Seely F.B. Analytical Mechanics for Engineers
Подождите немного. Документ загружается.
INERTIA
AND
MASS
291
Let
a
body
A
be acted
upon
by
a
single
force
FI
causing
a
change
in
its motion
such
that the acceleration of
the
body
is
found
to be
a\.
In the same
manner
let
the forces
F2,
F%,
etc.,
act on
the same
body,
in
turn,
causing
accelerations
which
are
found
to
be
ct2, as,
etc., respectively.
The
results
of
such
an
experiment
show
that,
FI
F2 FS
=
= =
etc.
=
a constant
=
C\
(say)
.
ai
a
2
as
If
any
other
body
is acted
upon
in
a
similar
manner
!
by
other
forces
(F
f
,
F"
',
F
fn
',
etc.),
one at
a
time,
such that
the
body
is
given
the
same series of
accelerations
(01, 02,
s,
etc.)
as
was
given
to the
first
body,
it will
be
found
that,
p,
pit
pin
=
= =
etc.
=
a constant
=
2
(say)
.
ai
a
2 as
That
is,
the
manner
in
which a
body
influences its
own motion
is
by
requiring
that
any
force
which is
impressed
on
it shall
bear
a constant
ratio to the
acceleration
which the
force
gives
to the
body.
Further,
it will be noted
that
the
particular
change
in
fhe
motion
of the
body
that
is related
directly
to the
forces
acting
on the
body
and the
kinetic
properties
of the
body
is the
accelera-
tion of
the
body.
Although
the ratio
of the
impressed
force
to the
acceleration
produced
is
constant for
any
given
body
it
has
different values
for
different
bodies.
That
is,
Ci
and
2
as used
above
are,
as
a
rule,
not
equal.
Now
Ci
and
2
are
measures of
the inertia
resistance
offered
by
the
particular
bodies
to a
change
in
their
motion
since
they
measure
the
force
required
to
overcome
the
inertia
at a
definite
rate
(acceleration).
The constants
Ci
and
2,
therefore,
are
proportional,
respectively,
to the
masses
of
the
two
bodies.
That
is,
Ci
=
kM\
and
2
=
kM2
in
which
k
is a
constant
and
MI
and
M2
are
the masses
of the bodies. If
now the
mass,
MI,
of the
second
body
is taken as the
unit
by
which
the
mass of
the
first
body
is
measured,
then the
ratio
(Ci/Cy
of
the
constants
expresses
the
number
of
units
of mass
in
the
first
body.
The
numerical
value
of this
ratio,
therefore,
depends
upon
the
choice of
the
value
of
2,
and
hence,
of the unit of mass.
1
It
is
assumed
in this
experiment
that
a
motion
of
rectilinear
translation
is
given
to the
body
in each
case and
hence the
only
cha.acteristic
of the force
that
changes
is
its
magnitude.
292
FORCE, MASS,
AND
ACCELERATION
TfJ
Now,
as shown
above,
2
is
expressed by
the
equation
2
=
.
a
Therefore,
the
unit of
mass will
depend upon
the
units
in
which
F and
a are
expressed.
The
various units of
force
and
accelera-
tion,
and the
resulting
units of mass are
discussed in
detail in
Art.
141. Mass
is
considered at this
point
only
to
show its
general
connection
with
the
other
quantities
(force
and
acceleration)
involved
in the kinetics
problem
and
to
show
a
method
of deter-
mining
it.
The
experiment
suggested
above for
determining
the mass
of a
body by
finding
the ratio of an
impressed
force
to
the acceleration
produced,
is difficult
and inconvenient
to
perform.
Fortunately,
a
much
simpler
experiment
serves the same
purpose.
Careful
experiments
have
shown
that
the
weights (earth-pulls)
of
bodies,
when
the
weights
(W)
are the
only
forces
acting
on
the
bodies,
pro-
duce
the same
acceleration,
g,
of all the bodies at
a
given
location
on the earth's
surface,
the
approximate
value of
g
for
most localities
being
32.2 ft.
per
sec.
per
sec.
