OF A PARTICLE
MTE 212/BME
212:
DYNAMICS
CHP.1
KINEMATICS OF A
PARTICLE
Dr. S. ISMAIL
12.1 Introduction
Statics deals with
the equilibrium of
bodies; those that
are either at rest or
move with a
constant velocity.
Mechanics is the branch of the physical
sciences concerned with the state of rest
or motion of bodies that are subjected to
the action of forces.
Dynamics is
concerned with the
accelerated motion
of bodies.
Statics
Kinematics
Rigid-body
Dynamics
Mechanics
Deformable-body
Fluid
Kinetics
Kinematics treats
only the geometric
aspects of the
motion.
Kinetics is the
analysis of the
forces causing the
motion.
2
12.1 Introduction
■ 1. Read the problem carefully and try to correlate the actual physical situation with
the theory you have studied.
■ 2. Draw any necessary diagrams and tabulate the problem data.
■ 3. Establish a coordinate system and apply the relevant principles, generally in
mathematical form.
■ 4. Solve the necessary equations algebraically as far as practical; then, use a
consistent set of units and complete the solution numerically. Report the answer with
no more significant figures than the accuracy of the given data.
■ 5. Study the answer using technical judgment and common sense to determine
whether or not it seems reasonable.
■ 6. Once the solution has been completed, review the problem. Try to think of other
ways of obtaining the same solution.
12.2 Rectilinear kinematics: continuous
motion
■ Kinematics of a particle that moves along a
rectilinear or straight-line path.
■ A particle has a mass but negligible size and shape.
■ Limit application to objects that have dimensions
that are of no consequence in the analysis of the
motion: the motion is characterized by the motion of
the object’s mass center and any rotation of the
body is neglected.
12.2 Rectilinear kinematics: continuous
motion
Rectilinear kinematics
■ The kinematics of a particle is characterized by specifying, at any given instant, the particle’s
position, velocity, and acceleration.
1- Position
-
+
Coordinate axis s
Origin on the path (fixed point) O
Position vector r (algebraic scalar s)
Particle P
r is along the axis s-> the direction of r never changes. What changes is its magnitude (m) and
its sense of direction.
12.2 Rectilinear kinematics: continuous
motion
Rectilinear kinematics
2- Displacement
The displacement of the particle is defined as the change in its
position:
∆𝑠 = 𝑠 ′ − 𝑠
If 𝑠 ′ > 𝑠, then ∆𝑠 is positive
Vector form: ∆𝐫 = 𝐫 ′ − 𝐫
The displacement of a particle is also a vector quantity, and it
should be distinguished from the distance the particle travels.
The distance traveled is a positive scalar that represents the
total length of path over which the particle travels.
12.2 Rectilinear kinematics: continuous
motion
Rectilinear kinematics
3- Velocity
Velocity is a measure of the rate of change in the position of a particle.
It is a vector quantity (it has both magnitude and direction). The
magnitude of the velocity is called speed, with units of m/s.
The average velocity of a particle during a time interval ∆𝑡: 𝑣𝑎𝑣𝑔 =
∆𝒓
∆𝑡
𝑑𝒓
The instantaneous velocity: 𝐯 = 𝑑𝑡
𝑑s
Speed is the magnitude of velocity: 𝑣 = 𝑑𝑡
Average speed is the total distance traveled divided by elapsed time:
𝑠𝑡
𝑣𝑠𝑝 𝑎𝑣𝑔 =
∆𝑡
12.2 Rectilinear kinematics: continuous
motion
Rectilinear kinematics
4-Acceleration
Acceleration is the rate of change in the velocity of a particle. It
is a vector quantity. Typical units are m/s2.
The average acceleration of a particle during a time interval ∆𝑡:
∆𝐯
𝑎𝑎𝑣𝑔 =
∆𝑡
The instantaneous acceleration is the time derivative of velocity:
Scalar form: 𝑎 =
𝑑𝑣
𝑑𝑡
;𝑎=
𝑑2 𝑠
𝑑𝑡
If ∆𝑣 =v’-v<0, the particle is slowing down-> decelerating.
If 𝑣 is constant, a=0.
