12'1 Overview

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Chap. 12 Linear Kinetics
12.1 Overview

• Kinetics based on kinematics, incorporated into
  the force effects causing motion
• Translational
• Rotational
• General
• Object for analysis
• Particles sufficiently small object with
  translational motion only
• Rigid bodies especially important in rotational
  motion
• Solving methods
• Newtons law of motion
• Energy method
• Impulse and Momentum method

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12.2 Equations of Motion

• Newtons 2nd law

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12.3 Special Cases of Translational Motion
12.3.1 Constant Force
12.3.2 Force as a function of time
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12.3 Special Cases of Translational Motion
(continued)
12.3.3 Force as a function of displacement
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12.4 Procedures for Problem Solving in Kinetics

• Draw a simple diagram of the system
• Isolate the interested body from the
  surroundings and draw a free-body diagrams of the
  parts
• Show all known and unknown forces and moments
• For unknown forces and moments, assume the
  direction as you want. But do not
• change during the analysis. The correct
  direction will appear in solutions, based on
• our assumptions.
• The general rule is the right-hand rule (?, ?,
  CCW is positive.)
• 3. Adopt proper coordinate system (x, y, z)
• Resolve all forces and moments in terms of
  their components.
• 4. For each free-body diagram, apply equations of
  translational and rotational motions.
• 5. Solve for unknowns and include their correct
  directions and units.

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12.4 Procedures for Problem Solving in Kinetics
Example
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12.5 Work and Energy Methods
To solve dynamics

• 1. Equations of motion
• Simple problems (generally constant
  acceleration)
• Differential equation
• 2. Work and Energy method
• Complicated problems (non-constant acceleration)
• Integration equation
• Without resorting equations of motion
• Mechanical work
• Energy (Potentional energy, Kinetic energy)

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12.6 Mechanical Work

• Mechanical Work the product of force and
  corresponding displacement

12.6.1 Work done by a constant force
?
?
?
?
?
?
Work done by the frictional force on the block
Net work done
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12.6 Mechanical Work (continued)
12.6.2 Work done by a varying force
?
?
area
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12.6 Mechanical Work (continued)
12.6.3 Work as a scalar product

• scalar product

?
?
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12.7 Mechanical Energy

• Energy capacity of a system to do work on
  another system
• Potential energy
• Kinetic energy

12.7.1 Potentional energy
?

• associated with position or elevation
• stored in the system
• Gravitational potentional energy

?
12.7.2 Kinetic energy

• Associated with motion

translational kinetic energy
rotational kinetic energy
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12.8 Work-Energy Theorem

• The net work done W12 (from 1 to 2) is equal to
  the change in kinetic energy ?Ek

12.9 Conservation of Energy

• Conservative
• gravitational force
• Conservation of energy
• Total energy of the system (EkEp) remains
  constant throughout the motion.
• Nonconservative
• Frictional force
• Dissipates energy into heat

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12.10 Dimension and Units of Work and Energy

• Mechanical work and energy have the same
  dimension and units.

joule in SI, dyne in cgs unit
12.11 Power

• Power P Time rate of work done

When F is const,
When T is const,
Watt joule/sec
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12.12 Application of Energy Method

• Example Pendulum

?
?
Conservation of energy
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Simple Pendulum
Moment equation about O,
for small
Figure 12.13 The pendulum
T time required for one cycle
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Example 12.4

• As illustrated in Figure 12.14, consider a ski
  jumper moving down a track to acquire sufficient
  speed to accomplish the ski jumping task. The
  length of the track is 25m and the track makes
  an angle 45 with the horizontal.
• If the skier starts at the top of the track with
  zero initial speed,
• determine the takeoff speed of the skier at the
  bottom of the track using
• the work-energy theorem
• the conservation of energy principle
• the equation of motion along with
• the kinematic relationships.
• Assume that the effects of friction and air
  resistance are negligible.

?
Figure 12. 4 A ski jumper
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Example 12.4
a) Work-Energy method
Since only component in the x direction, work
done by Wx
N
W
The free-body diagram of the ski jumper
Figure 12.4(a).
According to the work-energy theorem
0
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Example 12.4
b) Conservation of energy method
Since the effects of friction and air resistance
are assumed to be negligible,
Position 1 Position 2
Figure 12.4(b)
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Example 12.4
c) Using the equation of motion
0
0
The equation in the x direction
and
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Remark

• Work-Energy theorem and the conservation of
  energy principle are more straightforward. Thus,
  Try work-energy or conservation of energy first
  before resorting to the equations of motion !
• Since the effects of nonconservative forces due
  to friction and air resistance are neglected, the
  solution of the problem is independent of shape
  of the track or how the skier covers the distance
  between the top and bottom of the track.
• The most important parameter in this
  problem affecting the takeoff speed of the skier
  is the total vertical distance between 1 and 2.

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This implies that the problem could be simplified
by noting that the skier undergoes a free fall
between 1 and 2, which are
distance apart. This is illustrated in Figure
12.17. Applying the conservation of energy
between 1 and 2
?
?
Figure 12.17 The solution of the problem is
independent of the path of motion.