Assignment MasteringPhysics: Print View
Applying Newton's 2nd Law
Learning Goal: To learn a systematic approach to solving Newton's 2nd law problems using a simple example. Once you have decided to solve a problem using Newton's 2nd law, there are steps that will lead you to a solution. One such prescription is the following:
q q q q q q q q q
Visualize the problem and identify special cases. Isolate each body and draw the forces acting on it. Choose a coordinate system for each body. Apply Newton's 2nd law to each body. Write equations for the constraints and other given information. Solve the resulting equations symbolically. Check that your answer has the correct dimensions and satisfies special cases. If numbers are given in the problem, plug them in and check that the answer makes sense. Think about generalizations or simplfications of the problem.
As an example, we will apply this procedure to find the acceleration of a block of mass that is pulled up a frictionless plane inclined at angle with respect to the horizontal by a perfect string that is hanging vertically. that passes over a perfect pulley to a block of mass
First examine the problem by drawing a picture and visualizing the motion. Apply Newton's 2nd law, , to each body in your mind. Don't worry about which quantities are given. Think about the forces on each body: How are these consistent with the direction of the acceleration for that body? Can you think of any special cases that you can solve quickly now and use to test your understanding later? One special case in this problem is if , in which case block 1 would simply fall freely under the acceleration of gravity: . Part A
MasteringPhysics: Assignment Print View
Consider another special case in which the inclined plane is vertical ( what value of would the acceleration of the two blocks be equal to zero? and .
). In this case, for
Express your answer in terms of some or all of the variables ANSWER: =
A force diagram should include only real forces that act on the body and satisfy Newton's 3rd law. One way to check if the forces are real is to detrmine whether they are part of a Newton's 3rd law pair, that is, whether they result from a physical interaction that also causes an opposite force on some other body, which may not be part of the problem. Do not decompose the forces into components, and do not include resultant forces that are combinations of other real forces like centripetal force or fictitious forces like the "centrifugal" force. Assign each force a symbol, but don't start to solve the problem at this point. Part B Which of the four drawings is a correct force diagram for this problem? ANSWER: a b c d
Newton's 2nd law,
, is a vector equation. To add or subtract vectors it is often easiest to
decompose the vector into components. Whereas a particular set of vector components is only valid in a particular coordinate system, the vector equality holds in any coordinate system, giving you freedom to pick a coordinate system that most simplifies the equations that result from the component equations.
MasteringPhysics: Assignment Print View
It's generally best to pick a coordinate system with as many unknowns as possible along the coordinate axes. Vectors that lie along the axes appear in only one of the equations for each component, rather than in two equations with trigonometric prefactors. Note that it is sometimes advantageous to use different coordinate systems for each body in the problem. In this problem, you should use Cartesian coordinates and your axes should be stationary with respect to the inclined plane. Part C Given the criteria just described, what orientation of the coordinate axes should you use in this problem? In the answer options, "tilted" means with the x axis oriented parallel to the plane (i.e., at angle to the horizontal), and "level" means with the x axis horizontal. ANSWER: tilted for both block 1 and block 2 tilted for block 1 and level for block 2 level for block 1 and tilted for block 2 level for both block 1 and block 2 Part D What is , the sum of the x components of the forces acting on block 2? Take forces acting
up the incline to be positive. Part D.1 Part not displayed Express your answer in terms of some or all of the variables tension acceleration of gravity , and . ANSWER: = Part E , , the magnitude of the
MasteringPhysics: Assignment Print View
Now determine
, the sum of the y components of the forces acting on block 1.
Take forces acting upward as positive. Express your answer in terms of some or all of the variables ANSWER: = , , and .
Part F Write equations for the constraints and other given information In this problem, the fact that the length of the string does not change imposes a constraint on relative accelerations of the two blocks. Find a relationship between the x component of the acceleration of block 2, , and the acceleration of block 1. Pay careful attention to signs. Hint F.1 Hint not displayed Express in terms of ANSWER: = Part G Solve and check In the previous parts, you obtained the following equations using Newton's 2nd law and the constraint on the motion of the two blocks: and/or , the components of the acceleration vector of block 1.
and
Solve these equations to find
.
Before you enter your answer, make sure it satisfies the special cases you already identified:
MasteringPhysics: Assignment Print View
q q
if if
and and .
