Lecture Notes for Sections 13'113'4 Kinetics

1
NEWTONS LAWS OF MOTION (EQUATION OF MOTION)
(Sections 13.1-13.3)
Todays Objectives Students will be able
to a) Write the equation of motion for an
accelerating body. b) Draw the free-body and
kinetic diagrams for an accelerating body.
In-Class Activities Check homework, if
any Reading quiz Applications Newtons laws
of motion Newtons law of gravitational
attraction Equation of motion for a particle or
system of particles Concept quiz Group
problem solving Attention quiz
2
READING QUIZ
1. Newtons second law can be written in
mathematical form as ?F ma. Within the
summation of forces ?F, ________ are(is) not
included. A) external forces B) weight
C) internal forces D) All of the above.
2. The equation of motion for a system of
n-particles can be written as ?Fi ? miai maG,
where aG indicates _______. A) summation
of each particles acceleration B)
acceleration of the center of mass of the system
C) acceleration of the largest particle
D) None of the above.
3
APPLICATIONS
The motion of an object depends on the forces
acting on it.
A parachutist relies on the atmospheric drag
resistance force to limit his velocity.
Knowing the drag force, how can we determine the
acceleration or velocity of the parachutist at
any point in time?
4
APPLICATIONS (continued)
A freight elevator is lifted using a motor
attached to a cable and pulley system as shown.
How can we determine the tension force in the
cable required to lift the elevator at a given
acceleration?
Is the tension force in the cable greater than
the weight of the elevator and its load?
5
NEWTONS LAWS OF MOTION
The motion of a particle is governed by Newtons
three laws of motion.
First Law A particle originally at rest, or
moving in a straight line at constant velocity,
will remain in this state if the resultant force
acting on the particle is zero.
Second Law If the resultant force on the
particle is not zero, the particle experiences an
acceleration in the same direction as the
resultant force. This acceleration has a
magnitude proportional to the resultant force.
Third Law Mutual forces of action and reaction
between two particles are equal, opposite, and
collinear.
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NEWTONS LAWS OF MOTION (continued)
The first and third laws were used in developing
the concepts of statics. Newtons second law
forms the basis of the study of dynamics.
Mathematically, Newtons second law of motion can
be written F ma where F is the resultant
unbalanced force acting on the particle, and a is
the acceleration of the particle. The positive
scalar m is called the mass of the particle.
Newtons second law cannot be used when the
particles speed approaches the speed of light,
or if the size of the particle is extremely small
( size of an atom).
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NEWTONS LAW OF GRAVITATIONAL ATTRACTION
Any two particles or bodies have a mutually
attractive gravitational force acting between
them. Newton postulated the law governing this
gravitational force as
F G(m1m2/r2) where F force of
attraction between the two bodies, G
universal constant of gravitation ,
m1, m2 mass of each body, and r
distance between centers of the two bodies.
When near the surface of the earth, the only
gravitational force having any sizable magnitude
is that between the earth and the body. This
force is called the weight of the body.
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MASS AND WEIGHT
It is important to understand the difference
between the mass and weight of a body!
Mass is an absolute property of a body. It is
independent of the gravitational field in which
it is measured. The mass provides a measure of
the resistance of a body to a change in velocity,
as defined by Newtons second law of motion (m
F/a).
The weight of a body is not absolute, since it
depends on the gravitational field in which it is
measured. Weight is defined as W mg where g is
the acceleration due to gravity.
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UNITS SI SYSTEM
SI system In the SI system of units, mass is a
base unit and weight is a derived unit.
Typically, mass is specified in kilograms (kg),
and weight is calculated from W mg. If the
gravitational acceleration (g) is specified in
units of m/s2, then the weight is expressed in
newtons (N). On the earths surface, g can be
taken as g 9.81 m/s2. W (N) m (kg) g (m/s2)
gt N kgm/s2
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EQUATION OF MOTION
The motion of a particle is governed by Newtons
second law, relating the unbalanced forces on a
particle to its acceleration. If more than one
force acts on the particle, the equation of
motion can be written ?F FR ma where FR is
the resultant force, which is a vector summation
of all the forces.
To illustrate the equation, consider a particle
acted on by two forces.
First, draw the particles free-body diagram,
showing all forces acting on the particle. Next,
draw the kinetic diagram, showing the inertial
force ma acting in the same direction as the
resultant force FR.
