnitude and direction at every position ) is indicated by a series of identical vectors pointing to the left, parallel to the horizontal axis. The vectors are all identical only because the force is constant along the path. The magnitude of the force is , and the displacement vector from point B to point A is (of magnitude , making and angle (radians) with the positive x axis). Find the particle as it moves from point B to point A. , the work that the force performs on is independent of .
Hint A.1 by is .
Formula for work done by a constant force along a straight path represented by the displacement vector , the net work done
For a particle subjected to a constant force
Part A.2
Find the angle between
and in the direction of
You need to find the angle between the vector , which is directed horizontally to the left, and the vector the particle's motion (at an angle (radians) relative to the positive x axis). It may help to visualize negative x axis at the origin. What is the angle between Express your answer in radians, not degrees. ANSWER: = and ?
directed along the
Express the work in terms of , , and . Remember to use radians, not degrees, for any angles that appear in your answer. ANSWER: =
This result is worth remembering! The work done by a constant force of magnitude , which acts at an angle of with respect to the direction of motion along a straight path of length , is . This equation correctly gives the sign in this problem. Since is the angle with respect to the positive x axis (in radians), ; hence . Part B Now consider the same force
acting on a particle that travels from point A to point B. The displacement vector done by in this case.
now points
in the opposite direction as it did in Part A. Find the work
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Hint B.1
A physical argument Hint not displayed
Express your answer in terms of , , and . ANSWER: =
Workhorses on Erie Canal
Two workhorses tow a barge along a straight canal. Each horse exerts a constant force of magnitude , and the tow ropes make an angle with the direction of motion of the horses and the barge. Each horse is traveling at a constant speed .
Part A How much work Hint A.1 is done by each horse in a time ?
Formula for work Hint not displayed
Part the A.2
Find x component of the force Part not displayed
Part A.3
Find the distance traveled Part not displayed
Express the work in terms of the quantities given in the problem introduction. ANSWER: Part B How much power Hint B.1 does each horse provide? =
Formula for power .
Power is the time derivative of the work,
Express your answer in terms of the quantities given in the problem introduction. ANSWER: =
One way to compute the power provided by each horse is to first compute the work done by each horse during a time interval (as in Part A), then take the time derivative. However, an easier way to compute the power provided when a force acts on an object moving with velocity is to use the formula .
Vertical Spring Gun: Speed and Kinetic Energy
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MasteringPhysics: Assignment Print View
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The figure represents a multiflash photograph of a ball being shot straight up by a spring. The spring, with the ball atop, was initially compressed to the point marked and released. The point marked is the point where the ball would remain at rest if it were placed gently on the spring, and the ball reaches its highest point at the point marked . For most situations, including this problem, the point spring, where the ball loses contact with the spring. may be taken to be at the top of the
Part A Indicate whether the following statements are true or false. Assume that air resistance is negligible. The speed of the ball was greatest at point when it was still in contact with the spring. The speed of the ball was decreasing on its way from point to point . The speed of the ball was zero at point . The speed of the ball was the same for all points in its motion between points and . Enter t for true or f for false for each statement. Separate your responses with commas (e.g., t,f,f,t). ANSWER: t,t,t,f Part B Consider the kinetic energy of the ball. At what point or points is the ball's kinetic energy greatest? Hint B.1 What equation to use Hint not displayed ANSWER: only only only and and and and and
Work on a Sliding Block
A block of weight sits on a frictionless inclined plane, which makes an angle with respect to the horizontal, as shown. A force of magnitude , applied parallel to the incline, pulls the block up the plane at constant speed.
Part A The block moves a distance up the incline. The block does not stop after moving this distance but continues to move with constant speed. What is the total work done on the block by all forces? (Include only the work done after the block has started moving, not the work needed to start the block moving from rest.) Hint A.1 What physical principle to use
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Part B What is Hint B.1 , the work done on the block by the force of gravity as the block moves a distance up the incline? Force diagram
Part B.2
Force of gravity component
What is the component of the force of gravity in the direction of the block's displacement (along the inclined plane)? Hint B.2.a Relative direction of the force and the motion Remember that the force of gravity acts down the plane, whereas the block's displacement is directed up the plane. Express your answer in terms of ANSWER: = and any other quantities given in the problem introduction. and .
Express the work done by gravity in terms of the weight ANSWER: =
Part C What is Hint C.1 , the work done on the block by the applied force as the block moves a distance up the incline?
How to find the work done by a constant force Hint not displayed and other given quantities.
Express your answer in terms of ANSWER: Part D What is Part D.1 =
, the work done on the block by the normal force as the block moves a distance up the inclined plane? First step in computing the work Part not displayed
Express your answer in terms of given quantities. ANSWER: =0
Work-Energy Theorem Reviewed
