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,
,Build Skills
L earn basic and advanced skills that help
solve a broad range of physics problems.
Problem-Solving Strategies coach students in
how to approach specific types of problems.
Ż
Problem-Solving Strategy 5.2 Newton’s Second Law: Dynamics of Particles
IDENTIFY the relevant concepts: You have to use Newton’s second accelerate in different directions, you can use a different set of
law for any problem that involves forces acting on an accelerating axes for each body. S S
4. In addition to Newton’s second law, gF ⴝ ma, identify any
This text’s uniquely extensive set Ż body.
Identify the target variable—usually an acceleration or a force.
Example 5.17 Toboggan ride with friction II
other equations you might need. For example, you might need
If the target variable is something else, you’ll need to select another one or more of the equations for motion with constant accelera-
of Examples enables students concept to use. For example, suppose the target variable is how tion. If more than one body is involved, there may be relation-
The same toboggan with the same coefficient of friction as in From the second equation and Eq. (5.5) we get an expression for ƒk:
fast a sled is moving when it reaches the bottom of a hill. Newton’s ships among their motions; for example, they may be connected
to explore problem-solving Example 5.16 accelerates down a steeper hill. Derive an expres-
second law will let you find the sled’s acceleration; you’ll then use n = by
sion for the acceleration in terms of g, a, mk, and w.
mga cos
rope.a Express any such relationships as equations relating
the constant-acceleration relationships from Section 2.4 to find ƒk = the mknaccelerations
= mkmg cos of athe various bodies.
challenges in exceptional detail. S O LU T I O N
velocity from acceleration.
We substitute this into EXECUTEthe x-component
the solution and solve for ax:
as follows:
equation
SET UP the problem using the following steps: 1. For each body, determine the components of the forces along
IDENTIFY and SET UP: The toboggan is accelerating, so we must mg a + 1 - m mg a2 = max
Consistent 1. Draw a simple sketch of the situation that shows each moving
use Newton’s second law as given in Eqs. (5.4). Our target variable
body. For each body, draw a free-body diagram that shows
sin
= g1sinina terms
ax all
k
- mkofcos
cos
each of the body’s coordinate
its a2
axes. When you represent a force
components, draw a wiggly line through the orig-
is the downhill acceleration.
The Identify / Set Up / the forces acting on the body. (The acceleration of a body is inal force vector to remind you not to include it twice.
Our sketch and free-body diagram (Fig. 5.23) are almost the EVALUATE: As for the frictionless toboggan in Example 5.10, the
determined by the forces that act on it, not by the forces that it 2. Make a list of all the known and unknown quantities. In your
same as for Example 5.16. The toboggan’s y-component of accel- acceleration doesn’t depend on the mass m of the toboggan. That’s
Execute / Evaluate format, used in exerts on anything else.) Make sure you can answer the ques-
tion “What other body is applying this force?” forand
drawn the downhill component of weight as a longer vector than force,
eachkinetic
force infriction
list, identify the target variable or variables.
eration ay is still zero but the x-component ax is not, so we’ve because all of the forces that act on the toboggan (weight, normal
3. For
force)each arebody, write a separate
proportional to m. equation for each component of
S
all Examples, encourages students the (uphill) friction force.
your diagram. Never
diagram; it’s not a force!
include the quantity ma in your
Let’s free-body
check some Newton’s
special
so that sin a = 1 and cosadditional
cases. If
a = 0, we
second law,is as
the hill
have athat
equations
in Eqs.(a(5.4).
vertical = 90°
x = g (the toboggan
In) addition, write any
you identified in step 4 of “Set Up.”