Hence,
if
instead
of
using
the
ratio
of
any force,
F,
to
the
corresponding
acceleration, a,
the ratio
W
-
is
used,
the
only experiment
needed to
determine the mass
of
9
a
body
is that of
weighing
the
body
on an
ordinary spring
balance
since the value
of
g
is known.
Therefore,
But,
as shown
above,
the
ratio
of the
force
F to
the
accelera-
tion, a,
produced
by
the
force is
proportional
to the
mass
M
of
the
F
body,
that
is,
=
kM.
Therefore,
W
=
kM.
g
And
if units for
W,
g,
and
M
are so chosen that
k
equals unity,
as
will
be discussed
in
Art.
141,
then the
number
of
units
of
mass
in
a
body may
be found
from
its
weight
by
the
equation
4
NEWTON'S LAWS
293
140. Newton's Laws. Newton established
his laws
of motion
from a
study
of the
motion of
planets
whose orbits and
speeds
were then
well known.
Since
the dimensions of a
planet
are
negligible
in
comparison
with
the
range
of
its
motion,
Newton's
laws
really apply directly only
to a
particle,
that
is,
to
a
body,
all
points
of
which
may
be considered
at
any
instant to have the same
acceleration.
In
most
cases of motion
of
bodies,
however,
the
accelerations of different
particles
of the
body
are
not the same.
Thus,
the accelerations
of
various
parts
of
a
rotating flywheel
or
of the
connecting
rod of a steam
engine
are
quite
different. The
expression
"
acceleration
of
a
body
"
is, therefore,
indefinite
and
meaningless, except
for
a
body having
the motion
of translation.
Newton's
laws
may
be
stated
as
follows:
First
Law.
A
particle
remains at rest or
continues
to move
uniformly
in
a
straight
line unless
acted
upon
by
an
unbalanced
force.
Second Law.
When a
single
(unbalanced)
force acts
on
a
par-
ticle the
particle
is
accelerated;
the direction of
the acceleration
is
the same as that
of
the force and its
magnitude
is
directly
proportional
to the
force and
inversely proportional
to
the mass
of the
particle.
Third
Law.
There are
mutual actions
between
any
two
par-
ticles of
a
system (body)
such
that the action
of the
one
on
the
other
is
equal,
collinear,
and
opposite
to that of the
other on the
one.
1. The first
law
is
a
qualitative
one.
It
states that
a
particle
has
inertia,
that
is,
it
resists
having
its motion
changed.
It
states
that a force must act on the
particle
if its
motion
(velocity)
is
changed
either in
direction or
in
magnitude,
that
is,
if an
accel-
eration
is
produced.
2. The second
law
is a
quantitative
one. It
states
what the
magnitude
and
the
direction of
the
force
must be
in
order
to
pro-
duce a
given
acceleration of
a
given
particle,
and
shows that
although
a
particle
cannot,
of
itself,
change
its
state of motion
it
does nevertheless influence
the
change
of
motion
caused
by
the
force,
by regulating
or
governing
the manner in
which the
accelera-
tion shall
take
place;
namely,
that it
shall be
always inversely
proportional
to the mass of
the
particle.
3.
In
the first
two laws
it
is
assumed
that a
single particle
is
acted
upon
by
a
single
force.
But the
third
law
brings
out the
294
FORCE,
MASS,
AND
ACCELERATION
fact that a
single
force
does not
exist.
One
force
requires
the
pres-
ence
of another
force. A
push
cannot
be
exerted unless
there
is
a
resisting
force to
develop
the
push;
that
is,
forces
always
occur in
pairs.
A
pair
of
forces,
however,
requires
the
presence
of
a
pair
of
particles
each
of
which acts on
the
other,
and
hence
requires
that
each
offers a
resistance to
the other.