Learning Goal: Review the work-energy theorem and apply it to a simple problem. If you push a particle of mass in the direction in which it is already moving, you expect the particle's speed to increase. If you push with a constant force , then the particle will accelerate with acceleration (from Newton's 2nd law). Part A
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Enter a one- or two-word answer that correctly completes the following statement. If the force is applied for a fixed interval of time , then the _____ of the particle will increase by an amount . Hint A.1 Kinematic equations recalled Hint not displayed ANSWER: velocity Part B Enter a one- or two-word answer that correctly completes the following statement. If the force is applied over a given distance , along the path of the particle, then the _____ of the particle will increase by ANSWER: kinetic energy energy KE K The work Part C If the initial kinetic energy of the particle is , and its final kinetic energy is on the particle. ANSWER: = , express in terms of and the work done done on the particle by the force over the distance is . .
This is the work-energy theorem, often written . It is, essentially, a statement of energy conservation that does not include potential energy explicitly. All forces--even conservative forces like gravity--contribute to the work. Part D In general, the work done by a force is written as . Now, consider whether the following statements are true or false: The dot product assures that the integrand is always nonnegative. The dot product indicates that only the component of the force perpendicular to the path contributes to the integral. The dot product indicates that only the component of the force parallel to the path contributes to the integral. Enter t for true or f for false for each statement. Separate your responses with commas (e.g., t,f,t). ANSWER: f,f,t Part E Assume that the particle has initial speed . Find its final kinetic energy Part E.1 Find the initial kinetic energy in terms of the particle's initial velocity and its mass . in terms of , , , and .
Express the initial kinetic energy ANSWER: =
ANSWER:
=
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Part F What is the final speed of the particle? Express your answer in terms of ANSWER: = and .
Problem 6.1
You push your physics book a distance 1.53 of friction is 0.590 . Part A How much work does your force 2.37 ANSWER: Part B What is the work done on the book by the friction force? ANSWER: Part C What is the total work done on the book? ANSWER: 2.72 J -0.903 J 3.63 J do on the book? along a horizontal tabletop with a horizontal force of 2.37 . The opposing force
Problem 6.4
A factory worker pushes a crate of mass 28.9 a distance of 4.15 along a level floor at constant velocity by pushing horizontally on it. The coefficient of kinetic friction between the crate and floor is 0.253. Part A What magnitude of force must the worker apply? Take the free fall acceleration to be = 9.80 ANSWER: Part B How much work is done on the crate by this force? Take the free fall acceleration to be = 9.80 ANSWER: Part C How much work is done on the crate by friction? Take the free fall acceleration to be = 9.80 ANSWER: Part D -297 J . 297 J . 71.7 N .
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How much work is done by the normal force? ANSWER: Part E How much work is done by gravity? ANSWER: Part F What is the total work done on the crate? ANSWER: 0J 0J 0J
Problem 6.9
A ball of mass 0.755 Part A During one complete circle, starting anywhere, calculate the total work done on the ball by the tension in the string. Take the free fall acceleration to be = 9.80 ANSWER: Part B During one complete circle, starting anywhere, calculate the total work done on the ball by gravity. Take the free fall acceleration to be = 9.80 ANSWER: Part C Repeat part (a) for motion along the semicircle from the lowest to the highest point on the path. Take the free fall acceleration to be = 9.80 ANSWER: Part D Repeat part (b) for motion along the semicircle from the lowest to the highest point on the path. Take the free fall acceleration to be = 9.80 ANSWER: -23.8 J . 0J . 0J . 0J . is tied to the end of a string of length 1.61 and swung in a vertical circle.
Problem 6.10
Part A Compute the kinetic energy, in joules, of an automobile of mass 1520 ANSWER: Part B 1.41105 J traveling at a speed of 49.0 .
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By what factor does the kinetic energy change if the speed is doubled? ANSWER: 4
Problem 6.14
You throw a rock of weight 20.2 vertically into the air from ground level. You observe that when it is a height 15.3 the ground, it is traveling at a speed of 24.2 upward. Part A Use the work-energy theorem to find its speed just as it left the ground; Take the free fall acceleration to be = 9.80 ANSWER: Part B Use the work-energy theorem to find its maximum height. Take the free fall acceleration to be = 9.80 ANSWER: 45.2 m . 29.8 m/s . above
Problem 6.20
A sled with mass 7.80 moves in a straight line on a frictionless horizontal surface. At one point in its path, its speed is 3.90 ; after it has traveled a distance 2.40 beyond this point, its speed is 6.00 . Part A Use the work-energy theorem to find the force acting on the sled, assuming that this force is constant and that it acts in the direction of the sled's motion. ANSWER: 33.8 N
Problem 6.24
A 5.00 Part A Calculate the work done by gravity on the watermelon during its displacement from the roof to the ground. Take the free fall acceleration to be = 9.80 ANSWER: Part B What is the kinetic energy of the watermelon just before it strikes the ground? You can ignore air resistance. Take the free fall acceleration to be = 9.80 ANSWER: 1470 J 1470 J watermelon is dropped (zero initial speed) from the roof of a building of height 30.0 .
Summary
14 of 14 items complete (75.06% avg. score) 105.08 of 140 points
2 of 12
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CHAPTER 3 DIFFERENTIATION3.1 THE DERIVATIVE OF A FUNCTION 1. Step 1: f(x) oe 4 c x# and f(x b h) oe 4 c (x b h)# Step 2: oe c2x c h Step 3: f w (x) oe lim (c2x c h) oe c2x; f w (c$) oe 6, f w (0) oe 0, f w (1) oe c2h!# # # # # $ # # # #f(x b h)