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INERTIAL FRAME OF REFERENCE
This equation of motion is only valid if the
acceleration is measured in a Newtonian or
inertial frame of reference. What does this mean?
For problems concerned with motions at or near
the earths surface, we typically assume our
inertial frame to be fixed to the earth. We
neglect any acceleration effects from the earths
rotation.
For problems involving satellites or rockets, the
inertial frame of reference is often fixed to the
stars.
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KEY POINTS
1) Newtons second law is a Law of
Nature--experimentally proven and not the result
of an analytical proof.
2) Mass (property of an object) is a measure of
the resistance to a change in velocity of the
object.
3) Weight (a force) depends on the local
gravitational field. Calculating the weight of
an object is an application of F
ma, i.e., W m g.
4) Unbalanced forces cause the acceleration of
objects. This condition is fundamental to all
dynamics problems!
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PROCEDURE FOR THE APPLICATION OF THE EQUATION OF
MOTION
1) Select a convenient inertial coordinate
system. Rectangular, normal/tangential, or
cylindrical coordinates may be used.
2) Draw a free-body diagram showing all external
forces applied to the particle. Resolve forces
into their appropriate components.
3) Draw the kinetic diagram, showing the
particles inertial force, ma. Resolve this
vector into its appropriate components.
4) Apply the equations of motion in their scalar
component form and solve these equations for the
unknowns.
5) It may be necessary to apply the proper
kinematic relations to generate additional
equations.
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EXAMPLE
Given A crate of mass m is pulled by a cable
attached to a truck. The coefficient of kinetic
friction between the crate and road is mk.
Find Draw the free-body and kinetic diagrams of
the crate.
Plan 1) Define an inertial coordinate
system. 2) Draw the crates free-body diagram,
showing all external forces applied to the crate
in the proper directions. 3) Draw the crates
kinetic diagram, showing the inertial force
vector ma in the proper direction.
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EXAMPLE (continued)
Solution
1) An inertial x-y frame can be defined as fixed
to the ground.
2) Draw the free-body diagram of the crate
3) Draw the kinetic diagram of the crate
The crate will be pulled to the right. The
acceleration vector can be directed to the right
if the truck is speeding up or to the left if it
is slowing down.
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CONCEPT QUIZ 1
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CONCEPT QUIZ 2
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GROUP PROBLEM SOLVING
Given Each block has a mass m. The coefficient
of kinetic friction at all surfaces of contact is
m. A horizontal force P is applied to the bottom
block.
Find Draw the free-body and kinetic diagrams of
each block.
Plan 1) Define an inertial coordinate
system. 2) Draw the free-body diagrams for each
block, showing all external forces. 3) Draw the
kinetic diagrams for each block, showing the
inertial forces.
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GROUP PROBLEM SOLVING (continued)
Solution
1) An inertial x-y frame can be defined as fixed
to the ground.
2) Draw the free-body diagram of each block
The friction forces oppose the motion of each
block relative to the surfaces on which they
slide.
3) Draw the kinetic diagram of each block
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ATTENTION QUIZ
1. Internal forces are not included in an
equation of motion analysis because the internal
forces are_____. A) equal to zero B) equal and
opposite and do not affect the calculations
C) negligibly small D) not important
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EQUATIONS OF MOTION NORMAL AND TANGENTIAL
COORDINATES (Section 13.5)
Todays Objectives Students will be able to
apply the equation of motion using normal and
tangential coordinates.
In-Class Activities Check homework, if
any Reading quiz Applications Equation of
motion in n-t coordinates Concept quiz Group
problem solving Attention quiz
22
READING QUIZ
1. The normal component of the equation of
motion is written as ?Fnman, where ?Fn is
referred to as the _______. A) impulse B)
centripetal force C) tangential force D)
inertia force
2. The positive n direction of the normal and
tangential coordinates is ____________. A)
normal to the tangential component
B) always directed toward the center of
curvature C) normal to the
bi-normal component D) All of the above.
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APPLICATIONS
Race tracks are often banked in the turns to
reduce the frictional forces required to keep the
cars from sliding at high speeds.
If the cars maximum velocity and a minimum
coefficient of friction between the tires and
track are specified, how can we determine the
minimum banking angle (q) required to prevent the
