동역학
(Dynamics)
Kinematics of Particles: Rectilinear Motion of Particles
윤 헌 준 교수
숭실대학교 기계공학부
Prof. Heonjun Yoon
School of Mechanical Engineering, Soongsil University - Dynamics
Kinematics of Particles: Rectilinear Motion of Particles
<Review> Overview of Dynamics (1/2)
 Dynamics = Kinematics + Kinetics.
Kinetics
∑ F = ma
F.B.D
Kinematics
a
a=
dv
dt
Integration
Differentiation
v = ∫ adt
v=
dr
dt
Integration
Differentiation
r = ∫ vdt
r
 ( Kinematics ): How is it moving?
→ Only motions including ( position ), ( velocity ), ( acceleration ), ( orientation ), ( angular
velocity ), and ( angular acceleration ) are considered with respect to ( time ).
→ The ( geometry ) of motion without reference to the ( cause ) of the motion.
 ( Kinetics ): Why is it moving?
→ The relationship between the ( forces ) acting on a body, the ( mass ) of the body, and the
( motion ) of the body is studied.
Prof. Heonjun Yoon
School of Mechanical Engineering, Soongsil University - Dynamics
-2-
Kinematics of Particles: Rectilinear Motion of Particles
<Review> Overview of Dynamics (2/2)
 ( Particle ): A mass point that possesses a mass but has no ( size )
or ( shape ).
→ To consider only its ( motion ) as an entire unit.
→ To neglect any ( rotation ) about its own centers of mass.
 ( Rigid Body ): A body is said to be ( rigid ) if the distance
between any two points in the body remains ( constant ).
Particle motion of Earth
→ The laws of motion of a rigid body are the same as those of a ( particle ) if the rigid body
undergoes only ( translational ) motion.
Rectilinear translation
Prof. Heonjun Yoon
Curvilinear translation
School of Mechanical Engineering, Soongsil University - Dynamics
Rotation about a fixed axis
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Kinematics of Particles: Rectilinear Motion of Particles
Measure of Space and Time: Dimensions and Units (1/2)
 In this course, space and time are described strictly in the ( Newtonian ) sense.
 Philosophiæ Naturalis Principia Mathematica (Isaac Newton, 1687)
→ “Absolute, true and mathematical ( time ), of itself, and from its own
nature, flows equably, without relation to anything external.”
→ “Absolute ( space ), in its own nature, without relation to anything
external, remains always similar and immovable.“
 The unit of time
→ Time is measured relative to the ( duration ) of reoccurrences of a given configuration of a cyclical
system.
→ The ( second ) is the duration of 9,192,631,770 cycles of the radiation corresponding to the
transition between two hyperfine levels of the cesium-133 atom.
 The unit of length
→ The ( meter ) is defined as 1,650,763.73 wavelengths of the orange-red light corresponding to a
certain transition in an atom of krypton-86.
→ Since 1983, the ( meter ) is the distance light travels in 1/299,792,458 s in a vacuum.
Prof. Heonjun Yoon
School of Mechanical Engineering, Soongsil University - Dynamics
-4-
Kinematics of Particles: Rectilinear Motion of Particles
Measure of Space and Time: Dimensions and Units (2/2)
 The unit of mass
→ The ( kilogram ) is defined as the mass of a platinum-iridium standard kept at the International
Bureau of Weights and Measures at Sèvres, near Paris, France.
→ The units of length and time are based on ( atomic ) standards, but the unit of mass is not.
 No more than three fundamental quantities (mass, length, and time) are needed to completely
describe or characterize the behavior of any physical system.
→ The ( space ) that bodies occupy.
→ The ( matter ) of which they consist.
→ The ( time ) during which those bodies move.
→ In other words, ( dynamics ) deals with the motion
of physical objects through space and time.
 Mass, length, and time specify the three primary ( dimensions ) of all physical quantities.
→ Do not confuse the dimension of a quantity with the units chosen to express it!
−2
−3
γ
[a] [=
[ M ]α [ L ]β [T ]=
L ][T ] , [ ρ ] [ M ][ L ]
Prof. Heonjun Yoon
School of Mechanical Engineering, Soongsil University - Dynamics
-5-
Kinematics of Particles: Rectilinear Motion of Particles
Rectilinear motion
 A particle moves along a ( straight ) line.