𝑎𝑑𝑠 = 𝑣𝑑𝑣
12.2 Rectilinear kinematics: continuous
motion
Rectilinear kinematics
5-Constant acceleration
𝑎 = 𝑎𝑐
6-Velocity as a function of time
𝑣
𝑡
න 𝑑𝑣 = න 𝑎𝑐 𝑑𝑡
𝑣0
0
Constant acceleration: 𝑣 = 𝑣0 + 𝑎𝑐 𝑡
7-Position as a function of time
𝑠
𝑡
න 𝑑𝑠 = න (𝑣0 + 𝑎𝑐 𝑡) 𝑑𝑡
𝑠0
0
Constant acceleration: 𝑣 2 = 𝑣0 2 + 2𝑎𝑐 (𝑠 − 𝑠0 )
Remember that these equations are
useful only when the acceleration is
constant and when t = 0, s = s0 , v = v0 .
A typical example of constant
accelerated motion occurs when a body
falls freely toward the earth.
If air resistance is neglected and the
distance of fall is short, then the
downward acceleration of the body
when it is close to the earth is constant
and approximately 9.81 m>s 2
12.2 Rectilinear kinematics: continuous
motion
Rectilinear kinematics
12.2 Rectilinear kinematics: continuous
motion
Rectilinear kinematics
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12.1 CONTINUED
12.1 CONTINUED
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12.1CONTINUED
CONTINUED
12.1
example_12_
01
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example_12_
03
(continued)
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12.3 Rectilinear kinematics: erratic motion
■ When a particle has erratic or changing motion then its
position, velocity, and acceleration cannot be described by
a single continuous mathematical function along the entire
path. Instead, a series of functions will be required to
specify the motion at different intervals. For this reason, it
is convenient to represent the motion as a graph.
■ Graphing provides:
– a good way to handle complex motions that would be
difficult to describe with formulas.
– provide a visual description of motion and reinforce
the calculus concepts of differentiation and
integration as used in dynamics.
12.3 Rectilinear kinematics: erratic motion
Given the s-t graph, construct the v-t graph
■ 𝑣
=
𝑑𝑠
𝑑𝑡
velocity = slope of the s-t graph
■ The v-t graph can be constructed by finding the slope at
various points along the s-t graph.
12.3 Rectilinear kinematics: erratic motion
Given the v-t graph, construct the a-t graph
■ 𝑎=
𝑑𝑣
𝑑𝑡
acceleration = slope of the v-t graph
■ The a-t graph can be constructed by finding the slope at
various points along the v-t graph.
■ Note: the distance moved (displacement) of the particle is
the area under the v-t graph during time t.
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12.6 CONTINUED
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12.6 CONTINUED
12.3 Rectilinear kinematics: erratic motion
Given the a-t graph, construct the v-t graph
■
∆𝑣 = ‫𝑡𝑑𝑎 ׬‬
Change in velocity = area under a-t graph
𝑣 = 𝑣0 + ∆𝑣
■ We can construct a v-t graph from an a-t graph if we know
the initial velocity of the particle
12.3 Rectilinear kinematics: erratic motion
Given the v-t graph, construct the s-t graph
■
∆𝑠 = ‫𝑡𝑑𝑣 ׬‬
Change in displacement = area under v-t graph
𝑠 = 𝑠 + ∆𝑠
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12.3 Rectilinear kinematics: erratic motion
Given the a-s graph, construct the v-s graph
■ 𝑣𝑑𝑣 = 𝑎𝑑𝑠
■ ‫𝑣𝑑𝑣 ׬ = 𝑠𝑑𝑎 ׬‬
Area under a-s curve = change in velocity
𝑠
½ (v1² – vo²) = ‫ 𝑠׬‬2 𝑎𝑑𝑠
1
area under a-s graph
■ This equation can be solved for v1, allowing you to solve for
the velocity at a point. By doing this repeatedly, you can
create a plot of velocity versus distance.
12.3 Rectilinear kinematics: erratic motion
Given the v-s graph, construct the a-s graph
■ 𝑎𝑑𝑠 = 𝑣𝑑𝑣
■ 𝑎=𝑣
𝑑𝑣
𝑑𝑠
acceleration = velocity times slope of v-s graph
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12.8 CONTINUED
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12.8 CONTINUED
12.4 General curvilinear motion
Position
■ Curvilinear motion occurs when a particle moves along a
curved path. Since this path is often described in three
dimensions, vector analysis will be used to formulate the
particle’s position, velocity, and acceleration.
