Download - 制作 张昆实 Yangtze University
制作 张昆实制作 张昆实
Yangtze UniversityYangtze University
制作 张昆实制作 张昆实
Yangtze UniversityYangtze University
制作 张昆实制作 张昆实
Yangtze UniversityYangtze University
制作 张昆实制作 张昆实
Yangtze UniversityYangtze University
BilingualBilingual MechanicsMechanics BilingualBilingual MechanicsMechanics Chapter 2
MotionMotion
Chapter 2
MotionMotion
Chapter 2 Chapter 2 MotionMotion Chapter 2 Chapter 2 MotionMotion
2-1 What Is Physics
2-2 Vectors and Scalars
2-3 Multiplying Vectors
2-4 Motion
2-5 Position and Displacement
2-6 Average Velocity and Instantaneous
Velocity
2-7 Acceleration
2-1 What Is Physics
2-2 Vectors and Scalars
2-3 Multiplying Vectors
2-4 Motion
2-5 Position and Displacement
2-6 Average Velocity and Instantaneous
Velocity
2-7 Acceleration
Chapter 2 Chapter 2 MotionMotion Chapter 2 Chapter 2 MotionMotion
2-8 Constant Acceleration: A Special
Case
2-9 Graphical Integration of Motion
Analysis
2-10 Projectile Motion
2-11 Projectile Motion Analyzed
2-12 Uniform Circular Motion
2-13 Relative Motion
2-8 Constant Acceleration: A Special
Case
2-9 Graphical Integration of Motion
Analysis
2-10 Projectile Motion
2-11 Projectile Motion Analyzed
2-12 Uniform Circular Motion
2-13 Relative Motion
2-1 What Is Physics 2-1 What Is Physics
★ All objects in nature are in constant motion, nothing stands still absolutely.
★ All objects in nature are in constant motion, nothing stands still absolutely.
★ In this chapter we learn what is physicswhat is physics through the motion of particles
★ In this chapter we learn what is physicswhat is physics through the motion of particles
MechanicsMechanics KinematicsKinematics
DynamicsDynamics
Deals with the concepts discribing motion, without any reference to force.
Deals with the concepts discribing motion, without any reference to force.
Deals with the effect that forces have on motion . Deals with the effect that forces have on motion .
(chapter 2)(chapter 2)
(chapter 3)(chapter 3)
.
2-2 Vectors and Scalars2-2 Vectors and Scalars
★ Vectors are quantities that have both magnitude and direction and obey the vector law of addition.
★ Vectors are quantities that have both magnitude and direction and obey the vector law of addition.
★ Scalars are quantities that have magnitude only. ★ Scalars are quantities that have magnitude only.
length, time, masslength, time, mass , density, energy ,
temperature etc. are all scalars.
length, time, masslength, time, mass , density, energy ,
temperature etc. are all scalars.
Displacement, velocity, acceleration, Displacement, velocity, acceleration,
force, torque, force, torque, etc. are all vectors.
Displacement, velocity, acceleration, Displacement, velocity, acceleration,
force, torque, force, torque, etc. are all vectors.
.
2-2 Vectors and Scalars2-2 Vectors and Scalars
★Three notations of a vector★Three notations of a vector
a) unit-vector notationa) unit-vector notation
b) component notationb) component notation
c) magnitude-angle notationc) magnitude-angle notation
x ya a i a j
2 2x ya a a
cosxa a sinya a
tan y
x
a
a
a
xa
ya
x
y
o
2-3 Multiplying Vectors2-3 Multiplying Vectors
★ There are three ways in which vectors can be multiplied.
★ There are three ways in which vectors can be multiplied.
Multiplying a Vector by a ScalarMultiplying a Vector by a Scalar
Multiplying a Vector by a VectorMultiplying a Vector by a Vector
s a sa
a s
magnitudemagnitude
(new vector)(new vector)directiondirection
as )0(a s
as 0( )a s
1.1.