Also make sure that your answer has dimensions of acceleration. Hint G.1 Hint not displayed Express in terms of some or all of the variables given in the introduction. ANSWER: = Can you see how a simple generalization of the problem could be solved with a little extra work or how you could solve a nontrivial problem that is a subset of this one? For example, imagine that there is friction in this problem between the plane and block 2. This , where the normal force is given by would lead to an additional force on block 2: . This additional force would lead to a new term in the expression for the acceleration of block 1: .
Now, by choosing whether or not frictionless or not!
, you have a result that can be applied whether the plane is
Contact Forces Introduced
MasteringPhysics: Assignment Print View
Learning Goal: To introduce contact forces (normal and friction forces) and to understand that, except for friction forces under certain circumstances, these forces must be determined from: net Force = ma. Two solid objects cannot occupy the same space at the same time. Indeed, when the objects touch, they exert repulsive normal forces on each other, as well as frictional forces that resist their slipping relative to each other. These contact forces arise from a complex interplay between the electrostatic forces between the electrons and ions in the objects and the laws of quantum mechanics. As two surfaces are pushed together these forces increase exponentially over an atomic distance scale, easily becoming strong enough to distort the bulk material in the objects if they approach too close. In everyday experience, contact forces are limited by the deformation or acceleration of the objects, rather than by the fundamental interatomic forces. Hence, we can conclude the following: The magnitude of contact forces is determined by , that is, by the other
forces on, and acceleration of, the contacting bodies. The only exception is that the (although they can be smaller than this or even frictional forces cannot exceed zero).
Two types of contact forces operate in typical mechanics problems, the normal and frictional , or something similar) respectively. These are the forces, usually designated by and (or components of the overall contact force: perpendicular to and parallel to the plane of contact.
When one surface is sliding past the other, experiments show three things about the friction force (denoted ): 1. The frictional force opposes the relative motion at the point of contact, is proportional to the normal force, and 2. 3. the ratio of the magnitude of the frictional force to that of the normal force is fairly constant over a wide range of speeds.
The constant of proportionality is called the coefficient of kinetic friction, often designated long as the sliding continues, the frictional force is then (valid when the surfaces slide by each other).
. As
MasteringPhysics: Assignment Print View
When there is no relative motion of the surfaces, the frictional force can assume any value from zero up to a maximum , where is the coefficient of static friction. Invariably, is larger than , in agreement with the observation that when a force is large enough that something breaks loose and starts to slide, it often accelerates. The frictional force for surfaces with no relative motion is therefore (valid when the contacting surfaces have no relative motion). The actual magnitude and direction of the static friction force are such that it (together with other forces on the object) causes the object to remain motionless with respect to the contacting surface as long as the static friction force required does not . The equation is valid only when the surfaces are on the verge of sliding. exceed Part A When two objects slide by one another, which of the following statements about the force of friction between them, is true? ANSWER: The frictional force is always equal to . . The frictional force is always less than The frictional force is determined by other forces on the objects so it can be . either equal to or less than
Part B When two objects are in contact with no relative motion, which of the following statements about the frictional force between them, is true? ANSWER: The frictional force is always equal to . . The frictional force is always less than The frictional force is determined by other forces on the objects so it can be . either equal to or less than
For static friction, the actual magnitude and direction of the friction force are such that it, together with any other forces present, will cause the object to have the observed acceleration. The . If the magnitude of static friction needed to keep magnitude of the force cannot exceed acceleration equal to zero exceeds , then the object will slide subject to the resistance of unless you are considering a situation kinetic friction. Do not automatically assume that in which the magnitude of the static friction force is as large as possible (i.e., when determining at what point an object will just begin to slip). Whether the actual magnitude of the friction force is , or equal to depends on the magnitude of the other forces (if any) as well as 0, less than
MasteringPhysics: Assignment Print View
the acceleration of the object through Part C
.