■ A particle moves along a curve defined by the path
function, s.
■ The position of the particle at any instant is designated by
the vector r = r(t). Both the magnitude and direction of r
may vary with time.
12.4 General curvilinear motion
Displacement
■ If the particle moves a distance s along the curve during time interval t, the
displacement is determined by vector subtraction: ∆𝐫 = 𝐫 ′ − 𝐫
12.4 General curvilinear motion
Velocity
■ Velocity represents the rate of change in the position of a particle.
■ The average velocity of the particle during the time increment t is:
∆𝐫
𝐯𝐚𝐯𝐠 = ∆𝑡
■ The instantaneous velocity is the time-derivative of position:
𝑑𝐫
𝒗=
𝑑𝑡
■ The velocity vector, v, is always tangent to the path of motion.
■ The magnitude of v is called the speed. Since the arc length s
approaches the magnitude of r as t→0, the speed can be
obtained by differentiating the path function (v = ds/dt). Note that
this is not a vector!
12.4 General curvilinear motion
Acceleration
■ Acceleration represents the rate of change in the velocity of a
particle.
■ If a particle’s velocity changes from v to v’ over a time increment
t, the average acceleration during that increment is:
∆𝐯 𝐯 − 𝐯′
𝐚𝐚𝐯𝐠 =
=
∆𝑡
∆𝑡
■ The instantaneous acceleration is the time-derivative of velocity:
a=
𝑑𝐯 𝑑 2 𝐫
=
𝑑𝑡 𝑑𝑡 2
■ A plot of the locus of points defined by the arrowhead of the
velocity vector is called a hodograph. The acceleration vector is
tangent to the hodograph, but not, in general, tangent to the path
function.
12.5 Curvilinear motion: rectangular
components
Position
■ The motion of a particle can best be described along a path that
can be expressed in terms of its x, y, z coordinates or rectangular
components, relative to a fixed frame of reference.
■ The position of the particle can be defined at any instant by the
position vector
r= 𝑥𝐢 + 𝑦𝐣 + 𝑗𝐤
■ The x, y, z-components may all be functions of time:
x = x(t), y = y(t), and z = z(t)
■ The magnitude of the position vector is: r=
𝒙𝟐 + 𝒚𝟐 + 𝒛𝟐
1
𝑟
■ The direction of r is defined by the unit vector: 𝐮𝐫 = 𝐫
12.5 Curvilinear motion: rectangular
components
Velocity
■ The velocity vector is the time derivative of the position vector:
𝐯=
𝑑𝐫
𝑑𝑡
=
𝑑
𝑑𝑡
𝑥𝐢 +
𝑑
𝑑𝑡
𝑦𝐣 +
𝑑
𝑑𝑡
𝑧𝐤
■ Since the unit vectors i, j, k are constant in magnitude and
direction, this equation reduces to:
𝐯=
𝑑𝐫
𝑑𝑡
=𝑣𝑥 𝐢 + 𝑣𝑦 𝐣 + 𝑣𝑧 𝐤
where 𝑣𝑥 = 𝑥,ሶ 𝑣𝑦 = 𝑦ሶ and 𝑣𝑧 = 𝑧ሶ
■ The magnitude of the velocity vector is:
𝑣 = (𝑣𝑥 )2 +(𝑣𝑦 )2 +(𝑣𝑧 )2
■ The direction of 𝐯 is tangent to the path of motion.
12.5 Curvilinear motion: rectangular
components
Acceleration
■ The acceleration vector is the time derivative of the velocity
vector (second derivative of the position vector).
𝑑𝐯
𝐚=
= 𝑎𝑥 𝐢 + 𝑎𝑦 𝐣 + 𝑎𝑧 𝐤
𝑑𝑡
where 𝑎𝑥 = 𝑣ሶ𝑥 = 𝑥,ሷ 𝑎𝑦 = 𝑣ሶ𝑦 = 𝑦ሷ and 𝑎𝑧 = 𝑣ሶ𝑧 = 𝑧ሷ
■ The magnitude of the acceleration vector is
a = (𝑎𝑥 )2 +(𝑎𝑦 )2 +(𝑎𝑧 )2
■ The direction of a is usually not tangent to the path of the
particle.