The
resistance
which the
second
particle
offers to
the
first,
in
general,
may
arise
in
two
ways:
(1)
by
transmitting
to
the first
particle
the force
(static
reaction)
which a
third
particle (or
several
particles)
exerts on
it when
holding
it
in
equilibrium,
or
(2)
by
virtue of its
inertia
as
it
allows the first
particle
to
change
its motion
(kinetic
reaction).
Or,
the
resistance
may
be a
combination of
these
causes.
In
a
moving system
of
particles,
therefore,
the motion of each
particle
is
influenced
by
every
other
particle
in
the
system.
These
actions and
reactions which
influence
the
motion of
the
particles
of
the
body
are,
in
general, impossible
to
determine.
However,
since
they
occur
as collinear
pairs (third
law)
the
equations
of
motion
for
a
system
of
particles
(body) may
be found
without
introducing
them
into the
equations,
for,
in
finding
the
sums
of x-
and
y-com-
ponents
and the sum of
the
moments of
all
the forces
acting
on all
the
particles,
the mutual
actions
and
reactions between
the
pairs
of
particles
drop
out of
the
expressions,
since these
sums are
equal
to
zero.
Hence,
by
use of
Newton's third
law,
the second
law,
which
applies directly
only
to a
single particle
under the action
of a
single force, may
be
used to extend
the
equations
of motion
to a
system
of
particles
under
the action of
a
system
of
forces.
141.
Mathematical
Statement of
Newton's
Second Law.
Units.
Newton's
second
law
may
be
expressed mathematically by
the
equation
F
=
kma,
in
which
a is
the acceleration
of the
particle
of mass
m,
F
is
the
single
force
acting
on the
particle,
and
A;
is a
constant
factor
the
value
of which
depends
upon
the
units
used to
express
the
other
quantities
(F, m,
and
a)
in
the
equation.
But
in
Art. 139 it
was
F
W
shown
that
=
.
Hence for
any
particle,
F
W
= =
km.
a
g
NEWTON'S
SECOND
LAW. UNITS
295
It will
be
observed,
therefore,
that
although
the
mass, m,
of
F
a
particle
is
always
proportional
to
,
the
numerical
value
of
m
(that is,
the
number
of
units
of
mass
in
the
particle),
is
expressed
F
W
by (or
more
conveniently by
as discussed
in
Art.
139), only
when
units
are
chosen
for the force F and
the
acceleration a such
that
k
becomes
unity.
It is obvious
that such a
system
of units
is
desirable
for,
when
such
a
system
is
chosen,
Newton's second
law
may
be
written
simply
F
=
ma
=
a.
9
A
system
of
units which
gives
a value of
unity
to k
may
be
chosen
as
follows:
In
the
equation
F
=
kma
four
quantities
are
involved.
If a
particular
value
(unity)
is
assigned
to k then
the
units for
only
two
of the other
quantities
may
be
chosen
arbitrarily.
In
engineering,
the unit of
force and of acceleration
are
chosen
and
hence the
unit
of mass is derived from the units
selected for
force
and acceleration.
If
the
pound
is
chosen
as
the unit
of force
and the
foot
per
sec.
per
sec.
as the unit of
acceleration,
as is
usual
in
engineering,
it is evident
that the
body
which has a unit
of
mass is one
which,
when acted
on
by
a unit force
(pound),
will
be
given
a unit acceleration
(foot
per
second
2
).
That
is,
if in
the
equation
F
=
kma,
k
is made
unity,
F is
made
unity
(one
pound),
and
a is made
unity
(one
foot
per
second
2
),
then m
must be
unity
(one
unit).
The
body
which
is
given
an acceleration
of
1
ft.
per
sec.
per
sec.
by
a
1-lb.
force,
that
is,
a
body
of unit
mass,
may
be found
by experi-
ment.
The direct
method would be to
allow
a 1-lb.
force to
act
on different
bodies until one is
found
which
is
given
an
accelera-
tion of
1 ft.
per
sec.
per
sec.
However,
experiments
have
been
made
which
supply
sufficient
data from
which the
weight
of a
body
of unit
mass
may
be found.