Colorado State - MATH - 161

CHAPTER 4 APPLICATIONS OF DERIVATIVES4.1 EXTREME VALUES OF FUNCTIONS 1. An absolute minimum at x oe c# , an absolute maximum at x oe b. Theorem 1 guarantees the existence of such extreme values because h is continuous on [a b]. 2. An absolute minimu

Colorado State - MATH - 161

CHAPTER 5 INTEGRATION5.1 ESTIMATING WITH FINITE SUMS 1. faxb oe x# Since f is increasing on ! ", we use left endpoints to obtain lower sums and right endpoints to obtain upper sums.(a) ~x oe (b) ~x oe (c) ~x oe (d) ~x oe 2. faxb oe x$"c! # "c! %

Colorado State - MATH - 161

CHAPTER 6 APPLICATIONS OF DEFINITE INTEGRALS6.1 VOLUMES BY SLICING AND ROTATION ABOUT AN AXIS 1. (a) A oe 1(radius)# and radius oe 1 c x# A(x) oe 1 a1 c x# b (b) A oe width height, width oe height oe 21 c x# A(x) oe 4 a1 c x# b (d) A oe3 4(sid

Colorado State - MATH - 161

CHAPTER 7 TRANSCENDENTAL FUNCTIONS7.1 INVERSE FUNCTIONS AND THEIR DERIVATIVES 1. Yes one-to-one, the graph passes the horizontal test. 2. Not one-to-one, the graph fails the horizontal test. 3. Not one-to-one since (for example) the horizontal line

Colorado State - MATH - 161

CHAPTER 8 TECHNIQUES OF INTEGRATION8.1 BASIC INTEGRATION FORMULAS2.' 3 cos x dx '1 b 3 sin x3.3sin v cos v dv; "4.6.sec z dz tan z 4#z oe1 4du oe cln kukd 1 3 oe ln 3 c ln 1 oe ln 37.'dx x ^ x b 1 u oe x b " " ;

Colorado State - MATH - 161

CHAPTER 9 FURTHER APPLICATIONS OF INTEGRATION9.1 SLOPE FIELDS AND SEPARABLE DIFFERENTIAL EQUATIONS 1. (a) y oe e x y w oe ce x 2y w b 3y oe 2 ace x b b 3e x oe e x (b) y oe e x b e 3x 2 y w oe ce x c 3 e 3x 2 2y w b 3y oe 2 ^ce x c 3 e 3x 2 b 3

Colorado State - MATH - 161

CHAPTER 10 CONIC SECTIONS AND POLAR COORDINATES10.1 CONIC SECTIONS AND QUADRATIC EQUATIONS# # # #1. x oey 8 4p oe 8 p oe 2; focus is (2 0), directrix is x oe c2# #2. x oe c y 4p oe 4 p oe 1; focus is (c1 0), directrix is x oe 1 4 3. y o