12.5 Curvilinear motion: rectangular
components
Acceleration
12.5 Curvilinear motion: rectangular
components
Acceleration
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12.9 CONTINUED
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12.6 Motion of a projectile
■ The free-flight motion of a
projectile is often studied in
terms of its rectangular
components since the
projectile’s acceleration always
acts om the vertical direction.
■ Projectile motion can be treated
as two rectilinear motions, one
in the horizontal direction
experiencing zero acceleration
and the other in the vertical
direction experiencing constant
acceleration (i.e., from gravity).
12.6 Motion of a projectile
Horizontal motion
𝑎𝑥 = 0
+
՜ 𝑣 = 𝑣0 + 𝑎𝑐 𝑡 ; 𝑣𝑥 = (𝑣0 )𝑥
+
1
՜ 𝑥 = 𝑥0 + 𝑣0 𝑡 + 𝑎𝑐 𝑡 2 ; 𝑥 = 𝑥0 + (𝑣0 )𝑥 𝑡
2
+
՜ 𝑣 2 = 𝑣02 + 2𝑎𝑐 (𝑠 − 𝑠0 ) ; 𝑣𝑥 = (𝑣0 )𝑥
■ The horizontal component of velocity always remains constant
during the motion.
12.6 Motion of a projectile
Vertical motion
𝑎𝑦 = −𝑔
+↑ 𝑣 = 𝑣0 + 𝑎𝑐 𝑡 ; 𝑣𝑦 = (𝑣0 )𝑦 − 𝑔𝑡
1
1 2
2
+↑ 𝑦 = 𝑦0 + 𝑣0 𝑡 + 𝑎𝑐 𝑡 ; 𝑦 = 𝑦0 + (𝑣0 )𝑦 𝑡 − 𝑔𝑡
2
2
+↑ 𝑣 2 = 𝑣02 + 2𝑎𝑐 (𝑦 − 𝑦0 ) ; 𝑣𝑦2 = 𝑣0
2
𝑦
− 2𝑔(𝑦 − 𝑦0 )The
■ Note: only two of the above three equations are independent of one another.
■
Problems involving the motion of a projectile can have at most three unknowns since only three
independent equations can be written; that is, one equation in the horizontal direction and two in
the vertical direction. Once vx and vy are obtained, the resultant velocity v, which is always tangent
to the path, can be determined by the vector sum.
12.6 Motion of a projectile
Vertical motion
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12.12 CONTINUED
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12.12 CONTINUED
example_12_
12
(continued)
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12.7 Curvilinear motion: normal and
tangential components
Planar motion
■ When a particle moves along a curved path, it is sometimes
convenient to describe its motion using coordinates other than
Cartesian. When the path of motion is known, normal (n) and
tangential (t) coordinates are often used.
■ In the n-t coordinate system, the origin is located on the
particle (thus the origin and coordinate system move with the
particle).
■ The t-axis is tangent to the path (curve) at the instant
considered, positive in the direction of the particle’s motion.
■ The n-axis is perpendicular to the t-axis with the positive
direction toward the center of curvature of the curve.
■ The plane which contains the n and t axes is referred to as the
osculating plane.
12.7 Curvilinear motion: normal and
tangential components
Planar motion
■ The positive n and t directions are defined by the unit
vectors un and ut, respectively.
■ The center of curvature, O’, always lies on the concave side
of the curve.
■ The radius of curvature, r, is defined as the perpendicular
distance from the curve to the center of curvature at that
point.
■ The position of the particle at any instant is defined by the
distance, s, along the curve from a fixed reference point.
12.7 Curvilinear motion: normal and
tangential components
Velocity
■ s is function of time (the particle is moving).
■ The velocity vector is always tangent to the path of motion
(t-direction).
■ The magnitude is determined by taking the time derivative
of the path function, s(t).