Thus,
experiments
have
shown
that
the
earth-pull
on
any body,
if
it is the
only
force
acting, gives
the
body
an
acceleration
of
g,
the
value
of
which at
most
localities
is
32.2 ft.
/sec.
2
(approximately).
Hence,
if
the
earth-pull
on a
body
is
1
lb.,
that
is,
if
the
body weighs
1
lb.,
it
will
be
given
an
accelera-
tion of 32.2 ft.
/sec.
2
and
not
1
ft.
/sec.
2
And,
if
a
1-lb.
force
acts
on,
a free
body
which
weighs
more than 1
lb. the
acceleration
pro-
duced will
decrease
in
proportion
to
the
increase
in
weight
(New-
296
FORCE,
MASS,
AND ACCELERATION
ton's
second
law).
Therefore,
a
body
weighing
32.2
Ib.
will be
given
an acceleration of
1
ft.
per
sec.
per
sec.
by
a
1-lb.
force.
Therefore,
a
body
weighing
32.2 Ib.
is a
body
of unit
mass. A
body
weighing
64.4
Ib.
has
two
units of
mass,
or,
in
general,
a
W
body weighing
W
pounds
has
units of mass.
No
name for
the unit of
mass as here used has
gained
general
acceptance
as has
the name
pound
for the unit of force. The
name
slug
(from
sluggishness
which
suggests
inertia)
is
used to
some
extent,
particularly
in
England.
The
name
geepound
which
suggests
32.2
Ib. is also used
to some
extent, but,
in
general,
the
unit
of
mass here considered
is called
simply
a
unit of mass
(some-
times the
engineer's
unit of
mass as contrasted with
that used
by
the
physicist).
If
a
special
name
is
given
to
it,
such as a
slug
or
gee-
pound,
the
tendency
is to
think of the
unit of mass as an arbi-
trarily
chosen
unit like that
of force and
of
acceleration
and
to
forget
that it is a derived
unit
;
derived
from the units of
force and
acceleration.
Thus,
the
number of units of mass
in
a
body
when
W
expressed by
,
as discussed
above,
will
be
a certain
number
of
Ib
sec
2
'
,
for,
according
to the
dimensional
equation
for
mass,
it.
Force
F FT
2
mass
=
Acceleration
L^
JT2
Hence
the
mass
of
a
body
which
weighs
966
Ib. is
W
_
966
o
n
lb. sec.
2
g
32.2
ft.
Or,
if
each
of the
units
of
mass
is called
a
slug
or a
geepound
the
mass
of the
body
may
be
denoted
as 30
slugs
or
30
geepounds.
A
lr) SOP
quantity
expressed
in
the
units
--^
is
difficult
to visualize
it.
or
to
interpret
as a
physical
quantity,
which
accounts
for the fact
that
mass
as used
by
the
engineer
sometimes
fails to
appeal
to
common
experience.
142.
Other
Systems
of
Units.
If the
unit
of
force and unit
of
acceleration
are
chosen
arbitrarily
and
the
unit
of mass derived
therefrom,
as was
done
in the
preceding
article,
the
system
of
units
thus
obtained
is called
a
gravitational
system.
OTHER
SYSTEMS
OF
UNITS 297
If
the
unit
of
mass
and of
acceleration are chosen
arbitrarily
and
the
unit
of
force
derived
therefrom,
the
system
of units
thus
obtained
is
called
an
absolute
system.
If,
in either
the
gravitational
or
the absolute
system
of
units,
the
units are
so
chosen
that the value of
k in
the
equation
F
=
kma
becomes
unity,
the
system
is
called a
systematic
or
kinetic
system.
If
the
kilogram
is selected
as
the unit of force and the
meter
per
sec.
per
sec. as the
unit of
acceleration instead
of
selecting
the
pound
and the
foot
per
sec.
per
sec. as was done
in
the
preced-
ing
article,
then
the
number of units of
mass
in
the
body
will
be,
W
as
before, expressed
by ,
but each unit
of
the
mass will be ex-
y
pressed
in
terms
of
kilograms,
meters,
and
seconds,
instead
of
pounds, feet,
and
seconds.
