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12:30 MasteringPhysics
10/13/08 AM
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Introductory mechanics
Chapter 06 - Work And Kinetic Energy
Due at 11:59pm on Tuesday, October 14, 2008
View Grading Details
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 workenergy 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 . Part A Find the acceleration Hint A.1 of the particle. to , undergoing displacement given by
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
Express your answer in terms of ANSWER: =
and .
Part C 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 in terms of those quantities.
mass and the initial and final velocities. Give an expression for the work
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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. Express your answer in terms of ANSWER: = and . done on the particle by the external forces during the motion of the particle in terms of the initial and final
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 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 done on the particle is then given by . Now, using to
,
).
The net work
and , it can be shown that .
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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 from Hint G.1 Integrate Hint G.2 to Finding the work to calculate the work done on the particle. An integration formula is . What is , for some constant . The particle moves
, the change in kinetic energy of the particle during that time?
The formula for
.
Express your answer in terms of ANSWER: =
and
.
It can also be shown that the work-energy theorem is valid for two- and three-dimensional motion and for a varying net 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
are the initial and the final positions of the particle, is the net force acting on the particle.
is the vector representing a small
displacement, and
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 .
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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: = done on the block by the two forces?
Part D Imagine that the two forces are equal in magnitude, Determine the change Hint D.1 , and that there are no other horizontal forces acting on the block. to .
in the kinetic energy of the block as it moves from
If the forces are equal, how can the block be moving? Hint not displayed
ANSWER:
=0
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 (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 , the work that the . The force is independent of
(radians) with the positive x axis). Find force point A.
performs on the particle as it moves from point B to
Hint A.1
Formula for work done by a constant force
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Hint not displayed Part A.2 Find the angle between and , which is directed horizontally to the left, and the vector in the direction directed along the
You need to find the angle between the vector of the particle's motion (at an angle
(radians) relative to the positive x axis). It may help to visualize between and ?
negative x axis at the origin. What is the angle Express your answer in radians , not degrees. ANSWER: =
Express the work in terms of answer. ANSWER: =
,
, and . Remember to use radians , not degrees, for any angles that appear in your
This result is worth remembering! The work done by a constant force of magnitude respect to the direction of motion along a straight path of length the sign in this problem. Since , is
, which acts at an angle of
with
. This equation correctly gives ; hence
is the angle with respect to the positive x axis (in radians), .
Part B Now consider the same force acting on a particle that travels from point A to point B. The displacement vector now
points in the opposite direction as it did in Part A. Find the work done by in this case.
Hint B.1
A physical argument Hint not displayed
Express your answer in terms of ANSWER: =
,
, and .
Workhorses on Erie Canal
Two workhorses tow a barge along a straight canal. Each horse exerts a constant force of magnitude an angle , and the tow ropes make with the direction of motion of the horses and the barge. Each horse is traveling at a constant speed .
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Part A How much work Hint A.1 is done by each horse in a time ?
Formula for work Hint not displayed
Part A.2
Find the 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 Hint not displayed
Express answer your 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
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 may be
taken to be at the top of the spring, where the ball loses contact with the spring.
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. to point .
The speed of the ball was decreasing on its way from 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?
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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 force of magnitude sits on a frictionless inclined plane, which makes an angle , applied parallel to the incline, pulls the with respect to the horizontal, as shown. A
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 done on the block by all forces? (Include only the work done after the block has
constant speed. What is the total work
started moving, not the work needed to start the block moving from rest.) Hint A.1 What physical principle to use
To find the total work done on the block, use the work-energy theorem: . Part A.2 Find the change in kinetic energy
What is the change in the kinetic energy of the block, from the moment it starts moving until it has been pulled a distance ? Remember that the block is pulled at constant speed. Hint A.2.a Consider kinetic energy If the block's speed does not change, its kinetic energy cannot change. ANSWER: = Answer not displayed
Express your answer in terms of given quantities. ANSWER: =0
Part B What is Hint B.1 , the work done on the block by the force of gravity as the block moves a distance Force diagram Hint not displayed Part B.2 Force of gravity component Part not displayed Express the work done by gravity in terms of the weight and any other quantities given in the problem introduction. up the incline?
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ANSWER:
=
Part C What is Hint C.1 , the work done on the block by the applied force How to find the work done by a constant force Hint not displayed Express your answer in terms of ANSWER: = and other given quantities. as the block moves a distance up the incline?
Part D What is Part D.1 , the work done on the block by the normal force as the block moves a distance First step in computing the work Part not displayed Express your answer in terms of given quantities. ANSWER: =0 up the inclined plane?