When a board with a box on it is slowly tilted to larger and larger angle, common experience shows that the box will at some point "break loose" and start to accelerate down the board. The box begins to slide once the component of gravity acting parallel to the board equals the force of static friction. Which of the following is the most general explanation for why the box accelerates down the board? A. The force of kinetic friction is smaller than that of static friction, but B. Once the box is moving, force of kinetic friction. C. Once the box is moving, remains the same. is smaller than the force of static friction but larger than the
is larger than the force of static friction. D. When the box is stationary, equals the force of static friction, but once the box starts moving, the sliding reduces the normal force, which in turn reduces the friction. ANSWER: A B C D
At the point when the box finally does "break loose," you know that the component of the box's (i.e., this component of gravitational force on weight that is parallel to the board is equal to the box has just reached a magnitude such that the force of static friction, which has a maximum value of , can no longer oppose it.) For the box to then accelerate, there must be a net force on the box along the board. Thus, the component of the box's weight parallel to the board must be must be less greater than the force of kinetic friction. Therefore the force of kinetic friction than the force of static friction which implies , as expected. Part D
MasteringPhysics: Assignment Print View
Consider a problem in which a car of mass Select the best answer. ANSWER:
is on a road tilted at an angle . The normal force
is found using
The key point is that contact forces must be determined from Newton's equation. In the problem described above, there is not enough information given to determine the normal force (e.g., the acceleration is unknown). Each of the answer options is valid under some conditions ( , the car is sliding down an icy incline, or the car is going around a banked turn), but in fact none is likely to be correct if there are other forces on the car or if the car is accelerating. Do not memorize values for the normal force valid in different problems--you must determine from .
Newton's 1st and 2nd Laws
Learning Goal: To understand the meaning and the basic applications of Newton's 1st and 2nd laws. Newton's laws are fundamental in mechanics. Their mathematical expressions are very simple but conceptual understanding of Newton's laws, which is necessary for solving nontrivial problems, is not simple at all.
The common textbook statement of Newton's 1st law may seem rather straightforward. Here it is: An object has a constant velocity (possibly zero) if and only if the net force acting on the object is zero. In other words, if the vector sum of the forces applied to the object is zero, the object would be either at rest or at constant velocity (that is, the object would have zero acceleration). If such a sum is not zero, the object cannot possibly be moving at a constant velocity.
MasteringPhysics: Assignment Print View
The statement of Newton's 1st law becomes a bit more complicated in actual applications. Imagine yourself in a car. To understand Newton's 1st law fully, we need the concept of a frame of reference. A frame of reference is a set of coordinates used to measure distances and times. In your frame of reference, any distance would be measured relative to you. For example, the radio in the car is 0.75 m to the right of you. The radio is at rest in your frame of reference, because the radio doesn't change its distance or direction from you. In your frame of reference, the car is always at rest. It is entirely possible that the net force acting on the car is not zero: The car may (in the frame of reference of an observer standing on the ground) be accelerating, turning, or braking. Yet in your frame of reference, the car would remain at rest because, relative to you, it is not moving at all. So, the car is at rest or accelerating, depending upon who you ask.
It's tempting to ignore this difficulty by saying that the frame of reference attached to the car is somehow wrong. The observer on the ground, in contrast, is right: The observer sees the motion of the car as it really is. However, such a line of reasoning seems flawed, because it raises the question of how to determine which frames of reference are "right" and which ones are "wrong." This is what Newton's 1st law settles. Newton established the concept of an inertial frame of reference. An inertial frame of reference, by definiton, is one in which the statement of Newton's 1st law is, in fact, true.
It is important to know that the frame of reference being used is, in fact, inertial. Only then does Newton's 2nd law work in a simple and elegant form. Newton's 2nd law establishes the relationship between the net force acting on an object, the mass of the object, and its acceleration: , or
.
Note that Newon's 2nd law allows one to find the magnitude of the object's acceleration. It also establishes the fact that the acceleration of an object has the same direction as the net force acting
MasteringPhysics: Assignment Print View
on the object.