𝐯 = v𝐮𝐭 , 𝑣 = 𝑠 =
𝑑𝑠
𝑑𝑡
= 𝑠ሶ
■ Here v defines the magnitude of the velocity (speed) and
ut defines the direction of the velocity vector.
12.7 Curvilinear motion: normal and
tangential components
Acceleration
■ Acceleration is the time rate of change of velocity:
𝐚=
𝑑𝐯
𝑑𝑡
=
𝑑(v𝐮𝐭 )
𝑑𝑡
= 𝐯ሶ = 𝑣ሶ 𝐮𝐭 + 𝑣𝐮ሶ 𝐭
■ Here 𝐯ሶ represents the change in the magnitude of velocity
and 𝐮ሶ 𝐭 represents the rate of change in the direction of 𝐮𝐭 .
𝐚 = 𝑣ሶ 𝐮𝐭 + 𝑣 𝐮ሶ 𝐭 = 𝑣ሶ
𝑣=
ሶ 𝑎𝑡 ; 𝑎𝑡 𝑑𝑠 = 𝑣𝑑𝑣 and
𝑎=
𝑎𝑡 2 + 𝑎𝑛 2
𝑣2
𝐮𝐭 + 𝐮ሶ 𝐭
𝜌
𝑣2
𝑎𝑛 =
𝜌
= 𝑎𝑡 𝐮𝐭 + 𝑎𝑛 𝐮𝒏
12.7 Curvilinear motion: normal and
tangential components
Acceleration
■ Two special cases of motion.
–
1. If the particle moves along a straight line, then ρ->∞ and from Eq. :
𝑣2
𝑎𝑛 = ,
𝜌
𝑎𝑛 =0. Thus 𝑎 = 𝑎𝑡 = 𝑣ሶ , and we
can conclude that the tangential component of acceleration represents the time rate of change in the
magnitude of the velocity.
–
2
𝑣
2. If the particle moves along a curve with a constant speed, then 𝑎𝑡 = 𝑣 =ሶ 0 and 𝑎 = 𝑎𝑛 = . Therefore, the
𝜌
normal component of acceleration represents the time rate of change in the direction of the velocity. Since
𝑎𝑛 always acts towards the center of curvature, this component is sometimes referred to as the centripetal (or
center seeking) acceleration. As a result of these interpretations, a particle moving along the curved path in the
below figure will have accelerations directed as shown:
12.7 Curvilinear motion: normal and
tangential components
Three-dimensional motion
■ If a particle moves along a space curve, the n-t axes are
defined as before. At any point, the t-axis is tangent to
the path and the n-axis points toward the center of
curvature. The plane containing the n-t axes is called the
osculating plane.
■ A third axis can be defined, called the binomial axis, b.
The binomial unit vector, ub, is directed perpendicular to
the osculating plane, and its sense is defined by the cross
product u𝑏 = 𝐮𝑡 × 𝐮𝑛
■ There is no motion, thus no velocity or acceleration, in the
binomial direction.
12.7 Curvilinear motion: normal and
tangential components
Three-dimensional motion
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12.15 CONTINUED
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12.15 CONTINUED
example_12_
15
(continued)
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12.8 Curvilinear motion: cylindrical
components
Polar coordinates & Position
■ Sometimes the motion of the particle is constrained on a
path that is best described using cylindrical coordinates. If
motion is restricted to the plane, then polar coordinates
are used.
■ We can express the location of P in polar coordinates as:
𝐫 = 𝑟𝐮𝑟
■ Note that the radial direction, r, extends outward from the
fixed origin, O, and the transverse coordinate, q, is
measured counter-clockwise (CCW) from the horizontal.
180°
1 ra𝑑 =
𝜋
12.8 Curvilinear motion: cylindrical
components
Velocity
■ The instantaneous velocity:
𝐯 = 𝐫ሶ = 𝑟𝐮
ሶ 𝑟 + 𝑟𝐮𝒓 ሶ =
𝑑𝐫
𝑑𝑡
=
𝑑(𝑟𝐮𝑟 )
=
𝑑𝑡
𝑟𝐮
ሶ 𝑟 +𝑟
𝑑𝐮𝑟
𝑑𝑡
■ Using the chain rule:
𝑑𝐮𝑟
𝑑𝐮 𝑑𝜃
= 𝑟
𝑑𝑡
𝑑𝜃 𝑑𝑡
𝑑𝐮𝑟
= 𝐮𝜃
𝑑𝜃
𝑑𝐮𝑟
ሶ 𝜃
= 𝜃𝐮
𝑑𝑡
ሶ 𝜃
𝐯 = 𝑟𝐮
ሶ 𝑟 + 𝜃𝐮
■ The velocity vector has two components: r, called the radial
component, and rq called the transverse component.
