Useful Equations in Planar Rigid-Body Dynamics
Kinematics (Ch. 16)
Linear Motion
Angular Motion
r = r{t}
Constant Angular Acceleration
dv
a
l

dt
Constant linear Acceleration
s{t}  so  vot  12 aot 2
v{t}  vo  aot
v 2 vo 2  2ao s  so 
2D Rigid Body Kinematics, relative velocity and acceleration equations
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Useful Equations in Planar Rigid-Body Dynamics
Equations of Motion (Ch. 17)
Translation
 F = ma
Rotation about the axis thru the
center of gravity
 MG = IGα
Rotation about a fixed axis thru
point O
 MO = IOα
 MO =  (MO)k
Mass Moment of Inertia
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Parallel Axis Theorem
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Radius of Gyration
2
Work and Energy (Ch. 18)
Principle of Work and Energy
Total kinetic energy of a rigid
body rotating and translating
Work done by a force
Work done by a moment
Work done by gravity and spring
Due to gravity
Due to torsional spring
Due to linear spring
Conservation of
Energy
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Linear and Angular Momentum (Ch. 19)
Principle of Linear Impulse
and Momentum
Principle of Angular Impulse
and Momentum
Non-Centroidal
Rotation
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Solving Dynamic Problems
•
•
•
Newton’s second law (force and acceleration method) may
be the most thorough method, but can sometimes be
more difficult.
If all forces are conservatives (negligible friction forces),
the conservation of energy is easier to use than the
principle of work and energy.
Conservation of linear and angular impulse and momentum
can be used if the external impulsive forces are zero
(conservation of linear impulse and momentum, impact) or
the moment of the forces are zero (conservation of angular
momentum)
•
Energy method tends to be more intuitive and easier to
use.
•
Momentum method is less intuitive, but sometime
necessary.
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Solving Dynamic Problems
The type of unknowns and given information could point to
the best method to use.
 Acceleration suggests that the equations of motion
should be used, kinematic equations are used when there
are no forces or moments involved.
 Displacement or velocity (linear or angular) indicates that
the work and energy method is easier to use.
 Time suggests that the impulse and momentum method
is useful
 If springs (linear or torsional) exist, the work and energy
is a useful method.
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Example 1
A homogeneous hemisphere of mass M is released from
rest in the position shown. The moment of inertia of a
hemisphere about its center of mass is (83/320)mR2.
What is the angular velocity  when the object’s flat
surface is horizontal?
Conservation of energy
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Example 2
The angular velocity of a satellite can be altered by deploying small
masses attached to very light cables. The initial angular velocity of the
satellite is 1 = 4 rpm, and it is desired to slow it down to 2 = 1 rpm.
Known information:
IA = 500 kg·m2 (satellite)
mB = 2 kg (small weights)
What should be the extension length d to slow the satellite as required?
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Example 2
The only significant forces and moments in this problem are those
between the two bodies, so conservation of momentum applies. The
point about the deployed masses being small mean they have
insignificant I, and the (r x mv) for the main body is zero because it spins
about its center of mass (r = 0).
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Example 3
B
A
C
D
The left disk rolls at constant
2 rad/s clockwise. Determine
the linear velocities of joints
A and B, vA and vB. Also
determine the angular
velocities AB and BD .
Relative velocity equation
for points A and C
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B
Example 3
Relative velocity equation for points B and D
A
C
D
Relative velocity equation for points B and A
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Example 3
Setting the i and j component equal:
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Example 4
The system shown is released from rest with the
following conditions:
mA = 5 kg, mB = 10 kg, Ipulley = 0.2 kg·m2, R = 0.15 m
No moment is applied at the pivot. What is the velocity
of mass B when it has fallen a distance h = 1 m?
The only force or moment that exists and does work is gravity,
and it is a conservative force, so conservation of energy applies.
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Example 4
C
2
A
B
1
0
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Example 5
If bar AB rotates at 10 rad/s,
what is the rack velocity vR?
Relative velocity equations
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Example 5
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Example 5
i components
j components
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Example 6
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Example 6
Force and motion diagrams for the plate.
Kinematics
G
A
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Relative acceleration equation for points G and A
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Example 6
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Example 7
Force diagram
Motion diagram
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Example 7
G
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Example 8
vC/B  CB
vB = 21  AB
vC   slider dir.
 = (vC/B)/CB
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Example 8
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Example 9
At the instant shown, the disk is rotating with an
angular velocity of  and has an angular acceleration
of α. Determine the velocity and acceleration of
cylinder B at this instant. Neglect the size of the
pulley at C
Use position coordinate method
Determine the length s = AC in terms of the angle θ ( Law of Cosines)
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Example 9
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Example 10
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Example 10
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