If the frame of reference is not inertial, using Newton's 2nd law to calculate acceleration is still possible but may be far more complicated. Objects that experience zero net force may accelerate, and objects that move at constant velocity may experience a net force not equal to zero. The important question is: Which frames of reference are inertial and which ones are not? This also raises the following question: Are there any inertial frames of reference in this universe? Newton postulated that inertial frames of reference do exist. This statement, coupled with the definition of inertial frames of reference, may be considered a more proper way to state Newton's 1st law. Only an experiment can establish whether a particular frame of reference is inertial (or, to be precise, "inertial enough" for the purposes needed). Let us go back to the car example. The frame of reference attached to the ground, we would usually say, is inertial. That is, if we get an object and make sure that all external forces acting on it add up to zero, we can then observe that the object is, in fact, moving at constant velocity (or, possibly, remaining at rest). In most problems that we will be solving, the frame of reference of the earth will be considered an inertial frame of reference. For all practical purposes, this means that Newton's 2nd law will work in it. However, it is instructive to understand that the earth provides a reference frame that is less than "perfectly inertial." An observer on the sun, for instance, would notice that the object in question does, in fact, have an acceleration: the centripetal acceleration associated with the orbital motion of the earth around the sun! The best inertial frame of reference is the one assoicated with distant stars and any other frame of reference that is moving at a constant velocity relative to distant stars. The conceptual questions that follow should help you learn to apply Newton's 1st and 2nd laws properly. Note that, throughout this problem, we will assume that the frame of reference associated with the earth is perfectly inertial. Part A
MasteringPhysics: Assignment Print View
Which object provides an inertial frame of reference? ANSWER: the tip of the moving second hand of a clock a rock thrown vertically upward a pendulum swinging with no air resistance a skydiver falling at terminal velocity
Assuming that the earth provides an inertial frame of reference, an object moving at a constant velocity relative to the earth would also provide an inertial frame of reference. Part B You are conducting an experiment inside an elevator that can move in a vertical shaft. A load is hung vertically from the ceiling on a string, and is stationary with respect to you. The tension in the string is measured to be 10% less than the weight of the load. No other forces are acting on the load. Which of the following statements about the elevator are correct? A. B. C. D. E. F. The elevator is an inertial frame of reference. The elevator is not an inertial frame of reference. The elevator may be at rest for the duration of the entire experiment. The elevator may be moving at a constant velocity upward. The elevator may be moving at a constant velocity downward. The elevator must be accelerating.
Type the letters corresponding to all the correct answers. Do not use commas. For instance, if you think that only statements C and D are correct, type CD. ANSWER: BF Part C You are conducting an experiment inside an elevator that can move in a vertical shaft. A load is hung vertically from the ceiling on a string. The tension in the string is measured to be exactly equal to the weight of the load. No other forces are acting on the load. Which of the following statements about the elevator are correct? A. B. C. D. E. F. G. The elevator is an inertial frame of reference. The elevator is not an inertial frame of reference. The elevator may be at rest. The elevator may be moving at a constant velocity upward. The elevator may be moving at a constant velocity downward. The elevator may be accelerating. The elevator must be accelerating.
MasteringPhysics: Assignment Print View
Type the letters corresponding to all the correct answers. Do not use commas. For instance, if you think that only statements C and D are correct, type CD. ANSWER: ACDE Part D You are conducting an experiment inside a train car that may move horizontally along rail tracks. A load is hung from the ceiling on a string. The load is not swinging, and the string is observed to with the horizontal. No other forces are acting on the load. Which of make a constant angle of the following statements are correct? A. B. C. D. E. F. G. H. The train is an inertial frame of reference. The train is not an inertial frame of reference. The train may be at rest. The train may be moving at a constant speed in a straight line. The train may be moving at a constant speed in a The circle. train must be speeding up. The train must be slowing down. The train must be accelerating.
Type the letters corresponding to all the correct answers. Do not use commas. For instance, if you think that only statements C and D are correct, type CD. ANSWER: BEH Since the tension and the weight are not directed opposite to each other, the net force cannot possibly be zero--and yet the load is at rest relative to the train car. Therefore, the car is not an inertial frame of reference. It must be accelerating relative to the earth, although it is not clear exactly how. Part E Consider the train car described in the previous part. Another experiment is conducted in it: A net is applied to an object of mass . Can you determine the acceleration of the force of object, and, if so, what is its value? ANSWER: Yes; Yes; Yes; . . .
No; there is not enough information. The train car is not an inertial frame of reference, so would not work here.
MasteringPhysics: Assignment Print View
Part F A 1000-kg car is moving along a straight road down a What is the net force acting on the car? ANSWER: slope at a constant speed of .