■
The speed of the particle at any given instant is the sum of the
squares of both components or 𝑣 = (𝑟𝜃)2 +𝑟ሶ 2
12.8 Curvilinear motion: cylindrical
components
Acceleration
■ The instantaneous velocity:
ሶ 𝜃)
𝑑𝐯 𝑑(𝑟𝐮
ሶ 𝑟 + 𝜃𝐮
𝐚 = 𝐯ሶ =
=
𝑑𝑡
𝑑𝑡
ሶ 𝐮𝜃
𝐚 = (𝑟ሷ − 𝑟𝜃 2ሶ ) 𝐮𝑟 + (𝑟𝜃ሷ + 2𝑟ሶ 𝜃)
𝑟ሷ − 𝑟𝜃 2ሶ is the radial acceleration or 𝑎𝑟
𝑟𝜃ሷ + 2𝑟ሶ 𝜃ሶ is the transverse acceleration or 𝑎𝜃
The magnitude of acceleration:
𝑎=
ሶ 2
(𝑟ሷ − 𝑟𝜃 2ሶ )2 +(𝑟𝜃ሷ + 2𝑟ሶ 𝜃)
The direction of a is determined from the vector addition of its
components.
In general, a will not be tangent to the path.
12.8 Curvilinear motion: cylindrical
components
Cylindrical coordinates
■ If the particle P moves along a space curve, its position can be
written as:
𝐫𝒑 = 𝑟 𝐮𝑟 + 𝑧𝐮𝒛
■ Using the chain rule and taking the time derivatives:
ሶ 𝜃 + 𝑧𝐮
𝐯𝒑 = 𝑟𝐮
ሶ 𝑟 + 𝑟𝜃𝐮
ሶ 𝒛
ሶ 𝜃)
𝑑𝐯 𝑑(𝑟𝐮
ሶ 𝑟 + 𝜃𝐮
𝐚 = 𝐯ሶ =
=
𝑑𝑡
𝑑𝑡
ሶ 𝐮𝜃 + 𝑧ሷ 𝐮𝒛
𝐚 = (𝑟ሷ − 𝑟𝜃 2ሶ ) 𝐮𝑟 + (𝑟𝜃ሷ + 2𝑟ሶ 𝜃)
12.8 Curvilinear motion: cylindrical
components
Time derivates
12.8 Curvilinear motion: cylindrical
components
Time derivates
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12.20 CONTINUED
example_12_
20
(continued)
12.9 Absolute dependent motion
analysis of two particles
■ In many kinematics problems, the motion of one
object will depend on the motion of another
object.
■ The blocks in this figure are connected by an
inextensible cord wrapped around a pulley.
■ The motion of each block can be related
mathematically by defining position coordinates,
sA and sB. Each coordinate axis is defined from a
fixed point or datum line, measured positive
along each plane in the direction of motion of
each block.
12.9 Absolute dependent motion
analysis of two particles
■ In this example, position coordinates sA and sB
can be defined from fixed datum lines extending
from the center of the pulley along each incline to
blocks A and B.
■ If the cord has a fixed length, the position
coordinates sA and sB are related mathematically
by the equation
𝑠𝐴 + 𝑙𝐶𝐷 + 𝑠𝐵 = 𝑙 𝑇
■ Here 𝑙 𝑇 is the total cord length and 𝑙𝐶𝐷 is the
length of cord passing over the arc CD on the
pulley.
12.9 Absolute dependent motion
analysis of two particles
■ The velocities of blocks A and B can be related by
differentiating the position equation.