2. Label each force with an algebraic symbol for the force’s a
to tackle problems thoughtfully EXECUTE: It’s convenient to express the weight as w = mg. Then
Newton’s second law in component magnitude.
formUsually,
says one of the forces will be
falls freely).
the body’s
happens
For a certain (You
value need
of as many
the equations
acceleration asis there
zero;are target variables.)
this
if weight; 4. Do the easy part—the math! Solve the equations to find the tar-
it’s usually best to label this as = mg. get variable(s).
rather than skipping to the math. a Fx = mg 3. sin a + 1your
Choose - ƒkx-
2 = andma
y-coordinate
x
sin a = m cos a
axes for each body, and show k
and mk = tan a
EVALUATE your ans er: Does your answer have the correct units?
a Fy = n + 1 - mg cos a2 = 0 (When appropriate, use the conversion 1 N = 1 kg # m>s2.) Does it
them in its free-body diagram. Be sure to This indicate the positive
agrees with our result for the constant-velocity toboggan in
If the angle is even smaller, mk cos a is greater than
Focused direction for each axis. If you know the direction
5.23 Our sketches for this problem.
Example of the
a and axis
ation, it usually simplifies things to take onesinpositive
acceler-
5.16.
ax isalong
have the correct algebraic sign? When possible, consider particular
negative; if we give the toboggan an initial down-
values or extreme cases of quantities and compare the results with
that direction. If your problem involves hill
two or push bodies
more to startthat
it moving, it will slow down and stop. Finally, if
All Examples and Problem- (a) The situation (b) Free-body diagram for toboggan i t iti t ti
the hill is frictionless so that mk = 0, we retrieve the result of
A k “D thi lt k ?”
Example 5.10: ax = g sin a.
Solving Strategies are revised Notice that we started with a simple problem (Example 5.10)
and extended it to more and more general situations. The general
to be more concise and focused. result we found in this example includes all the previous ones as
special cases. Don’t memorize this result, but do make sure you
understand how we obtained it and what it means.
Visual Suppose instead we give the toboggan an initial push up the
hill. The direction of the kinetic friction force is now reversed, so
Most Examples employ a diagram— the acceleration is different from the downhill value. It turns out
that the expression for ax is the same as for downhill motion except
often a pencil sketch that shows that the minus sign becomes plus. Can you show this?
what a student should draw.
Ż
NEW! Video Tutor Solution for Every Example NEW! Mathematics Review Tutorials
Each Example is explained and solved by an instructor MasteringPhysics offers an extensive set of assignable mathematics
in a Video Tutor solution provided in the Study Area review tutorials—covering differential and integral calculus as well
of MasteringPhysics® and in the Pearson eText. as algebra and trigonometry.
, Build Confidence
NEW! Bridging Problems D evelop problem-solving confidence through a range
of practice options—from guided to unguided.
At the start of each problem set, a
Ż
Bridging Problem helps students
BRIDGING PROBLEM Billiard Physics
make the leap from routine A cue ball (a uniform solid sphere of mass m and radius R) is at 3. Draw two free-body diagrams for the ball in part (b): one show-
exercises to challenging problems rest on a level pool table. Using a pool cue, you give the ball a
sharp, horizontal hit of magnitude F at a height h above the center
ing the forces during the hit and the other showing the forces
after the hit but before the ball is rolling without slipping.
of the ball (Fig. 10.37). The force of the hit is much greater
with confidence and ease. than the friction force ƒ that the table surface exerts on the ball.
The hit lasts for a short time ¢t. (a) For what value of
4. What is the angular speed of the ball in part (b) just after the
hit? While the ball is sliding, does vcm increase or decrease?
Does v increase or decrease? What is the relationship between
h will the ball roll without slipping? (b) If you hit the ball dead vcm and v when the ball is finally rolling without slipping?
Each Bridging Problem poses a center 1h = 02, the ball will slide across the table for a while, but
eventually it will roll without slipping. What will the speed of its EXECUTE
5. In part (a), use the impulse–momentum theorem to find the
moderately difficult, multi-concept center of mass be then?
speed of the ball’s center of mass immediately after the hit.
Then use the rotational version of the impulse–momentum the-
SOLUTION GUIDE
problem, which often draws on earlier See MasteringPhysics® study area for a Video Tutor solution.
orem to find the angular speed immediately after the hit. (Hint:
To write down the rotational version of the impulse–momentum
chapters. In place of a full solution, IDENTIFY and SET UP
1. Draw a free-body diagram for the ball for the situation in part (a),
theorem, remember that the relationship between torque and
angular momentum is the same as that between force and linear
momentum.)
it provides a skeleton solution guide including your choice of coordinate axes. Note that the cue
exerts both an impulsive force on the ball and an impulsive
torque around the center of mass.