car from sliding?
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APPLICATIONS (continued)
Satellites are held in orbit around the earth by
using the earths gravitational pull as the
centripetal force the force acting to change
the direction of the satellites velocity.
Knowing the radius of orbit of the satellite, how
can we determine the required speed of the
satellite to maintain this orbit?
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NORMAL TANGENTIAL COORDINATES
When a particle moves along a curved path, it may
be more convenient to write the equation of
motion in terms of normal and tangential
coordinates.
The normal direction (n) always points toward the
paths center of curvature. In a circle, the
center of curvature is the center of the circle.
The tangential direction (t) is tangent to the
path, usually set as positive in the direction of
motion of the particle.
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EQUATIONS OF MOTION
Since the equation of motion is a vector equation
, ?F ma, it may be written in terms of the n
t coordinates as ?Ftut ?Fnun mat man
Here ?Ft ?Fn are the sums of the force
components acting in the t n directions,
respectively.
This vector equation will be satisfied provided
the individual components on each side of the
equation are equal, resulting in the two scalar
equations ?Ft mat and ?Fn man .
Since there is no motion in the binormal (b)
direction, we can also write ?Fb 0.
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NORMAL AND TANGENTIAL ACCERLERATIONS
The tangential acceleration, at dv/dt,
represents the time rate of change in the
magnitude of the velocity. Depending on the
direction of ?Ft, the particles speed will
either be increasing or decreasing.
The normal acceleration, an v2/r, represents
the time rate of change in the direction of the
velocity vector. Remember, an always acts toward
the paths center of curvature. Thus, ?Fn will
always be directed toward the center of the path.
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SOLVING PROBLEMS WITH n-t COORDINATES
Use n-t coordinates when a particle is moving
along a known, curved path.
Establish the n-t coordinate system on the
particle.
Draw free-body and kinetic diagrams of the
particle. The normal acceleration (an) always
acts inward (the positive n-direction). The
tangential acceleration (at) may act in either
the positive or negative t direction.
Apply the equations of motion in scalar form
and solve.
It may be necessary to employ the kinematic
relations at dv/dt v dv/ds an v2/r
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EXAMPLE
Given At the instant q 60, the boys center
of mass G is momentarily at rest. The boy has a
weight of 300 N (30 kg). Neglect his size and
the mass of the seat and cords.
Find The boys speed and the tension in each of
the two supporting cords of the swing when q
90.
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EXAMPLE (continued)
Solution
1) The n-t coordinate system can be established
on the boy at some arbitrary angle q.
Approximating the boy and seat together as a
particle, the free-body and kinetic diagrams can
be drawn.
T tension in each cord W weight of the boy
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EXAMPLE (continued)
2) Apply the equations of motion in the n-t
directions.
(a) ?Fn man gt 2T W sin q man
Using an v2/r v2/10, w 60 lb, and m w/g
(60/32.2), we get 2T 60 sin q
(60/32.2)(v2/10) (1)
(b) ?Ft mat gt W cos q mat
gt 60 cos q (60/32.2) at Solving for at
at 32.2 cos q (2)
Note that there are 2 equations and 3 unknowns
(T, v, at). One more equation is needed.
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EXAMPLE (continued)
3) Apply kinematics to relate at and v.
v dv at ds where ds r dq 3 dq
gt v dv 9.81 cosq ds 9.81 cosq (3 dq )
This v is the speed of the boy at q 90?. This
value can be substituted into equation (1) to
solve for T.
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CONCEPT QUIZ
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GROUP PROBLEM SOLVING
Given A 200 kg snowmobile with rider is
traveling down the hill. When it is at point A,
it is traveling at 4 m/s and increasing its speed
at 2 m/s2.
Find The resultant normal force and resultant
frictional force exerted on the tracks at point A.
Plan 1) Treat the snowmobile as a particle.
Draw the free-body and kinetic diagrams. 2) Apply
the equations of motion in the n-t
directions. 3) Use calculus to determine the
slope and radius of curvature of the path at
point A.
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GROUP PROBLEM SOLVING (continued)
Solution
1) The n-t coordinate system can be established
on the snowmobile at point A. Treat the
snowmobile and rider as a particle and draw the
free-body and kinetic diagrams
W mg weight of snowmobile and passenger N
resultant normal force on tracks F resultant
friction force on tracks
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GROUP PROBLEM SOLVING (continued)
2) Apply the equations of motion in the n-t
directions
? Fn man gt W cos q N man