2. The Scalar Product2. The Scalar Product a b
3. The Vector Product3. The Vector Product a b
2-3 Multiplying Vectors2-3 Multiplying Vectors
Multiplying a Vector by a VectorMultiplying a Vector by a Vector
2. The Scalar Product ( dot product )2. The Scalar Product ( dot product ) a b
cosa b ab
a b c
3. The Vector Product ( cross product )3. The Vector Product ( cross product ) a b
“a dot b”:“a dot b”:
“a cross b”:“a cross b”:
sinc ab
(scalar)(scalar)
(vector)(vector)
magnitudemagnitude
Direction right hand screw ruleDirection right hand screw rule
c
a
b
c
2-4 Motion2-4 Motion
★ Objects move in one, two or three
dimension(s).
★ Objects move in one, two or three
dimension(s).
★The moving object is either a particle or an object that moves like a particle.
★The moving object is either a particle or an object that moves like a particle.
★To describ the motion of an object a
reference frame ( 参考系 ) must be chosen and a coordinate system ( 坐标系 ) must be constructed on it.
★To describ the motion of an object a
reference frame ( 参考系 ) must be chosen and a coordinate system ( 坐标系 ) must be constructed on it.
2-5 Position and Displacement2-5 Position and Displacement
★ Position vector a vector extends fromthe origin of a coordinate system to the particle
● unit vector
● scalar components
★ Position vector a vector extends fromthe origin of a coordinate system to the particle
● unit vector
● scalar components
r * P
x
y
z x
z
y
o
222 z yxr
magnitude :magnitude :
r xi y j zk , ,i j k
, ,x y z
(2-25)
★ Position vector
★ Position vector
rr * P
x
y
z x
z
y
o
r
direction :direction :
,r
xcos
rzcos,
r
ycos
222 z yxr
magnitude :magnitude :
2-5 Position and Displacement2-5 Position and Displacement
r xi y j zk
★Displacement
★Displacement
2 1r r r
r
x
y2
2r
1r 1
r
o2 1x x
2 1y y
1x 2x
2 2 2
1 1 1
( )
( )
r x i y j z k
x i y j z k
2 1 2 1 2 1( ) ( ) ( )r x x i y y j z z k
r xi y j zk
2-5 Position and Displacement2-5 Position and Displacement
(2-27)
(2-28)
(2-26)
P20 Sample Problem 2-4P20 Sample Problem 2-4
2-5 Position and Displacement2-5 Position and Displacement
(2-29)(2-29)
P20 Sample Problem 2-4P20 Sample Problem 2-4A rabbit runs across a parking plot on which a set of coor-dinate axes has been drawn. The coordinates of the rabbit’s position as function of time are given by
A rabbit runs across a parking plot on which a set of coor-dinate axes has been drawn. The coordinates of the rabbit’s position as function of time are given by
20.22 9.1 30y t t
20.31 7.2 28x t t (2-30)(2-30)
With in seconds and and in meters. With in seconds and and in meters. t
15t s
t
2(0.22)(15) (9.1)(15) 30 57y m
2( 0.31)(15) (7.2)(15) 28 66x m
x y(a) At , What is the rabbit’s position vector in unit-vector notation and in magnitude-angle notation?(a) At , What is the rabbit’s position vector in unit-vector notation and in magnitude-angle notation?
15t s r
( ) ( ) ( ) (66 ) (57 )r t x t i y t j m i m j unit-vector notationunit-vector notation
(Answer)(Answer)
SolutionSolution
2-5 Position and Displacement2-5 Position and Displacement
P20 Sample Problem 2-4P20 Sample Problem 2-4
(a) At , What is the rabbit’s position vector in magnitude-angle notation?(a) At , What is the rabbit’s position vector in magnitude-angle notation?