6. Use your results from step 5 to find the value of h that will
cause the ball to roll without slipping immediately after the hit.
consisting of questions and hints. 2. The cue force applied for a time ¢t gives the ball’s center of
mass a speed vcm, and the cue torque applied for that same
7. In part (b), again find the ball’s center-of-mass speed and
angular speed immediately after the hit. Then write Newton’s
time gives the ball an angular speed v. What must be the second law for the translational motion and rotational motion
of the ball as it is sliding. Use these equations to write
A full solution is explained in relationship between vcm and v for the ball to roll without
slipping? expressions for vcm and v as functions of the elapsed time
t since the hit.
a Video Tutor, provided in the 8. Using your results from step 7, find the time t when vcm and v
have the correct relationship for rolling without slipping. Then
10.37 find the value of vcm at this time.
Study Area of MasteringPhysics® EVALUATE
and in the Pearson eText. h mass m
9. If you have access to a pool table, test out the results of parts
(a) and (b) for yourself!
R
10. Can you show that if you used a hollow cylinder rather than a
solid ball, you would have to hit the top of the cylinder to
cause rolling without slipping as in part (a)?
Ż In response to professors, the Problem Sets now include more
biomedically oriented problems (BIO), more difficult problems
requiring calculus (CALC), and more cumulative problems that
draw on earlier chapters (CP).
About 20% of problems are new or revised. These revisions are
driven by detailed student-performance data gathered nationally
through MasteringPhysics.
Problem difficulty is now indicated by a three-dot ranking
system based on data from MasteringPhysics.
NEW! Enhanced End-of-Chapter Problems in MasteringPhysics
Select end-of-chapter problems will now offer additional support such
as problem-solving strategy hints, relevant math review and practice,
and links to the eText. These new enhanced problems bridge the gap
between guided tutorials and traditional homework problems.
Physics 13th Edition install download
https://ebookluna.com/product/ebook-pdf-university-physics-with-
modern-physics-13th-edition/
Download more ebook instantly today - Get yours now at ebookluna.com
,
,Build Skills
L earn basic and advanced skills that help
solve a broad range of physics problems.
Problem-Solving Strategies coach students in
how to approach specific types of problems.
Ż
Problem-Solving Strategy 5.2 Newton’s Second Law: Dynamics of Particles
IDENTIFY the relevant concepts: You have to use Newton’s second accelerate in different directions, you can use a different set of
law for any problem that involves forces acting on an accelerating axes for each body. S S
4. In addition to Newton’s second law, gF ⴝ ma, identify any
This text’s uniquely extensive set Ż body.
Identify the target variable—usually an acceleration or a force.
Example 5.17 Toboggan ride with friction II
other equations you might need. For example, you might need
If the target variable is something else, you’ll need to select another one or more of the equations for motion with constant accelera-
of Examples enables students concept to use. For example, suppose the target variable is how tion. If more than one body is involved, there may be relation-
The same toboggan with the same coefficient of friction as in From the second equation and Eq. (5.5) we get an expression for ƒk:
fast a sled is moving when it reaches the bottom of a hill. Newton’s ships among their motions; for example, they may be connected
to explore problem-solving Example 5.16 accelerates down a steeper hill. Derive an expres-
second law will let you find the sled’s acceleration; you’ll then use n = by
sion for the acceleration in terms of g, a, mk, and w.
mga cos
rope.a Express any such relationships as equations relating
the constant-acceleration relationships from Section 2.4 to find ƒk = the mknaccelerations
= mkmg cos of athe various bodies.
challenges in exceptional detail. S O LU T I O N
velocity from acceleration.
We substitute this into EXECUTEthe x-component
the solution and solve for ax:
as follows:
equation
SET UP the problem using the following steps: 1. For each body, determine the components of the forces along
IDENTIFY and SET UP: The toboggan is accelerating, so we must mg a + 1 - m mg a2 = max
Consistent 1. Draw a simple sketch of the situation that shows each moving
use Newton’s second law as given in Eqs. (5.4). Our target variable
body. For each body, draw a free-body diagram that shows
sin
= g1sinina terms
ax all
k
- mkofcos
cos
each of the body’s coordinate
its a2
axes. When you represent a force
components, draw a wiggly line through the orig-
is the downhill acceleration.