A
systematic
absolute
system
of
units
commonly
used
by physi-
cists
in accurate
scientific work
and
in
engineering
electrical meas-
urements
is
a
system
in
which
the
gram, centimeter,
and second
are used
as
the units of
mass, length,
and
time, respectively.
Thus,
if
the
gram
is chosen
for
the unit of mass and the centimeter
per
sec.
per
sec. for
the
unit
of
acceleration
and if
k
is
to have a
value of
unity,
then
in the
equation
F
=
kma,
F
must be
unity.
Hence,
the force that
will
give
a mass of
1
gram
an
acceleration
of
1 cm.
per
sec.
per
sec.
is the unit of force
and
is called
a
dyne.
A
dyne
is
g-Jy
of a
gram.
This fact
may
be shown as follows:
The
earth-pull
on
a
body
of unit
mass
(gram)
produces
an
acceler-
ation of
981
cm.
per
sec.
per
sec. instead of 1 cm.
per
sec.
per
sec.
Hence,
from Newton's
law,
it follows that a
force of
^3-
gram
acting
on the same
body (unit
mass)
will
produce
an
acceleration
of 1 cm.
per
sec.
per
sec.
That
is,
^^
gram
is the unit of
force
in the
absolute
system
of units.
If the
pound
is used
for
the unit
of
mass and
the foot
per
sec.
per
sec.
for the
unit of
acceleration,
an
absolute
system
of
units is
obtained
in
which
the unit
of
force is
called the
poundal,
corre-
sponding
to the
dyne
in
the
metric absolute
system.
The
poundal
is
3^2
of a
pound.
This
fact
may
be shown
by
reasoning
in
a
way
similar to that
used
above
in
showing
that
a
dyne
is
-g-Jy
of
a
gram.
143.
Equations
of
Motion for
a
Particle. Since
Newton's
laws
apply directly only
to a
particle,
they
are limited to the
action
of a
single
force or of
a force
system
which
has
a
single
force
(not
a
298
FORCE,
MASS,
AND
ACCELERATION
couple)
for a
resultant,
for
the reason
that
the
force
system
which
acts
on
a
particle
is
a
concurrent
system
having
as its
resultant
a
single
force
acting through
the
particle
(point
of
concurrence).
Newton's
laws, therefore,
state the relation
between
(1)
the
result-
ant
force
acting
on a
particle,
(2)
the
mass of
the
particle,
and
(3)
the
acceleration
of
the
particle.
As noted
above,
however,
by
means
of the third
law,
this relation
may
be extended
so as to
apply
to the motion
of
a
system
of
particles
under
the action
of
a
system
of forces.
From Newton's second
law the
equation
of motion
of a
par-
ticle,
when
systematic
units are
used, is,
R=ma
in
which R
is the resultant of all
the
(concurrent)
forces
acting
on
the
particle,
m
is the mass
of
the
particle,
and
a
is the
acceleration
of the
particle.
The action line
of R
passes through
the
particle
since the
particle
is
the
point
of concurrence
of the forces
acting
on it. Both R and a
are
directed
quantities
and their directions
always
agree;
hence,
both sides
of the
equation
may
be resolved
into
components.
For
convenience,
three
rectangular
axes,
Xj
y,
and
z,
are
chosen
and,
since R
X
=2F
X
,
R
V
=2F
V
, etc.,
three
equations
of
motion
may
be
written
as follows:
in
which
2F
X,
2F
y
,
2F
Z
are the
algebraic
sums,
in
the
specified
directions,
of all the forces
acting
on the
particle.
It is
important
to note
that all of
the forces
acting
on the
particle
are external to the
particle.
But,
when
the
particle
considered is
one
of a
system
of
particles,
some of
the
forces
acting
on it are exerted
by
the
other
(neighboring)
particles
of
the
system
and
are,
therefore,
internal forces
when
considering
the motion
of
the
whole
system
of
particles.