1 1 57tan tan
66
y m
x m
15t s r
41
87m
magnitude-angle notationmagnitude-angle notation
(Answer)(Answer)
2 2 2 2(66 ) ( 57 )r x y m m
2-5 Position and Displacement2-5 Position and Displacement
P20 Sample Problem 2-4P20 Sample Problem 2-4
(b) Graph the rabbit’s path for to(b) Graph the rabbit’s path for to 25 .t s0t
magnitude-angle notationmagnitude-angle notation
★ Average Velocity If a particle moves through a displacement in a timeInterval ,Then its
★ Average Velocity If a particle moves through a displacement in a timeInterval ,Then its
avg
rv
t
avg
x y zv i j k
t t t
rt
avgv
tr
Average displacement Velocity time intervalAverage displacement Velocity time interval
2-6 Average Velocity and Instantaneous Velocity2-6 Average Velocity and Instantaneous Velocity
(2-33)
(2-32)
★ Instantaneous Velocity
★ Instantaneous Velocity
0
dlim
dt
r rv
t t
avgv v
dx dy dzv i j k
dt dt dt
drdt
0t
0t 0r
v
The direction of is always tangent to the particle’s path at the particle’s position The direction of is always tangent to the particle’s path at the particle’s position
x y zv v i v j v k
2-6 Average Velocity and Instantaneous Velocity2-6 Average Velocity and Instantaneous Velocity
●
●
●
(2-34)
(2-35)
★ Instantaneous Velocity and its components
★ Instantaneous Velocity and its components
x
dxv
dt
y
dyv
dt
z
dzv
dt
Fig.2-18 The velocity of a particle, along with the scalar components of
Fig.2-18 The velocity of a particle, along with the scalar components of v
v
2-6 Average Velocity and Instantaneous Velocity2-6 Average Velocity and Instantaneous Velocity
x
y
o
vyv
xv
(2-36)
P22 Sample Problem 2-5P22 Sample Problem 2-5
(2-37)(2-37)
P22 Sample Problem 2-5P22 Sample Problem 2-5
For the rabbit in Sample Problem 2-4, Find the velocity at time , in unit-vector notation and in magnitude-angle notation.
For the rabbit in Sample Problem 2-4, Find the velocity at time , in unit-vector notation and in magnitude-angle notation.
2( 0.31 7.2 28) 0.62 7.2x
dx dv t t t
dt dt
(2-38)(2-38)
15t sv
2(0.22 9.1 30 0.44 9.1y
dy dv t t t
dt dt
unit-vector notationunit-vector notation
(Answer)(Answer)
2-6 Average Velocity and Instantaneous Velocity2-6 Average Velocity and Instantaneous Velocity
SolutionSolution
15t s 2.1 / ,xv m s 2.5 /yv m s( 2.1 / ) ( 2.5 / )v m s i m s j
taking derivetives of the components of to get the components oftaking derivetives of the components of to get the components of v
r
P22 Sample Problem 2-5P22 Sample Problem 2-5
Finding the velocity at time , in magnitude-angle notation.Finding the velocity at time , in magnitude-angle notation.
(2-38)(2-38)
15t sv
3.3 /m s
magnitude-angle notation
magnitude-angle notation
(Answer)(Answer)
2-6 Average Velocity and Instantaneous Velocity2-6 Average Velocity and Instantaneous Velocity
SolutionSolution2 2x yv v v
1 1 2.5 /tan tan
2.1 /y
x
v m s
v m s
The magnitude of :The magnitude of :v
1tan 1.19 130
2 2( 0.21 / ) ( 0.25 / )m s m s
3.3 /m s
(in the third quadrant)(in the third quadrant)
★ Average AccelerationWhen a particle’s velocitychanges from to in a time interval ,Then its
★ Average AccelerationWhen a particle’s velocitychanges from to in a time interval ,Then its
1v
2v
t
x
y
O
1v
t1
2v
t2
v
1v
2v
yx zavg
vv vva i j k
t t t
t
2 1avg
v v va
t t
2-7 Acceleration2-7 Acceleration
average change in velocityacceleration time intervalaverage change in velocityacceleration time interval
(2-39)
★ Instantaneous acceleration
★ Instantaneous acceleration
0
dlim
dta
t t
v v
0t
2-7 Acceleration2-7 Acceleration
(2-40)
(2-41)
xx
dva
dt y
y
dva
dt z
z
dva
dt (2-42)
P23 Sample Problem 2-6P23 Sample Problem 2-6
dd d
d d dyx za i j k
t t t vv v
x y za a i a j a k
P23 Sample Problem 2-6P23 Sample Problem 2-6
For the rabbit in Sample Problem 2-4 and 2-5, Find the acceleration at time , in unit-vector notation and in magnitude-angle notation.
For the rabbit in Sample Problem 2-4 and 2-5, Find the acceleration at time , in unit-vector notation and in magnitude-angle notation.