 Position, velocity, and acceleration can be treated as ( scalar )
quantities.
 In rectilinear motion, the velocity has a ( known ) and ( fixed )
direction.
→ We need only to specify its ( magnitude ).
Rectilinear motion
Curvilinear motion
 A particle moves along a ( curved ) line in two or three
dimensions.
 Position, velocity, and acceleration must be treated as ( vectors ).
 Some curvilinear motions can be represented as a superposition of
independent ( rectilinear ) motions.
Prof. Heonjun Yoon
School of Mechanical Engineering, Soongsil University - Dynamics
Curvilinear motion
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Kinematics of Particles: Rectilinear Motion of Particles
Position Coordinate
 The position can be defined only if a ( reference ) point is
specified.
 The position can be treated as a ( scalar ) quantity
with ( plus ) or ( minus ) signs.
 Step 1: Choose a fixed origin O on the ( straight ) line.
 Step 2: Set a ( positive ) direction along the line.
Position measured from a fixed origin
 When we know the position coordinate x of a particle for every value of time t, we say that
the motion of the particle is ( known ).
 Examples:
Running
Prof. Heonjun Yoon
Eccentric circular cam
School of Mechanical Engineering, Soongsil University - Dynamics
Piston that travels in a cylinder
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Kinematics of Particles: Rectilinear Motion of Particles
Velocity of a Particle Along a Line
 During the time interval, the ( average ) velocity is defined as:
=
vavg
∆x
 m ⋅ s −1 
∆t
A small displacement
 By taking the limit as the time interval approaches zero, the ( instantaneous ) velocity is defined as:
∆x
∆t →0 ∆t
v = lim
=
v
dx
= x  m ⋅ s −1 
dt
 If a particle moves from A to B, the ( average ) velocity is constant but the ( instantaneous ) velocity
continuously changes.
 The ( magnitude ) of the velocity is the ( speed ).
 The velocity direction is the same with the ( motion ) direction.
 The velocity can be treated as a ( scalar ) quantity
with ( plus ) or ( minus ) signs.
Prof. Heonjun Yoon
School of Mechanical Engineering, Soongsil University - Dynamics
Velocity direction
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Kinematics of Particles: Rectilinear Motion of Particles
Acceleration of a Particle Along a Line (1/2)
 For the time interval, the ( average ) acceleration is defined as:
aavg
=
∆v
 m ⋅ s −2 
∆t
 By taking the limit as the time interval approaches zero,
A change in velocity
the ( instantaneous ) acceleration is defined as:
∆v
a = lim
∆t →0 ∆t
dv
d 2x
−2



a=
= v=
=
x
m
⋅
s
2

dt
dt
 Another expression for the acceleration:
=
a
dv dx dv
dv
 m ⋅ s −2 
=
= v
dt dt dx
dx
→ It is useful when the acceleration is a given function of the ( position ).
Prof. Heonjun Yoon
School of Mechanical Engineering, Soongsil University - Dynamics
-9-
Kinematics of Particles: Rectilinear Motion of Particles
Acceleration of a Particle Along a Line (2/2)
 A positive value of a: The velocity ( increases ).
→ The particle is moving ( faster ) in the positive direction.
→ Or, the particle is moving more ( slowly ) in the negative direction.
 A negative value of a: The velocity ( decreases ).
→ The particle is moving more ( slowly ) in the positive direction.
→ Or, the particle is moving ( faster ) in the negative direction.
Prof. Heonjun Yoon
School of Mechanical Engineering, Soongsil University - Dynamics
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Kinematics of Particles: Rectilinear Motion of Particles
Summary of Kinematic Variables
 If a kinematic variable is given as a function of time, the other variables can be found.
x = h(t )
dh(t )
v=
dt
v = g (t=
)
x
v
a = f (=
t)
Position
Prof. Heonjun Yoon
∫
t
∫
t
t0 = 0
t0 = 0
d 2 h(t )
a=
dt 2
g (t )dt + x0
a=
f (t )dt + v0 =
x
dg (t )
dt
∫
t
t0 = 0
vdt + x0
Velocity
School of Mechanical Engineering, Soongsil University - Dynamics
Acceleration
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Kinematics of Particles: Rectilinear Motion of Particles
Concept Application
 Consider a particle moving in a ( straight ) line.