The car has zero acceleration; therefore, it experiences zero net force. According to Newton's 1st law, no net force is required to maintain a constant velocity (in an inertial frame of reference, of course). The car has a constant veclocity relative to the earth; therefore, the car is also an inertial frame of reference. Part G Consider two cars moving along the same straight road in opposite directions. Car A has a mass of and has a constant speed of ; car B has a mass of and a constant speed of . Whar can you say about the net forces on the cars? ANSWER: Car A experiences greater net force than car B. Car B experiences greater net force than car A. Both cars experience equal net forces.
Each car has zero acceleration; therefore, the net force on each car, according to Newton's 1st law, is zero. Part H
MasteringPhysics: Assignment Print View
In an inertial frame of reference, a series of experiments is conducted. In each experiment, two or three forces are applied to an object. The magnitudes of these forces are given. No other forces are acting on the object. In which cases may the object possibly remain at rest? The forces applied are as follows: A. B. C. D. E. F. G. H. 2 N; 2 N 200 N; 200 N 200 N; 201 N 2 N; 2 N; 4 N 2 N; 2 N; 2 N 2 N; 2 N; 3 N 2 N; 2 N; 5 N 200 N; 200 N; 5 N
Hint H.1 Hint not displayed Type the letters corresponding to all the correct answers. Do not use commas. For instance, if you think that only cases C and D are correct, type CD. ANSWER: ABDEFH Part I In an inertial frame of reference, a series of experiments is conducted. In each experiment, two or three forces are applied to an object. The magnitudes of these forces are given. No other forces are acting on the object. In which cases may the object possibly move at a constant velocity of ? The forces applied are as follows: A. B. C. D. E. F. G. H. 2 N; 2 N 200 N; 200 N 200 N; 201 N 2 N; 2 N; 4 N 2 N; 2 N; 2 N 2 N; 2 N; 3 N 2 N; 2 N; 5 N 200 N; 200 N; 5 N
MasteringPhysics: Assignment Print View
Hint I.1 Hint not displayed Type the letters corresponding to all the correct answers. Do not use commas. For instance, if you think that only cases C and D are correct, type CD. ANSWER: ABDEFH You should have noticed that the sets of forces applied to the object are the same as the ones in the prevous question. Newton's 1st law (and the 2nd law, too) makes no distinction between the state of rest and the state of moving at a constant velocity (even a high velocity). In both cases, the net force applied to the object must equal zero. Although some of the questions in this problem may have seemed tricky and unfair, the subtleties here are important in improving conceptual understanding. That understanding, in turn, will enable you to correctly solve complex computational problems using Newton's laws.
PSS 5.1: Don't Rock the Block
Learning Goal: To practice Problem-Solving Strategy 5.1 for equilibrium problems. A pair of students are lifting a heavy trunk on move-in day . Using two ropes tied to a small ring at the center of the top of the trunk, they pull the trunk straight up at a constant velocity . Each rope makes an angle with respect to the vertical. The magnitude of the weight of the trunk is . Find the tension in each rope.
MODEL: Make simplifying assumptions. VISUALIZE:
q
q
Physical representation--Identify all forces acting on the object and show them on a free-body diagram. Pictorial representation--The free-body diagram is usually sufficient as a picture for equilibrium problems, but you still must translate words into symbols and identify what the
MasteringPhysics: Assignment Print View
problem is trying to find. It is acceptable practice to go back and forth between these two steps as you visualize the situation.
SOLVE: The mathematical representation is based on Newton's 1st law:
The vector sum of the forces is found directly from the free-body diagram.
ASSESS: Check if your result has the correct units, is reasonable, and answers the question.
Start by choosing an appropriate model for the problem and by making appropriate simplifying assumptions. Part A In this problem, the trunk should be modeled as ANSWER: a particle in static equilibrium. a particle in dynamic equilibrium. a particle not in equilibrium.