Colorado State - MATH - 161

CHAPTER 11 INFINITE SEQUENCES AND SERIES11.1 SEQUENCES 1. a" oe 2. a" oe 3.1 c1 1 1 1!#oe 1, a# oe#" #!oe" 2, a$ oe$1 3!oe1 6, a% oe%1 4!oe" 51 244. a" oe 2 b (c1)" oe 1, a# oe 2 b (c1)# oe 3, a$ oe 2 b (c1)$ oe 1, a%

Colorado State - MATH - 161

CHAPTER 12 VECTORS AND THE GEOMETRY OF SPACE12.1 THREE-DIMENSIONAL COORDINATE SYSTEMS 1. The line through the point (# $ !) parallel to the z-axis 2. The line through the point (c1 0 !) parallel to the y-axis 3. The x-axis 4. The line through the po

Colorado State - MATH - 161

CHAPTER 13 VECTOR-VALUED FUNCTIONS AND MOTION IN SPACE13.1 VECTOR FUNCTIONS 1. x oe t b 1 and y oe t# c 1 y oe (x c 1)# c 1 oe x# c 2x; v oe at t oe 1 2. x oe t# b 1 and y oe 2t c 1 x oe ^ y b 1 b " x oe # v oe i b 2j and a oe 2i at t oe 3. x o

Colorado State - MATH - 161

CHAPTER 14 PARTIAL DERIVATIVES14.1 FUNCTIONS OF SEVERAL VARIABLES 1. (a) (b) (c) (d) (e) (f) 2. (a) (b) (c) (d) Domain: all points in the xy-plane Range: all real numbers level curves are straight lines y c x oe c parallel to the line y oe x no boun

Colorado State - MATH - 161

CHAPTER 15 MULTIPLE INTEGRALS15.1 DOUBLE INTEGRALS 1.'03 '02 a4 c y# b dy dx oe '03 '4y c y3 " # dx oe 16 '03 dx oe 16 3$!oe ' 1 (2y b 2) dy oe cy# b 2yd c" oe 104.' 2 '0 (sin x b cos y) dx dy oe ' 2 c(c cos x) b (cos y)xd 1 dy ! 2 #1 oe

Colorado State - MATH - 161

CHAPTER 16 INTEGRATION IN VECTOR FIELDS16.1 LINE INTEGRALS 1. r oe ti b (" c t)j x oe t and y oe 1 c t y oe 1 c x (c) 2. r oe i b j b tk x oe 1, y oe 1, and z oe t (e) 3. r oe (2 cos t)i b (2 sin t)j x oe 2 cos t and y oe 2 sin t x# b y# oe 4

Cornell - BIO G - 109

BioG 109September 25, 2003Prelim #1Part 1. Multiple choice. Questions 1-45. CHOOSE THE ONE BEST ANSWER. CIRCLE IT ON YOUR QUESTION PAPER. THEN MARK YOUR CHOICE CAREFULLY ON THE ANSWER SHEET. A CORRECT STATEMENT IS NOT NECESSARILY THE RIGHT (BES

Cornell - BIO G - 109

Biological Sciences 009 10Concepts of BiologyPractice Prelim #1Page 1 ofFall 2007 Biology Learning Skills Center Concepts of Biology: Analysis, Enrichment, and Review Biological Sciences 009: Section for Non-Majors Allen D. MacNeill, Instruct

Cornell - BIO G - 109

Prokaryotes=no nucleus but DNA, plasma membrane, no membrane-enclosed organelles, unicellular,5,000 species so very diverse, simple and small, some have pili which are hairlike appendages that allow prokaryotes to stick to substrate, divide by binary

Cornell - BIO G - 109

1. cell, tissue, organ, community, ecosystem, biosphere 2. a community 3. the scientific method 4. eat mainly marine protistans 5. Life or Eukaryotes 6. 5 taxa, 2 nodes, 4 branches 7. all of the above choices are correct 8. put taxa together that uni

Cornell - BIO G - 109

Prelim 2 Review Deuterostomes-blastopore formed during gastrulation turns into the anus Echninodermata=sea stars, sea urchins, sand dollars -spiny skin -radial symmetry in adults -endoskeleton -water vascular system=suction cup-like tube feet used fo

Cornell - BIO G - 109

Biological Sciences 009 Concepts of Biology Practice Prelim #1 Page 1 of 8Fall 2007 Biology Learning Skills Center Concepts of Biology: Analysis, Enrichment, and Review Biological Sciences 0