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Analytical dynamics of a particles
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ANALYTICAL
DYNAMICS
A PARTICLE
AND OF RIGID BODIES
S.R.
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GUPTA
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ELEMENTARY
ANALYTICAL DYNAMICS
OF A PARTICLE
AND OF RIGID
BODIES
(INCLUDING MOMENT OF INERTIA, COMPOUND PENDULUM AND
MOTION OF A RIGID BODY IN TWO DIMENSIONS UNDER
FINITE AND IMPULSIVE FORCES)
FOR
B. A. Pass
&
Honours Students of Indian Universities
BY
S.
R.
GUPTA,
M.A., P.E.S.
(Retd.)
Formerly Senior Lecturer in Applied Mathematics 9
Government College, LaJiore.
1963
S.
CHAND
NEW
DELHI
BOMBAY
LUCKNOW
DELHI
JULLUNDUR
& CO.
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S.
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& GO.
Fountain
DELHI
Ram
Nagar
Mai Hiran Gate
Hazratganj
Lmninjjton Road
NEW DELHI
JCJLLTJNDCJA
LUOKNOW
BOMBAY
THIRTEENTH EDITION
(Revised & Enlarged)
Price
:
Rs. 7.00
Published by 8. Chand
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PREFACE TO THE THIRTEENTH EDITION
The present
to
book has given me an opportunity
of minor corrections and to add Chapter XVI
edition of this
make a number
containing suitable solved examples illustrating the
rigid
body under
The
finite
end impulsive
motion of a
forces.
number of examples always a popular feature of
has been supplemented by some additional problems
large
the book
taken from recent University Papers.
present form the book will be still more
useful and will continue to serve the cause of Mathematical teaching
It
is
hoped that in
its
in Indian Universities.
4,
Daryaganj, Delhi-6
S.
;
R.
GUPTA
September, 1963
PREFACE TO THE FIRST EDITION
This book is meant to be an introduction to the study of Dynamics of a Particle, and is primarily intended for the use of students
preparing for the B.A. degree examination of Indian Universities, but
be used by students preparing for the Engineering and
other competitive examinations. It includes a short account of the
may
also
Cycloidal Pendulum and the Planetary Motion
students preparing for the Honours course.*
This work differs from others of
its
for the benefit of
kind in the method of treat-
ment of the subject. Elementary \*orks already existing in the
market treat the subject geometrically, whereas I have tried to
popularize the analytical method, the employment of which has
hitherto been restricted to advanced works only.
This preference in
treatment
in accordance with
my
conviction (which, in turn, is the
my personal experience of teaching the subject for the
last twelve years and the various discussions which 1 have had with
students and teachers of Indian and Foreign Universities) that no
is
result both of
other method but the analytical can enable the beginners to grasp
the subject clearly. Thus the present volume is designed to supply
a real need of the students.
*The Treatment of Moment of Inertia, Compound Pendulum, Linear
motion in a resisting medium, Motion of a rigid body in two dimensions under
finite
and impulsive forces, centre of Percussion, Torsional vibrations and
Pendulum have been added in subsequent editions.
Ballistic
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(
To
fr
)
afford a fuller idea of the scope of this work, I
may mention
that Newton's laws of motion, motion of a particle attached to elastic
strings, motion of trains and cars on curved paths, the principle of
work and energy, the principle of relative velocity and the graphical
method of solving Dynamical problems all these are treated in
detail and profusely illustrated by carefully selected solved examples.
Besides, other examples, which are scattered throughout the book
also particularly designed to assist the student to understand
the principles of Dynamics. The ready made examples are taken
mostly from questions set at the Indian and Foreign Universities.
arc
and examples which may present some difficulty to
a beginner are marked with an asterisk and may conveniently be
omitted by him during the first reading.
Certain articles
acknowledgment is generally due to modern writers on the
subject whose works I have freely used. But I am particularly
My
indebted to Professor A.T. Banerjee of the Hindu College Delhi, for
his kindness in going through the manuscript and making valuable
suggestions. I shall be thankful to others who may suggest improvements for future editions, or even point out errors in this volume
which
Lahore
may have
escaped
my
scrutiny.
S.
;
April 1031.
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R.
GUPTA
CONTENTS
CHAPTKB
I.
PAO
Velocity, Acceleration
Velocity at a Point
Acceleration at a Point
Distance-Time Graph
Velocity-Time Graph
II*
Motion In a Straight Line
i
Motion with Constant Acceleration
Bodies falling vertically
Bodies projected vertically upwards
Space-average and time-average of velocity
Graphical Methods
Revision Questions
III.
Motion
f
Revisici
IV.
.
Questions
III
. .
. .
2
. .
. .
. .
. ;
..
..
3
7
7
. .
.
8
. .
. .
8
. .
. .