2( 0.62 7.2) 0.62 /xx
dv da t m s
dt dt
15t sa
2(0.44 9.1) 0.44 /yy
dv da t m s
dt dt
unit-vector notationunit-vector notation
(Answer)(Answer)
SolutionSolution
2 2( 0.62 / ) (0.44 / )a m s i m s j
2-7 Acceleration2-7 Acceleration
taking derivetives of the components of to get the components oftaking derivetives of the components of to get the components of a
v
P23 Sample Problem 2-6P23 Sample Problem 2-6
in magnitude-angle notation.in magnitude-angle notation.
SolutionSolution
2-7 Acceleration2-7 Acceleration
magnitude-angle notation magnitude-angle notation
(Answer)(Answer)
2 2x ya a a
2 2 2 2( 0.62 / ) (0.44 / )m s m s
20.76 /m s
1 1 0.44 /tan tan 145
6.2 /y
x
a m s
a m s
(in the second quadrant)(in the second quadrant)
The magnitude of :The magnitude of :a
0
0avg
v va a
t
0v v at (2-43)
(2-44)
0
0avg
x xv
t
0 avgx x v t
102 ( )avgv v v (2-45)
10 2avgv v at (2-46)
(2-47)21
0 0 2x x v t at
★ In many cases the acceleration is constant
★ In many cases the acceleration is constant
The average velocityThe average velocity
2-8 Constant Acceleration: A Special Case2-8 Constant Acceleration: A Special Case
Eq. 2-43 and 2-47: are the basic equations for constant acceleration.
Eq. 2-43 and 2-47: are the basic equations for constant acceleration.
Recast this equation:Recast this equation:
SimilarlySimilarly
Substituting 2-43 for :Substituting 2-43 for :
Substituting 2-46 into 2-44 :Substituting 2-46 into 2-44 :
v
0v v at (2-43) (2-47)21
0 0 2x x v t at
2 20 02 ( )v v a x x (2-48)
(2-49)10 02 ( )x x v v t
(2-50)21
0 2x x vt at
2-8 Constant Acceleration: A Special Case2-8 Constant Acceleration: A Special Case
Eliminate :Eliminate :t
Eliminate : Eliminate :
a
Eliminate :Eliminate :0v
the basic equations 2-43 and 2-47 for constant acceleration can be combined in three ways to yield additional equations:
the basic equations 2-43 and 2-47 for constant acceleration can be combined in three ways to yield additional equations:
P26 Table 2-1
Eq. 2-43 and 2-47 can be ontained by integration
Eq. 2-43 and 2-47 can be ontained by integration
(2-43)
dv adt
dv adt a dt Indefinite integralIndefinite integral
v at C To determine constant C, let
To determine constant C, let 0t
0v v 0C v
0v v at
dx vdtIndefinite integralIndefinite integral
0( )dx vdt v at dt 21
0 2x v t at C
To determine constant C’, letTo determine constant C’, let 0t
0x x 0C x
(2-47)21
0 0 2x x v t at
2-8 Constant Acceleration: A Special Case2-8 Constant Acceleration: A Special Case
Toss an object either up or down (eliminate the air resistance) the object accelerates downward at a certain constant rate: free-fall acceleration
Toss an object either up or down (eliminate the air resistance) the object accelerates downward at a certain constant rate: free-fall acceleration g
2-8 Constant Acceleration: A Special Case2-8 Constant Acceleration: A Special Case
The value of varies slightly with latitude and with elevation. The value of varies slightly with latitude and with elevation.
*** Note: For free fall ***
(2) The free-fall acceleration, being downward, is now negative!
(2) The free-fall acceleration, being downward, is now negative!