 The position is defined by:=
x
6t 2 − t 3
→ t is in seconds.
→ x is in meters.
 The velocity by differentiating the position with respect to time as:
v= x= 12t − 3t 2
 The acceleration by differentiating the velocity with respect to time as:
a= v= 12 − 6t
 The total distance traveled from t = 0 to t = 6 s is ( 64 ) m.
Position
Prof. Heonjun Yoon
Velocity
School of Mechanical Engineering, Soongsil University - Dynamics
Acceleration
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Kinematics of Particles: Rectilinear Motion of Particles
Determination of Motion of a Particle (1/2)
 Case 1: The acceleration a is a given function of t.
dv
a=
dt
∫
dv = f (t )dt
v
t
dv = ∫
∴ v = v0 + ∫
t
t0 = 0
∫
dx = vdt
x
x0
dx = ∫
t
t0 = 0
v − v0 =
∫ f (t )dt
f (t )dt
t0 = 0
v0
t
t0 = 0
f (t )dt
( Initial ) values
t
x − x0 =
∫ vdt
vdt
t0 = 0
x x0 + ∫
=
t
t0 = 0
vdt
 Case 2: The acceleration a is a given function of x.
dv
a=v
dx
vdv = f ( x)dx
∴ v=
dt =
Prof. Heonjun Yoon
dx
v
∫
t
t0 = 0
∫
v
v0
vdv = ∫ f ( x)dx
x
x0
x
1 2 2
v − v0 ) = ∫x f ( x)dx
(
0
2
x
v + 2 ∫ f ( x)dx
2
0
1
dx =
t
x0 v
dt = ∫
x
x0
∫
x
x0
{
x
}
v02 + 2 ∫ f ( x)dx
x0
School of Mechanical Engineering, Soongsil University - Dynamics
−
1
2
dx
- 13 -
Kinematics of Particles: Rectilinear Motion of Particles
Determination of Motion of a Particle (2/2)
 Case 3: The acceleration a is a given function of v.
dv
a=
dt
dv
dt =
f (v )
dv
a=v
dx
vdv
dx =
f (v )
∫
t
∫
x
t0 = 0
x0
dt = ∫
v
v0
dx = ∫
dv
f (v )
vdv
f (v )
v
v0
t=∫
v
v0
dv
f (v )
=
x x0 + ∫
v
v0
vdv
f (v )
 Case 4: The velocity v is a given function of x.
=
a
dv dv dx
dv
dg ( x)
=
= v = g ( x)
dt dx dt
dx
dx
dx
v=
dt
Prof. Heonjun Yoon
dx
dt =
g ( x)
∫
t
t0 = 0
dt = ∫
x
x0
dx
g ( x)
t=∫
School of Mechanical Engineering, Soongsil University - Dynamics
x
x0
dx
f ( x)
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Kinematics of Particles: Rectilinear Motion of Particles
Uniform Rectilinear Motion
 The acceleration of the particle is ( zero ) for every value of time.
 The velocity is ( constant ).
v
=
dx
= const.
dt
 The position of the particle:
x
=
∫ dx
t
vdt v ∫
∫=
t
=
x0
t0 0 =
t0 0
dt
x − x0 =
vt
=
x x0 + vt
 Example: Steady flight (unaccelerated flight or equilibrium flight)
→ A special case in flight dynamics where the aircraft’s ( linear )
and ( angular ) velocities are ( constant ) in a body-fixed
reference frame.
→ An aircraft is in steady flight when all forces are ( balanced ).
Forces in flight dynamics
Prof. Heonjun Yoon
School of Mechanical Engineering, Soongsil University - Dynamics
- 15 -
Kinematics of Particles: Rectilinear Motion of Particles
Uniformly Accelerated Rectilinear Motion
 The acceleration of the particle is ( constant ) for every value of time.