The Identify / Set Up / the forces acting on the body. (The acceleration of a body is inal force vector to remind you not to include it twice.
Our sketch and free-body diagram (Fig. 5.23) are almost the EVALUATE: As for the frictionless toboggan in Example 5.10, the
determined by the forces that act on it, not by the forces that it 2. Make a list of all the known and unknown quantities. In your
same as for Example 5.16. The toboggan’s y-component of accel- acceleration doesn’t depend on the mass m of the toboggan. That’s
Execute / Evaluate format, used in exerts on anything else.) Make sure you can answer the ques-
tion “What other body is applying this force?” forand
drawn the downhill component of weight as a longer vector than force,
eachkinetic
force infriction
list, identify the target variable or variables.
eration ay is still zero but the x-component ax is not, so we’ve because all of the forces that act on the toboggan (weight, normal
3. For
force)each arebody, write a separate
proportional to m. equation for each component of
S
all Examples, encourages students the (uphill) friction force.
your diagram. Never
diagram; it’s not a force!
include the quantity ma in your
Let’s free-body
check some Newton’s
special
so that sin a = 1 and cosadditional
cases. If
a = 0, we
second law,is as
the hill
have athat
equations
in Eqs.(a(5.4).
vertical = 90°
x = g (the toboggan
In) addition, write any
you identified in step 4 of “Set Up.”
2. Label each force with an algebraic symbol for the force’s a
to tackle problems thoughtfully EXECUTE: It’s convenient to express the weight as w = mg. Then
Newton’s second law in component magnitude.
formUsually,
says one of the forces will be
falls freely).
the body’s
happens
For a certain (You
value need
of as many
the equations
acceleration asis there
zero;are target variables.)
this
if weight; 4. Do the easy part—the math! Solve the equations to find the tar-
it’s usually best to label this as = mg. get variable(s).
rather than skipping to the math. a Fx = mg 3. sin a + 1your
Choose - ƒkx-
2 = andma
y-coordinate
x
sin a = m cos a
axes for each body, and show k
and mk = tan a
EVALUATE your ans er: Does your answer have the correct units?
a Fy = n + 1 - mg cos a2 = 0 (When appropriate, use the conversion 1 N = 1 kg # m>s2.) Does it
them in its free-body diagram. Be sure to This indicate the positive
agrees with our result for the constant-velocity toboggan in
If the angle is even smaller, mk cos a is greater than
Focused direction for each axis. If you know the direction
5.23 Our sketches for this problem.
Example of the
a and axis
ation, it usually simplifies things to take onesinpositive
acceler-
5.16.
ax isalong
have the correct algebraic sign? When possible, consider particular
negative; if we give the toboggan an initial down-
values or extreme cases of quantities and compare the results with
that direction. If your problem involves hill
two or push bodies
more to startthat
it moving, it will slow down and stop. Finally, if
All Examples and Problem- (a) The situation (b) Free-body diagram for toboggan i t iti t ti
the hill is frictionless so that mk = 0, we retrieve the result of
A k “D thi lt k ?”
Example 5.10: ax = g sin a.
Solving Strategies are revised Notice that we started with a simple problem (Example 5.10)
and extended it to more and more general situations. The general
to be more concise and focused. result we found in this example includes all the previous ones as
special cases. Don’t memorize this result, but do make sure you
understand how we obtained it and what it means.
Visual Suppose instead we give the toboggan an initial push up the
hill. The direction of the kinetic friction force is now reversed, so
Most Examples employ a diagram— the acceleration is different from the downhill value. It turns out
that the expression for ax is the same as for downhill motion except
often a pencil sketch that shows that the minus sign becomes plus. Can you show this?
what a student should draw.