Work Done by a Spring
Consider a spring, with spring constant , one end of which is attached to a wall. The spring is initially unstretched, with the unconstrained end of the spring at position .
Part A The spring is now compressed so that the unconstrained end moves from , find the work done by the spring as it is compressed. Hint A.1 Spring force as a function of position as a function of displacement from the spring's equilibrium position, is given by to . Using the work integral
The spring force vector
where
is the spring constant and
is a unit vector in the direction of the displacement of the spring (in this case, towards
the right). Part A.2 Integrand of the work integral Part not displayed Part A.3 Upper limit of the work integral . What will be the integral's upper limit?
The lower limit of the work integral will be at ANSWER: =
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Express the work done by the spring in terms of ANSWER: =
and
.
Holding Force of a Nail
A hammer of mass is moving at speed when it strikes a nail of negligible mass that is stuck in a wooden block. The deeper into the block. hammer is observed to drive the nail a distance Part A Find the magnitude of the force that the wooden block exerts on the nail, assuming that this force is independent of the , so that the change in the hammer's
depth of penetration of the nail into the wood. You may also assume that gravitational potential energy, as it drives the nail into the block, is insignificant. Hint A.1 How to approach the problem Hint not displayed Part A.2 Find the work done in terms of Part not displayed Part A.3 Find the change in kinetic energy of the hammer Part not displayed Express the magnitude of the force in terms of ANSWER: = , , and .
Part B Now evaluate the magnitude of the holding force of the wooden block on the nail by assuming that the force necessary to pull the nail out is the same as that needed to drive it in, which we just derived. Assume a relatively heavy hammer (about 18 ounces), moving with speed rise 5 m). Take the penetration depth . (If such a hammer were swung this hard upward and released, it would to be 2 cm, which is appropriate for one hit on a relatively heavy construction nail. .)
Express your answer to the nearest pound. (Note: ANSWER: = 281 lb
Pulling a Block on an Incline with Friction
A block of weight sits on an inclined plane as shown. A force of magnitude is applied to pull the block up the incline at constant speed. The coefficient of kinetic friction between the plane and the block is .
Part A What is the total work Hint A.1 How to start Hint not displayed done on the block by the force of friction as the block moves a distance up the incline?
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Part A.2
Find the magnitude of the friction force Part not displayed
Express the work done by friction in terms of any or all of the variables ANSWER: =
,
,,,
, and
.
Part B What is the total work done on the block by the applied force , as the block moves a distance ,,, , and . up the incline?
Express your answer in terms of any or all of the variables ANSWER: =
Now the applied force is changed so that instead of pulling the block up the incline, the force pulls the block down the incline at a constant speed.
Part C What is the total work done on the block by the force of friction as the block moves a distance , ,,, , and . down the incline?
Express your answer in terms of any or all of the variables ANSWER: =
Part D What is the total work done on the box by the appled force in this case? , ,,, , and .
Express your answer in terms of any or all of the variables ANSWER: =
Combining Truck Power
A loaded truck (truck 1) has a maximum engine power a maximum engine power shown. To solve this problem, assume that each truck, when not attached to another truck, has a speed that is limited only by wind resistance. Also assume (not very realistically) A) That the wind resistance is a constant force (a different constant for each truck though). i.e. It is independent of the speed at which the truck is going. B) That the wind resistance force on each truck is the same before and after the cable is connected, and, C) That the power that each truck's engine can generate is independent of the truck's speed. and is able to attain a maximum speed . Another truck (truck 2) has . The two trucks are then connected by a long cable, as and can attain a maximum speed of
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Part A Find power. Hint A.1 , the maximum speed of the two trucks when they are connected, assuming both engines are running at maximum
Method for solving Hint not displayed
Part A.2
Resistance force on truck 1 Part not displayed
Part A.3
Net wind resistance on the two trucks Part not displayed
Hint A.4
Net power of the two trucks Hint not displayed
Hint A.5
Solving for Hint not displayed
Express the maximum speed in terms of . ANSWER: =
Note that truck 1 is going faster when in tow than when under its own power, and that truck 2 is going slower. This is consistent with having the cable connecting the trucks being subject to a tension. Anyone who has ever driven a truck, or closely watched one being driven, will know that this sort of arrangement is very unsafe and consequently is never used. However, train locomotives, which can be coupled together without cables, can combine their power in this way.
Summary
10 of 10 items complete (101.51% avg. score) 10.15 of 10 points
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