NOTE. In the
problems
which
follow,
the
assumption
is
m<ade that
the
bodies
having
the motions
described
may
be
considered
to be
particles
without
introducing
serious errors in the
analysis
of
the motion.
The
problems may
be divided
roughly
into two
classes; (1)
the forces
acting
on the
particle
are
given
and
the
characteristics
(distance,
time, velocity,
acceleration)
of the
resulting
motion
are
required,
and
(2)
the
characteristics
of
the
motion
are
EQUATIONS
OF MOTIONS
FOR
A PARTICLE
299
given
and
the forces
required
to
produce
the motion are
required.
In
solving
the
problems,
therefore,
the fact
should be
kept
in
mind that the
kinematics
equations
of
Chapter
VII
are to be
used,
when
needed,
in
conjunction
with the
equations
of motion
of the
particle,
since
the
acceleration of the
particle
is
involved
both in the kinematics and
the
kinetics
of the
particle.
In
solving
problems
in kinetics it is
important
to follow a rather definite
procedure
the
main
steps
in which
may
be
outlined as follows:
1. Determine
carefully
what
is
given
in the
problem.
>
2. Determine
carefully
what is
required
in the
problem.
These two
steps,
as a
rule,
require
a sketch
with
the known and unknown
quantities
either
indicated on the sketch
or
briefly
listed.
3.
Draw a
complete free-body
diagram
of the
body
(particle).
That
is,
show the
actions
of all other
bodies
on
the
particle
considered.
This
diagram
is of
particular importance
in
determining
what
shall be used for the left-hand
member of each of the
equations
of motion for the
particle.
4. Write the three
equations
of motion for the
particle
and
select the
three coordinate axes
tp
be used
in
applying
the
equations.
Frequently, by
a
proper
choice
of
axes, only
two or
even
one of
the
equations
of motion will
be sufficient for the solution.
5. Write
any necessary
additional
equations
such as
kinematics
equa-
tions, definitions,
equations
of
equilibrium,
etc.,
which
apply
to
the
particular
problem, keeping
in mind that there
must be as
many
equations
as there are
unknown
quantities.
6. Solve the
equations,
using
the known
quantities
stated
in the
prob-
lem.
ILLUSTRATIVE PROBLEMS
342. A box
weighing
16.1
Ib. rests on the floor of an elevator.
If the ele-
vator starts
up
with an
acceleration of
8
ft.
per
sec.
2
,
what
is the
pressure
on
the
floor of the
elevator?
,
Solution.
'The box
is
given
the acceleration
of
8 ft.
per
sec.
per
sec.
by
the
upward
pressure,
P,
of
the
elevator and the
downward
weight,
W,
as
indicated in the
freebody
diagram
(Fig.
323).
Taking
the
y-axis
as
vertical,
we
need
only
one of
the
equations
of
motion,
namely,
^Fy
=
ma
v,
for
the solution of
the
problem.
[
Hence
using
this
equation
we
hav
a
v
,
Therefore,
20.1 Ib.
w
'I,
FIG.
323.
300
FORCE,
MASS,
AND
ACCELERATION
343. A
small
body weighing
4
Ib.
is
attached
to
one end of
a
string
5
ft.
long
and is made to revolve
as a
conical
pendulum
with
a
constant
angular
velocity, to,
so
that the
string
is
inclined
30
with the vertical
as shown in
Fig.
324.
What is the
tension,
T,
in the
string
and the
linear
velocity,
v,
of
the
body?
W-4
Ib.
FIG. 324.
Solution.
The
body
moves on a
circular
path
in
a
horizontal
plane
under
the
influence
of
two
forces
T
and W
as
shown
in
the
free-body
diagram
(Fig.
324).
The acceleration
of the
body
is ro>
2
,
or
,
toward the
center of the circle.
The
equations
of
motion
are:
From
(1),
From
(3),
Tcos
60
=