Despite the fact that , the velocity of the trunk is constant. Therefore, the acceleration of the trunk is zero, and the trunk is in dynamic equilibrium. Part B Does your model of this problem need to include air resistance? ANSWER: yes no
Draw a free-body diagram that includes all the forces acting on the trunk. Use a coordinate system with the y axis vertical and the ropes both in the xy plane. Use your diagram to answer the following question. Part C How many forces are acting on the trunk? ANSWER: one two three four
MasteringPhysics: Assignment Print View
Part D Which of the following statements are correct? A. B. C. D. E. F. G. H. I. J. The x component of the weight of the trunk is zero. The y component of the weight of the trunk is zero. The x components of the forces of tension in the ropes have the same magnitude and sign. The x components of the forces of tension in the ropes have the same magnitude and opposite sign. The y components of the forces of tension in the ropes have the same magnitude and sign. The y components of the forces of tension in the ropes have the same magnitude and opposite sign. The x component of the acceleration of the trunk is zero. The y component of the acceleration of the trunk is zero. The angle between the ropes is . The angle between the ropes is .
Enter the letters corresponding to the right answers in alphabetical order. Do not use commas. For instance, if options C and D are correct, enter CD. ANSWER: ADEGHJ Starting with Newton's 1st law, construct the necessary mathematical expressions and derive the solution. Part E Find the force of tension Part E.1 Part not displayed Part E.2 Part not displayed Express your answer in terms of variables given in the problem introduction. ANSWER: = in each rope.
Now answer the following questions to verify that the answer you just obtained is reasonable. Part F
MasteringPhysics: Assignment Print View
Intuitively, what will happen to the value of that all other conditions remain the same. ANSWER: will increase. will decrease. will remain the same. Of course, the mathematical expression for Part G
if the students try to lift a heavier trunk? Assume
predicts this result: As
increases, so does
.
Intuitively, what will happen to the value of if the students try to stand farther apart so that the ropes make a greater angle with the vertical? Assume that all other conditions remain the same. ANSWER: will increase. will decrease. will remain the same. As the angle increases, a greater fraction of the total force exerted by the students is being applied in the sideways ( ) direction rather than upward. Since the weight of the trunk is constant, this means that the students need to pull harder to lift it. Of course, the mathematical expression for predicts this result: As increases, decreases and increases. Part H Intuitively, what should the value of virtually vertical? be if the students stand close together so that the ropes are
Express your answer in terms of the given quantities. You may or may not use all of them. ANSWER: = Of course, the mathematical expression for . predicts this result: If is zero, and
PSS 5.2: The Rope Pull
MasteringPhysics: Assignment Print View
Learning Goal: To practice Problem-Solving Strategy 5.2 for dynamics problems. On a bright sunny day, two girls play tug of war while standing on the opposite sides of a pond. Initially, each stands at a distance from her edge of the pond. Jackie, who weighs less than her opponent Sue, loses the tug of war and is forced to take a dive in the pond. (Most often, the tug of war is won by the heaviest, not the strongest, team--you will find out why when you learn more about friction.) Jackie's mass is , the tension in the rope is , and the opposing force of friction between Jackie's
feet and the ground is . Find the time it took Sue to win the competition.
MODEL: Make simplifying assumptions. VISUALIZE:
q
q
Pictorial representation--Show important points in the motion with a sketch, establish a coordinate system, define symbols, and identify what the problem is trying to find. This is the process of translating words into symbols. Physical representation--Use a motion diagram to determine the object's acceleration vector . Then identify all forces acting on the object and show them on a free-body diagram.
It is perfectly accepable to go back and forth between these two steps as you visualize the situation.
SOLVE: The mathematical representation is based on Newton's 2nd law:
. The vector sum of the forces is found directly from the free-body diagram. Depending on the problem, either
s s
solve for acceleration, then use kinematics to find velocities and positions, or use kinematics to determine the acceleration, then solve for unknown forces.
ASSESS: Check if your result has the correct units, is reasonable, and answers the question.
MasteringPhysics: Assignment Print View
Make simplifying assumptions and choose an appropriate model. Part A Which of the following describes an appropriate model for this problem? ANSWER: Jackie can be treated as a particle moving with constant velocity. Jackie can be treated as a particle moving with constant nonzero acceleration. Jackie can be treated as a particle moving with varying acceleration.
Part B Which of the following assumptions or interpretations are reasonable? A. B. C. D. E. F. G. Air resistance is substantial. Air resistance is negligible. The ground near the pond is slanted toward the water. The ground near the pond is horizontal. The mass of the rope equals Jackie's mass. The mass of the rope is negligible. The rope is unstretchable.