29.8 /a g m s
(1) The directions of motion are now along a verticle axis with the positive direction of upward.(1) The directions of motion are now along a verticle axis with the positive direction of upward.y y
g
0v v at (2-43)
(2-47)
210 0 2x x v t at
2 20 02 ( )v v a x x
(2-48)
Horizontal motion Vertical motion Horizontal motion Vertical motion
0v v gt (2-43)
210 0 2y y v t gt
(2-47)
2-8 Constant Acceleration: A Special Case2-8 Constant Acceleration: A Special Case
2 20 02 ( )v v g y y
(2-48)
2-9 Graphical Integration of Motion Analysis2-9 Graphical Integration of Motion Analysis
Using a graph of an object, the object’s can be found by integrating on the graph:Using a graph of an object, the object’s can be found by integrating on the graph:
a tv
a dv dt1
01 0
t
tv v adt
1
0
t
tadt Area between acceleration curve
and time axis, from toArea between acceleration curve
and time axis, from to0t 1t
1t0t
Area
t
a When the acceleration curve is above the time axis, the area is positive;When the acceleration curve is above the time axis, the area is positive;
When the acceleration curve is below the time axis, the area is negative;When the acceleration curve is below the time axis, the area is negative;
(definite integral)(definite integral)
2-9 Graphical Integration of Motion Analysis2-9 Graphical Integration of Motion Analysis
Using a graph of an object, the object’s can be found by integrating on the graph:Using a graph of an object, the object’s can be found by integrating on the graph:
v tx
v dx dt1
01 0
t
tx x vdt
1
0
t
tvdt Area between velocity curve
and time axis, from toArea between velocity curve
and time axis, from to0t 1t
1t0t
Area
t
v When the velocity curve is above the time axis, the area is positive;When the velocity curve is above the time axis, the area is positive;
(definite integral)(definite integral)
When the velocity curve is belowthe time axis, the area is negative;When the velocity curve is belowthe time axis, the area is negative;
★ Projectile Motion : A particle moves in a vertical planewith some initial velocity and an angle with respect to the horizontal axis but its acceleration is always the free-fall acceleration , which is downward.
★ Projectile Motion : A particle moves in a vertical planewith some initial velocity and an angle with respect to the horizontal axis but its acceleration is always the free-fall acceleration , which is downward.
00v
g
0d
y
0v
xv
yv v
xv
yv vx
o0xv
0yv
0
0 0 0cosxv v 0 0 0sinyv v 0 0 0x yv v i v j
(2-58)
(2-59)
2-10 Projectile Motion2-10 Projectile Motion
Fig. 2-29 The projectile bullet always hits the falling coconutFig. 2-29 The projectile bullet always hits the falling coconut
2-10 Projectile Motion2-10 Projectile Motion
★ In projectile motion , the horizontal motion andthe vertical motion are independent of each other.
★ In projectile motion , the horizontal motion andthe vertical motion are independent of each other.
0xa
xv
yv
v
xv
yv
v0dx
y
o
0v
x0v
y0v
0 0 0 0( cos )xx x v t v t (2-60)
0xa ★ the horizontal motion:★ the horizontal motion:
2-11 Projectile Motion Analyzed2-11 Projectile Motion Analyzed
★ the vertical motion:
★ the vertical motion:
0xa xv
yv
v
xv
yv
v0dx
y
o
0v
x0v
y0v
210 0 2yy y v t gt
(2-61)
ya g
210 0 2( sin )v t gt
0 0sinyv v gt 2 2
0 0 0( sin ) 2 ( )yv v g y y (2-62)
(2-63)
2-11 Projectile Motion Analyzed2-11 Projectile Motion Analyzed
★ The Equation of the Path★ The Equation of the Path
0 0 0 0( cos )xx x v t v t 21
0 0 2yy y v t gt 210 0 2( sin )v t gt (2-61)
(2-60)
Let Solving Eq.2-60 for t and
Substituting into Eq.2-61:
Let Solving Eq.2-60 for t and
Substituting into Eq.2-61:0 0,x 0 0y
This is the equation of a parabola, so the path (trajectory) is parabolic.This is the equation of a parabola, so the path (trajectory) is parabolic.
2-11 Projectile Motion Analyzed2-11 Projectile Motion Analyzed
2
0 20
(tan )2( cos )
gxy x
0v(2-64)(trajectory)(trajectory)
The Horizontal Range: the horizontal distance theProjectile has traveled when it returns to its initial(launch) height.
The Horizontal Range: the horizontal distance theProjectile has traveled when it returns to its initial(launch) height.
0 0 0( cos )R x x v t
00 y y 210 0 2( sin )v t gt (2-61)
(2-60)
Eliminating t between these two equations yieldsEliminating t between these two equations yields2 20 0
0 0 0
2sin cos sin 2
v vR
g g (2-65)
The horizontal distance R is maximum for a launch angle of The horizontal distance R is maximum for a launch angle of 045
0sin 2 1 002 90 0
0 45
2-11 Projectile Motion Analyzed2-11 Projectile Motion Analyzed
The Effects of the Air (Air resistance):In vacuum: thepath(trajectory) is parabolic.The Effects of the Air (Air resistance):In vacuum: thepath(trajectory) is parabolic.
x
y
o0d
In Air In vacuum
d
In air: the horizontal range, the maximum
height of the path are much less.
