📘 ❞ Statics and Dynamics - Andy Ruina ❝ كتاب

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Introduction to
STATICS and
DYNAMICS
Andy Ruina and Rudra Pratap
Oxford University Press (Preprint)
Most recent modifications on August 21, 2010.

Filename:Summaryofmechanics
Reference Tables: The front and back tables concisely summarize much of the text material.
0) The laws of mechanics apply to any collection of material or ‘body.’ This body could be the overall system of study
or any part of it. In the equations below, the forces and moments are those that show on a free body diagram. Interacting
bodies cause equal and opposite forces and moments on each other.
I) Linear Momentum Balance (LMB)/Force Balance
Equation of Motion Fi L The total force on a body is equal
to its rate of change of linear
momentum.
(I)
Impulse-momentum
(integrating in time)
t2
t1
Fi ·dt L Net impulse is equal to the change in
momentum.
(Ia)
Conservation of momentum
(if Fi 0 )
L = 0
L = L2 L1 0
When there is no net force the linear
momentum does not change.
(Ib)
Statics
(if L is negligible)
Fi 0 If the inertial terms are zero the
net force on system is zero.
(Ic)
II) Angular Momentum Balance (AMB)/Moment Balance
Equation of motion MC HC The sum of moments is equal to the
rate of change of angular momentum.
(II)
Impulse-momentum (angular)
(integrating in time)
t2
t1
MCdt HC The net angular impulse is equal to
the change in angular momentum.
(IIa)
Conservation of angular momentum
(if MC 0)
HC 0
HC HC2 HC1 0
If there is no net moment about point
C then the angular momentum about
point C does not change.
(IIb)
Statics
(if HC is negligible)
MC 0 If the inertial terms are zero then the
total moment on the system is zero.
(IIc)
III) Power Balance (1st law of thermodynamics)
Equation of motion Q P EK EP Eint
E
Heat flow plus mechanical power
into a system is equal to its change
in energy (kinetic + potential +
internal).
(III)
for finite time
t2
t1
Qdt
t2
t1
Pdt E The net energy flow going in is equal
to the net change in energy.
(IIIa)
Conservation of Energy
(if Q P 0)
E 0
E E2 E1 0
If no energy flows into a system,
then its energy does not change.
(IIIb)
Statics
(if EK is negligible)
Q P EP Eint If there is no change of kinetic energy
then the change of potential and
internal energy is due to mechanical
work and heat flow.
(IIIc)
Pure Mechanics
(if heat flow and dissipation
are negligible)
P EK EP In a system well modeled as purely
mechanical the change of kinetic
and potential energy is due to mechanical
work on the system.
(IIId)
Summary of Mechanics

Some definitions (Also see the index and back tables)
*r
or *x
Position e.g.,*r
i  *r
i=O is the position of a point i
relative to the origin, O.
*v

d*r
dt
Velocity e.g., *v
i  *v
i=O is the velocity of a point i
relative to O, measured in a non-rotating reference
frame.
*a

d*v
dt D
d2*r
dt2 Acceleration e.g., *a
i *a
i=O is the acceleration of a point i
relative to O, measured in a Newtonian frame.
*
F Force e.g., the force on A from B is FA from B.
*
M or *
MC D
*
M=C Moment or Torque e.g., the moment of a collection of forces
about point C.
*!
Angular velocity A measure of rotational velocity of a rigid object.
*!
B = angular velocity of rigid object B.
*
 P *!
Angular acceleration A measure of rotational acceleration of a rigid
object.
*
L  8<:
P mi
*v
i discrete
R*v
dm continuous
Linear momentum A measure of a system’s net translational rate
(weighted by mass).
D mtot
*v
cm
P *
L  8<:
P mi
*a
i discrete
R*a
dm continuous
Rate of change of linear momentum
The aspect of motion that balances the net
force on a system.
D mtot
*a
cm
*
H=C  8<:
P*r
i=C  mi
*v
i discrete
R*r
=C *v
dm continuous
Angular momentum about point C A measure of the rotational rate of a system
about a point C (weighted by mass and distance
from C).
P *
H=C  8<:
P*r
i=C  mi
*a
i discrete
R*r
=C *a
dm continuous
Rate of change of angular momentum
about point C
The aspect of motion that balances the net
torque on a system about a point C.
EK  8<:
1
2 P mi v2
i discrete
1
2 R v2dm continuous
Kinetic energy A scalar measure of net system motion.
Eint D (heat-like terms) Internal energy The non-kinetic non-potential part of a system’s
total energy.
P  P *
Fi
*v
i CP *
Mi
*!
i Power of forces and torques The mechanical energy flow into a system.
Also, P  WP , rate of work.
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I cm
xx I cm
xy I cm
xz
I cm
xy I cm
yy I cm
yz
I cm
xz I cm
yz I cm
zz
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Moment of inertia matrix about
center of mass (cm)
A measure of the mass distribution in a rigid
object.