Enter the letters corresponding to all the correct answers in alphabetical order. Do not use commas. For instance, if you think that only assumptions C and D are correct, type CD. ANSWER: BDFG Now draw all the elements listed in the problem-solving strategy. Be sure that your pictorial representation is clear and includes all necessary symbols, both known and unknown; your freebody diagram should include all relevant forces, clearly drawn and labeled. Don't forget the motion diagram. Use your sketches to answer the following questions. Part C Assuming that in the tug of war Jackie is being pulled to the right, which of the free-body diagrams shown is correct? Note that the forces are not drawn to scale. Also, they are not labeled; however, they should be labeled on your own diagram.
MasteringPhysics: Assignment Print View
ANSWER:
A C E
B D F
Part D Based on your motion diagram, which statements about Jackie's vertical acceleration true? A. The magnitude of is greater than that of . must be
is zero. B. The magnitude of C. The magnitude of is infinitely large. D. is directed downward. E. is directed upward. Type the letters corresponding to all the correct answers in alphabetical order. Do not use commas. For instance, if you think that only answers C and D are correct, type CD. ANSWER: B Part E
MasteringPhysics: Assignment Print View
Look at your pictorial representation. Which of the following are known quantities in this problem? A. B. C. D. E. F. Jackie's initial horizontal position The distance Jackie is pulled before reaching the edge of the pond Jackie's horizontal acceleration Jackie's initial horizontal velocity Jackie's horizontal velocity at the moment she reaches the edge of the pond The time it takes for Sue to pull Jackie into the pond
Type the letters corresponding to all the correct answers in alphabetical order. Do not use commas. For instance, if you think that only answers C and D are correct, type CD. ANSWER: ABD Be sure to choose appropriate symbols for each important quantity that doesn't already have one. Here is one possibility following the conventions of your textbook:
q q q q q
for Jackie's initial horizontal position, for Jackie's horizontal position at the moment she reaches the edge of the pond, for Jackie's horizontal acceleration, for Jackie's initial horizontal velocity, and for Jackie's horizontal velocity at the moment she reaches the edge of the pond.
Note that Jackie's acceleration does not need a subscript 0 or 1 because it is constant throughout the problem.
Now use the information and the insights that you have accumulated to construct the necessary mathematical expressions and to derive the solution. Part F
MasteringPhysics: Assignment Print View
Find the time Hint F.1
it would take Sue to pull Jackie into the pond.
Hint not displayed Part F.2 Part not displayed Part F.3 Part not displayed Express your answer in terms of the quantities given in the problem introduction. You may or may not use all of them. ANSWER: =
When you work on a problem on your own, without the computer-provided feedback, only you can assess whether your answer seems right. The following questions will help you practice the skills necessary for such an assessment. Part G What are the units of the expression ANSWER: ?
Part H
MasteringPhysics: Assignment Print View
Which of these mathematical expressions have dimensions of time? A. B. C.
D.
E.
Enter the letters corresponding to all the correct answers in alphabetical order. Do not use commas. For instance, if you think that only expressions C and D have dimensions of time, type CD. ANSWER: BCD Part I Intuitively, what could you do to increase the amount of time it takes Sue to pull Jackie into the pond? A. Increase the force . B. Decrease the force . C. Increase the force . D. E. F. G. Decrease the force . Increase the mass . Decrease the mass . Increase the distance . H. Decrease the distance .
Type the letters corresponding to all the correct answers in alphabetical order. Do not use commas. For instance, if you think only C and D are correct, type CD. ANSWER: BCEG
MasteringPhysics: Assignment Print View
Note that the formula that you obtained,
, does agree with your "common-
sense" predictions. If you increase a quantity in the numerator, or decrease a quantity in the denominator, the time will get bigger.