In air: the horizontal range, the maximum
height of the path are much less. Air resistance force:
density of air, the cross section area of the projectile, mainly the velocity of the body.
Air resistance force:
density of air, the cross section area of the projectile, mainly the velocity of the body.
200 / ,v m s 2f v
2-11 Projectile Motion Analyzed2-11 Projectile Motion Analyzed
Sample Problem 2-8 (P32)Sample Problem 2-8 (P32)
Figure 2-32 shows a pirate ship 560m from a fort defending the harbor entrance of an island. A defense canno, located at sea level, fires balls at initial speed .
Figure 2-32 shows a pirate ship 560m from a fort defending the harbor entrance of an island. A defense canno, located at sea level, fires balls at initial speed .0 82 /v m s
2-11 Projectile Motion Analyzed2-11 Projectile Motion Analyzed
(a) at what angle from the horizontal must a ball be fired to hit the ship?
(a) at what angle from the horizontal must a ball be fired to hit the ship?
0
(b) How far should the pirate ship be from the canno if it is to be beyond the maximum range of the cannoballs?
(b) How far should the pirate ship be from the canno if it is to be beyond the maximum range of the cannoballs?
Sample Problem 2-8 (P32)Sample Problem 2-8 (P32)
2-11 Projectile Motion Analyzed2-11 Projectile Motion Analyzed
(a) at what angle from the horizontal must a ball be fired to hit the ship? (a) at what angle from the horizontal must a ball be fired to hit the ship?
0
SolutionSolution The fired cannoball is a projectileThe fired cannoball is a projectile
(2-65)
0 82 /v m s 560R mWe know:We know:20
0sin 2v
Rg
(2-65)
Which gives usWhich gives us
1 10 2
0
2 sin sin 0.816gR
v
0 02 54.7 2 125.3and
0 027 63and
Solution: The maximum range corresponds to an elevation angle ofSolution: The maximum range corresponds to an elevation angle of
2-11 Projectile Motion Analyzed2-11 Projectile Motion Analyzed
(b) How far should the pirate ship be from the canno if it is to be beyond the maximum range of the cannoballs?
(b) How far should the pirate ship be from the canno if it is to be beyond the maximum range of the cannoballs?
0 45
20
0sin 2v
Rg
(2-65)
Sample Problem 2-8 (P32)Sample Problem 2-8 (P32)
2
2
(82 / )
9.8 /
m s
m s
686 690m m
★ Uniform circular Motion : A particle travel around a circle or a circular arc at constant (uniform) speed. The velocity changes only in direction, there are still an acceleration– centripetal acceleration.
★ Uniform circular Motion : A particle travel around a circle or a circular arc at constant (uniform) speed. The velocity changes only in direction, there are still an acceleration– centripetal acceleration.
2va
r (2-70)
(2-71)2 r
Tv
Period: the time for going around a circle exactly once(circumference of the circle)
Period: the time for going around a circle exactly once(circumference of the circle)
2-12 Uniform Circular Motion2-12 Uniform Circular Motion
Fig.2-34
2-13 Relative Motion2-13 Relative Motion
In three dimensions:Two observers are watchinga moving particle P from theorigins of frames A and B,while B moves at , Thecorresponding axes of frame
A and B remain parallel position vecter: B to A :
P to A :
P to B :(4-41)
Frame A
y
P
Frame B
y
x
x
d
dtd
dt
Frame A
y
P
Frame B
y
x
x
(4-41)
d
dt
Take the time derivativeTake the time derivative
Get :
(4-42)
Take the time derivativeTake the time derivative
Since:Since: (4-43)
The acceleration of the particle measured from frames A and B are the same! The acceleration of the particle measured from frames A and B are the same!
2-13 Relative Motion2-13 Relative Motion
(4-38)
d
dtd
dt
d
dt
(4-42)
Since:
(4-43)
The acceleration of the particle measured from frames A and B are the same! The acceleration of the particle measured from frames A and B are the same!
in Two or Three Dimensions in One Dimension
(4-41)
d
dtd
dt
d
dt
(4-39)
Since:
(4-40)
2-13 Relative Motion2-13 Relative Motion