• Using W mg and an v2/r (4)2/r
• gt (200)(9.81) cos q N (200)(16/r)
• gt N 1962 cosq 3200/r
  (1)

? Ft mat gt W sinq F mat

• Using W mg and at 2 m/s2 (given)
• gt (200)(9.81) sin q F (200)(2)
• gt F 1962 sinq 400
  (2)

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GROUP PROBLEM SOLVING (continued)
3) Determine r by differentiating y f(x) at x
10 m
y -5(10-3)x3 gt dy/dx (-15)(10-3)x2 gt
d2y/dx2 -30(10-3)x
Determine q from the slope of the curve at A
From Eq.(1) N 1962 cos(56.31) 3200/19.53
924 N
From Eq.(2) F 1962 sin(56.31) 400 1232 N
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ATTENTION QUIZ
1. The tangential acceleration of an
object A) represents the rate of change of the
velocity vectors direction. B) represents the
rate of change in the magnitude of the
velocity. C) is a function of the radius of
curvature. D) Both B and C.
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EQUATIONS OF MOTION CYLINDRICAL COORDINATES
(Section 13.6)
Todays Objectives Students will be able to
analyze the kinetics of a particle using
cylindrical coordinates.
In-Class Activities Check homework, if
any Reading quiz Applications Equations of
motion using cylindrical coordinates Angle
between radial and tangential directions Concept
quiz Group problem solving Attention quiz
40
READING QUIZ

• The normal force which the path exerts on a
  particle is always perpendicular to the
  _________.
• A) radial line B) transverse direction
• C) tangent to the path D) None of the above.

• Friction forces always act in the __________
  direction.
• A) radial B) tangential
• C) transverse D) None of the above.

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APPLICATIONS
The forces acting on the 45 kg boy can be
analyzed using the cylindrical coordinate
system. If the boy slides down at a constant
speed of 2 m/s, can we find the frictional force
acting on him?
42
APPLICATIONS (continued)
When an airplane executes the vertical loop shown
above, the centrifugal force causes the normal
force (apparent weight) on the pilot to be
smaller than her actual weight. If the pilot
experiences weightlessness at A, what is the
airplanes velocity at A?
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EQUATIONS OF MOTION CYLINDRICAL COORDINATES
This approach to solving problems has some
external similarity to the normal tangential
method just studied. However, the path may be
more complex or the problem may have other
attributes that make it desirable to use
cylindrical coordinates.
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EQUATIONS OF MOTION (continued)
Note that a fixed coordinate system is used, not
a body-centered system as used in the n t
approach.
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TANGENTIAL AND NORMAL FORCES

If a force P causes the particle to move along a
path defined by r f (q ), the normal force N
exerted by the path on the particle is always
perpendicular to the paths tangent. The
frictional force F always acts along the tangent
in the opposite direction of motion. The
directions of N and F can be specified relative
to the radial coordinate by using angle y .
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DETERMINATION OF ANGLE y
If y is positive, it is measured counterclockwise
from the radial line to the tangent. If it is
negative, it is measured clockwise.
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EXAMPLE
Plan Draw a FBD. Then develop the kinematic
equations and finally solve the kinetics problem
using cylindrical coordinates.
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EXAMPLE (continued)
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EXAMPLE (continued)
tan y r/(dr/dq) where dr/dq -2rc
sinq tan y (2rc cosq)/(-2rc sinq)
-1/tanq ? y 120?
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CONCEPT QUIZ

• If needing to solve a problem involving the
  pilots weight at Point C, select the approach
  that would be best.
• Equations of Motion Cylindrical Coordinates
• B) Equations of Motion Normal Tangential
  Coordinates
• C) Equations of Motion Polar Coordinates
• No real difference all are bad.
• Toss up between B and C.

51
GROUP PROBLEM SOLVING
Given A plane flies in a vertical loop as
shown. vA 80 m/s (constant) W 60
kg Find Normal force on the pilot at A.
At A
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GROUP PROBLEM SOLVING (continued)
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GROUP PROBLEM SOLVING (continued)
Kinetics ? Fr mar gt -mg N mar N
-60 (9.81) 60 (53.3) gt N 3786.6 N

3.8 kN
Notice that the pilot would experience
weightlessness when his radial acceleration is
equal to g.
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ATTENTION QUIZ

• For the path defined by r q2 , the angle y at
  q .5 rad is
• A) 10 º B) 14 º
• C) 26 º D) 75 º

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End of the Lecture
Let Learning Continue