Board Pulled Out from under a Box
A small box of mass is sitting on a board of mass frictionless horizontal surface. The coefficient of static friction between the board and the box is . The coefficient of kinetic friction between the board and the box is, as usual, less than . Throughout the problem, use for the magnitude of the acceleration due to for the gravity. In the hints, use magnitude of the friction force between the board and the box. Part A , the constant force with the Find least magnitude that must be applied to the board in order to pull the board out from under the the box (which will then fall off of the opposite end of the board). Hint A.1 Hint not displayed Part A.2 Part not displayed Part A.3 Part not displayed Part A.4 Part not displayed Part A.5 Part not displayed and length . The board rests on a
MasteringPhysics: Assignment Print View
Hint A.6 Hint not displayed Express your answer in terms of some or all of the variables , , , , and . Do not include in your answer. ANSWER: =
Block on an Incline
A block lies on a plane raised an angle the force of gravity; from the horizontal. Three forces act upon the block: ,
, the normal force; and
, the force of friction. The coefficient of friction is
large enough to prevent the block from sliding . Part A Consider coordinate system a, with the x axis along the plane. Which forces lie along the axes? ANSWER: only only only and and and and and
Part B
MasteringPhysics: Assignment Print View
Which forces lie along the axes of the coordinate system b, in which the y axis is vertical? ANSWER: only only only and and and and and
Now you are going to ignore the general rule (actually, a strong suggestion) that you should pick the coordinate system with the most vectors, especially unknown ones, along the coordinate axes. You will find the normal force, the magnitude function. Part C , using vertical coordinate system b. In these coordinates you will find appearing in both the x and y equations, each multiplied by a trigonometric
Because the block is not moving, the sum of the y components of the forces acting on the block must be zero. Find an expression for the sum of the y components of the forces acting on the block, using coordinate system b. Part C.1 Part not displayed Part C.2 Part not displayed Express your answer in terms of some or all of the variables ANSWER: , , , and .
Part D
MasteringPhysics: Assignment Print View
Because the block is not moving, the sum of the x components of the forces acting on the block must be zero. Find an expression for the sum of the x components of the forces acting on the block, using coordinate system b. Part D.1 Part not displayed Express your answer in terms of some or all of the variables ANSWER: , , , and .
Part E To find the magnitude of the normal force, you must express involving Hint E.1 Hint not displayed ANSWER: = Congratulations on working this through. Now realize that in coordinate system a, which is aligned with the plane, the y-coordinate equation is immediately to the result obtained here for . , which leads and but not . in terms of since is an
unknown. Using the equations you found in the two previous parts, find an expression for
CONCLUSION: A thoughtful examination of which coordinate system to choose can save a lot of algebra.
A Gymnast on a Rope
MasteringPhysics: Assignment Print View
A gymnast of mass 68.0 hangs from a vertical rope attached to the ceiling. You can ignore the weight of the rope and assume that the rope does not stretch. Part A Calculate the tension Hint A.1 Hint not displayed Part A.2 Part not displayed Express your answer in newtons. ANSWER: = 667 Part B Calculate the tension Hint B.1 Hint not displayed Express your answer in newtons. ANSWER: = 667 Does it surprise you that the answers to Parts A and B are the same? In both cases, the gymnast is not accelerating. Therefore, the net force acting on the gymnast is zero. Since the only two forces acting on the gymnast are tension and weight, the tension in the rope in each case is equal in magnitude (and opposite in direction) to the gymnast's weight. Part C in the rope if the gymnast climbs the rope at a constant rate. in the rope if the gymnast hangs motionless on the rope.
MasteringPhysics: Assignment Print View
Calculate the tension of magnitude 0.500 Hint C.1
in the rope if the gymnast climbs up the rope with an upward acceleration .
Hint not displayed Part C.2 Part not displayed Part C.3 Part not displayed Express your answer in newtons. ANSWER: = 701 Part D Calculate the tension in the rope if the gymnast slides down the rope with a downward .
acceleration of magnitude 0.500 Hint D.1
Hint not displayed Part D.2 Part not displayed Part D.3 Part not displayed Express your answer in newtons. ANSWER: = 633 In this problem, the directions of velocity and acceleration happened to be the same. If they are different, it is the direction of the acceleration, not the direction of velocity, that determines the magnitudes of the forces. Newton's 2nd law has nothing to do with the object's velocity.
Suspending a Speaker
MasteringPhysics: Assignment Print View
A loudspeaker of mass 15.0 Part A What is the tension cables? Hint A.1
is suspended a distance of
= 1.00
below the ceiling by two .
cables that make equal angles with the ceiling. Each cable has a length of = 2.90
in each of the
Hint not displayed Part A.2 Part not displayed Part A.3 Part not displayed Part A.4 Part not displayed Use 9.80 for the magnitude of the
acceleration due to gravity. ANSWER: = 213