=
a
dv
= const.
dt
 The velocity of the particle:
v
=
∫ dv
t
adt a ∫
∫=
t
=
v0
t0 0 =
t0 0
dt
v − v0 =
at
v= v0 + at
 The position of the particle:
x
=
∫ dx
x0
t
∫t =0 ( v0 + at ) dt
0
1
x − x0 = v0t + at 2
2
1
x = x0 + v0t + at 2
2
 The velocity of the particle with respect to the ( position ):
v
vdv
∫=
v0
Prof. Heonjun Yoon
x
x
x0
x0
adx a ∫ dx
∫=
1 2 2
a ( x − x0 )
v − v=
(
0 )
2
School of Mechanical Engineering, Soongsil University - Dynamics
v2 =
v02 + 2a ( x − x0 )
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Kinematics of Particles: Rectilinear Motion of Particles
Uniformly Accelerated Rectilinear Motion - Free Fall
 Velocity and elevation:
dv = adt
dy = vdt
 The highest elevation:
∫
v
∫
y
10
20
dv
=
t
∫ (−9.81)dt
0
=
v 10 − 9.81t
t
dy =
− ∫ (10 − 9.81t )dt
v=
10 − 9.81t =
0
0
y = 20 + 10t − 4.905t 2
t = 1.019s
ymax =
20 + 10(1.019) − 4.905(1.019) 2 =
25.1m
 Time and velocity when the ball hits the ground:
y=0
−1.243s or t =
+3.28s
t=
−22.2m/s
v=
10 − 9.81(3.28) =
Prof. Heonjun Yoon
School of Mechanical Engineering, Soongsil University - Dynamics
- 17 -
Kinematics of Particles: Rectilinear Motion of Particles
Motion of Several Particles
 When several particles move ( independently ) along the ( same ) line, we can write ( independent )
equations of motion for each particle.
 Consider two particles moving along the ( same )
straight line.
 The ( relative ) coordinate of B with respect to A:
xB /=
xB − x A
A
→ xB / A
→ xB / A
> 0 : B is to the (
< 0 : B is to the (
right ) of A.
Example of motion of several particles
left ) of A.
 The ( relative ) velocity of B with respect to A:
v B /=
vB − v A
A
→ vB / A
→ vB / A
Prof. Heonjun Yoon
Two particles in motion
> 0 : B is observed from A to move in the (
< 0 : B is observed from A to move in the (
positive ) direction.
negative ) direction.
School of Mechanical Engineering, Soongsil University - Dynamics
- 18 -
Kinematics of Particles: Rectilinear Motion of Particles
Dependent Motion of Particles (1/3)
 Sometimes, the position of a particle ( depends ) upon the
position of another particle.
 This ( dependency ) commonly occurs if the particles are
interconnected by ( inextensible ) cords which are wrapped
around pulleys.
 Each of the coordinate axes is measured from a ( fixed ) point or ( fixed ) datum line.
 Each of the coordinate axes is measured along each inclined plane in the direction of ( motion )
of each block.
 Each of the coordinate axes has a ( positive ) sense from the fixed datums to A and to B.
s A + lCD + sB =
lT
ds A dsB
dl
 dl

+
= 0,  CD = T = 0 
dt
dt
dt
 dt

Prof. Heonjun Yoon
vB = − v A
School of Mechanical Engineering, Soongsil University - Dynamics
aB = − a A
- 19 -
Kinematics of Particles: Rectilinear Motion of Particles
Dependent Motion of Particles (2/3)
 The position coordinates have their ( origin ) at fixed datums.
 The position coordinates from the fixed datums are positive to the
( right ) for sA and positive ( downward ) for sB.
 During the motion, the length of the red colored segments of the
cord remains ( constant ).
2 sB + h + s A =
l
2
dsB ds A
 dh dl

+
=0, 
= =0 
dt
dt
 dt dt

2v B = − v A
2aB = − a A
Prof. Heonjun Yoon
School of Mechanical Engineering, Soongsil University - Dynamics
- 20 -
Kinematics of Particles: Rectilinear Motion of Particles
Dependent Motion of Particles (3/3)
Prof. Heonjun Yoon
One degree of freedom
Two degrees of freedom
x A + 2 xB =
const.
2 x A + 2 xB + xC =
const.
2v B = − v A
2v A + 2vB + vC =
0
School of Mechanical Engineering, Soongsil University - Dynamics
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