Ż
NEW! Video Tutor Solution for Every Example NEW! Mathematics Review Tutorials
Each Example is explained and solved by an instructor MasteringPhysics offers an extensive set of assignable mathematics
in a Video Tutor solution provided in the Study Area review tutorials—covering differential and integral calculus as well
of MasteringPhysics® and in the Pearson eText. as algebra and trigonometry.
, Build Confidence
NEW! Bridging Problems D evelop problem-solving confidence through a range
of practice options—from guided to unguided.
At the start of each problem set, a
Ż
Bridging Problem helps students
BRIDGING PROBLEM Billiard Physics
make the leap from routine A cue ball (a uniform solid sphere of mass m and radius R) is at 3. Draw two free-body diagrams for the ball in part (b): one show-
exercises to challenging problems rest on a level pool table. Using a pool cue, you give the ball a
sharp, horizontal hit of magnitude F at a height h above the center
ing the forces during the hit and the other showing the forces
after the hit but before the ball is rolling without slipping.
of the ball (Fig. 10.37). The force of the hit is much greater
with confidence and ease. than the friction force ƒ that the table surface exerts on the ball.
The hit lasts for a short time ¢t. (a) For what value of
4. What is the angular speed of the ball in part (b) just after the
hit? While the ball is sliding, does vcm increase or decrease?
Does v increase or decrease? What is the relationship between
h will the ball roll without slipping? (b) If you hit the ball dead vcm and v when the ball is finally rolling without slipping?
Each Bridging Problem poses a center 1h = 02, the ball will slide across the table for a while, but
eventually it will roll without slipping. What will the speed of its EXECUTE
5. In part (a), use the impulse–momentum theorem to find the
moderately difficult, multi-concept center of mass be then?
speed of the ball’s center of mass immediately after the hit.
Then use the rotational version of the impulse–momentum the-
SOLUTION GUIDE
problem, which often draws on earlier See MasteringPhysics® study area for a Video Tutor solution.
orem to find the angular speed immediately after the hit. (Hint:
To write down the rotational version of the impulse–momentum
chapters. In place of a full solution, IDENTIFY and SET UP
1. Draw a free-body diagram for the ball for the situation in part (a),
theorem, remember that the relationship between torque and
angular momentum is the same as that between force and linear
momentum.)
it provides a skeleton solution guide including your choice of coordinate axes. Note that the cue
exerts both an impulsive force on the ball and an impulsive
torque around the center of mass.
6. Use your results from step 5 to find the value of h that will
cause the ball to roll without slipping immediately after the hit.
consisting of questions and hints. 2. The cue force applied for a time ¢t gives the ball’s center of
mass a speed vcm, and the cue torque applied for that same
7. In part (b), again find the ball’s center-of-mass speed and
angular speed immediately after the hit. Then write Newton’s
time gives the ball an angular speed v. What must be the second law for the translational motion and rotational motion
of the ball as it is sliding. Use these equations to write
A full solution is explained in relationship between vcm and v for the ball to roll without
slipping? expressions for vcm and v as functions of the elapsed time
t since the hit.
a Video Tutor, provided in the 8. Using your results from step 7, find the time t when vcm and v
have the correct relationship for rolling without slipping. Then
10.37 find the value of vcm at this time.
Study Area of MasteringPhysics® EVALUATE
and in the Pearson eText. h mass m
9. If you have access to a pool table, test out the results of parts
(a) and (b) for yourself!
R
10. Can you show that if you used a hollow cylinder rather than a
solid ball, you would have to hit the top of the cylinder to
cause rolling without slipping as in part (a)?
Ż In response to professors, the Problem Sets now include more
biomedically oriented problems (BIO), more difficult problems
requiring calculus (CALC), and more cumulative problems that
draw on earlier chapters (CP).
About 20% of problems are new or revised. These revisions are
driven by detailed student-performance data gathered nationally
through MasteringPhysics.
Problem difficulty is now indicated by a three-dot ranking
system based on data from MasteringPhysics.
NEW! Enhanced End-of-Chapter Problems in MasteringPhysics
Select end-of-chapter problems will now offer additional support such
as problem-solving strategy hints, relevant math review and practice,
and links to the eText. These new enhanced problems bridge the gap
between guided tutorials and traditional homework problems.