■ Note that 𝑙𝐶𝐷 and 𝑙 𝑇 remain constant,
so
𝑑𝑠𝐴
𝑑𝑡
+
𝑑𝑠𝐵
=0
𝑑𝑡
𝑑𝑙𝐶𝐷
𝑑𝑡
=
𝑑𝑙𝑇
=0
𝑑𝑡
or 𝑣𝐵 = − 𝑣𝐴
■ The negative sign indicates that as A moves down
the incline (positive sA direction), B moves up the
incline (negative sB direction).
■ Accelerations can be found by differentiating the
velocity expression; in a similar manner we have:
𝑎𝐵 = − 𝑎𝐴
12.9 Absolute dependent motion
analysis of two particles
Example:
Consider a more complicated example. Position
coordinates (sA and sB) are defined from fixed datum
lines, measured along the direction of motion of each
block.
Note that sB is only defined to the center of the pulley
above block B, since this block moves with the pulley.
Also, h is a constant.
The red-colored segments of the cord remain constant
in length during motion of the blocks.
12.9 Absolute dependent motion
analysis of two particles
Example:
The position coordinates are related by the equation
2𝑠𝐵 + ℎ + 𝑠𝐴 = 𝑙 𝑇
where 𝑙 𝑇 is the total cord length minus the lengths of
the red segments.
Since 𝑙 𝑇 and ℎ remain constant during the motion, the
velocities and accelerations can be related by two
successive time derivatives:
2𝑣𝐵 = − 𝑣𝐴 and 2𝑎𝐵 = − 𝑎𝐴
When block B moves downward (+ 𝑠𝐵 ), block A moves to
the left (- 𝑠𝐴 ).
Remember to be consistent with your sign convention!
12.9 Absolute dependent motion
analysis of two particles
Example:
This example can also be worked by defining the
position coordinate for B (𝑠𝐵 ) from the bottom pulley
instead of the top pulley.
The position, velocity, and acceleration relations then
become:
2(ℎ − 𝑠𝐵 ) + ℎ + 𝑠𝐴 = 𝑙 𝑇 and 2𝑣𝐵 = 𝑣𝐴 and 2𝑎𝐵 = 𝑎𝐴
12.9 Absolute dependent motion
analysis of two particles
■ These procedures can be used to relate the dependent motion of particles moving along
rectilinear paths (only the magnitudes of velocity and acceleration change, not their line of
direction).
1.
Define position coordinates from fixed datum lines, along the path of each particle.
Different datum lines can be used for each particle.
2.
Relate the position coordinates to the cord length. Segments of cord that do not
change in length during the motion may be left out.
3.
If a system contains more than one cord, relate the position of a point on one cord to
a point on another cord. Separate equations are written for each cord.
4.
Differentiate the position coordinate equation(s) to relate velocities and
accelerations. Keep track of signs!
12.9 Absolute dependent motion
analysis of two particles
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12.21 CONTINUED
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12.22 CONTINUED
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12.10 Relative-motion analysis of two
particles using translating axes
Position
■ The absolute positions of two particles A and B with
respect to the fixed x, y, z-reference frame are given by rA
and rB. The position of B relative to A is represented by
𝐫𝐁/𝐀 = 𝐫𝐁 − 𝐫 𝐀
Example: if
rB = (10 i + 2 j ) m and rA = (4 i + 5 j ) m,
𝐫𝐁/𝐀 = 𝐫𝐁 − 𝐫 𝐀 = (6 i – 3 j ) m.
12.10 Relative-motion analysis of two
particles using translating axes
Velocity
■ To determine the relative velocity of B with respect to A,
the time derivative of the relative position equation is
taken.
𝐯𝐁/𝐀 = 𝐯𝐁 − 𝐯 𝐀
In these equations, 𝐯𝐁 and 𝐯𝑨 are called absolute velocities
and 𝐯𝐁/𝐀 is the relative velocity of B with respect to A.
■ Note that 𝐯𝐁/𝐀 =- 𝐯𝐀/𝑩
12.10 Relative-motion analysis of two
particles using translating axes
Acceleration
■ The time derivative of the relative velocity equation yields
a similar vector relationship between the absolute and
relative accelerations of particles A and B.
𝐚𝐁 = 𝐚𝑨 − 𝐚𝑩/𝑨
12.10 Relative-motion analysis of two
particles using translating axes
Acceleration
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