13
18
. .
% .
17
. .
. .
17
..
.
22
. .
.
26
. .
. .
. .
.
34
42
..
*&
Motion with Variable Acceleration
..
Harmonic
Motion
.
Simple
Motion of a particle attached to an elastic string
Further Examples of Motion with Variable
.
. .
.
Acceleration
///
Revision Questions
V.
Composition and Resolution of
Accelerations and Forces
VI.
..
^2
63
Velocities,
66
* .
. .
Motion on an Inclined Plane
. .
. .
in
81
k,
*juuir-- 1
83
. .
Equations of motion of a Particle moving in a plane
Motion of a Projectile
. .
Range on an Inclined Plane
. .
VII,
V
Motion of a Particle in a Plane-II
.
Angular Velocity and Acceleration
in
two
a
of
line
moving
jbining
points
Angular Velocity
*
a plane
for
Expressions for Tangential and Normal Accelerations
motion along a circle
.
Conical
Pendulum
Motion of a bicyclLfc on a curved path
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101
106
*
..
84
86
. .
Rtvifrion Questions
74
M
IV
Motion of a Particle
45
^7
54
. .
Vector quantities
Revision Questions
25
.
//
Motion in a Straight Line
I
..
II
Connected Particles
..
* .
I
Motion in * Straight Line
Newton's Laws of Motion
..
..
109
. .
10*
H2
vi
PAGE
Motion of a carriage on a curved level track
Upsetting of a carriage on a curved level track
Motion on a banked -up track
.
.
.
.
Revision Questions-- VI
YIII.
Loss or
Gam
..
Revision Questions
IX.
VIJ
Work, Energy and Impulse
127
.
..
129
. .
130
. .
131
133
. .
J
..
..
138
. .
.
.
H2
.
37
1*5
. .
.
. .
HO
..
..
150
Conservation of Energy
. .
.
.
161
.
.
.
Variable Force
.
Impulse and Energy
Impulse of a Variable Force
Motion of two bodies
Conservation of Linear
.
.
.
.
171
. .
. .
173
.
.
Direct Impact of two Smooth Spheres
Oblique Impact of two Smooth Spheres
84
18S
.
194
.
.
.
.
.
Miscellaneous Problems
196
..
.
.
200
.
201
. .
203
-
206
.
.
. .
X
84
1
.
Relative Acceleration
1
. .
..
Relative Velocity
Revision Questions
.
.
. .
Impact against a Fixed Piano
IX
181
.1
.
A projectile impinging on a
Revision Questions
17^
.
,
. .
.
plane
Kinetic energy lost by impact
Impulses of Compression and Restitution
170
..
. .
Motion of the centre of mass of a systom of particles
..
Revision Questions VIII
Collision of Elastic Bodies
166
168
.
.
..
Momentum
Further Examples on Impulse and Energy
XI.
123
125
Power
Work done by a
X.
.
. .
.
of Oscillations
.
.
Motion on the outside of a smooth vertical circle
Motion on the inside of a smooth vertical circle
Motion of a particle attached to the end of a string
Simple Pendulum
.
.
.
Motion Along a Smooth Ve^ical Circle
123
.
..
..
..
..
..
20G
213
214
..
Relative Angular Velocity of two points moving in concentric
circles.
Lines
XII*
.
,
2
.
1
T
. .
218
Hodograph
,.
..
222
Units and Dimensions
. .
.
%?
Motion along a Straight Line -IV
Bodies falling vertically in a Resisting
XIII.
.
.
.
# Quickest and Slowest Descent
Motion of a Particle
Elliptic
in
a PlaneIll
Harmonic Motion
.
.
Medium
. .
.
.
Geometrical Properties of an Ellipse
Expressions for Tangential and Normal Accelerations
.
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*
.
.
.
2
.
.
2!J2
. .
>~
2
.
240
. ,
24 1
. ,
243
.
vn
Jil
PA0S
AFTER
Motion of a Hoavy Particle on a Smooth Curve in a Vertical
'
Piano
..
..246
The Cycloidal Pendulum
. .
Motion under the action of a Central Force
Expressions for Radial and Transverse Components
of Velocity and Acceleration
..
. .
Central Orbits
Planetary Motion
Apse and Apaidal Distance
Revision Questions
XIV.
Moment
of Inertia.
248
251
..
253
. .
. .
2tJO
. .
, ,
. .
. .
264
270
.
XI
Compound Pendulum
Moment of Inertia about Perpendicular Axea
Moment of Inertia about Parallel Axes
Momental Ellipse
Momental Ellipsoid
. .
. .
. .
,
274
. .
276
. .
280
.
281
. .
.
. .
. .
. .
. .
;
. .
. .
. .
284
288
288
290
202
. .
296
296
...
Equimomental Bodies or Particles
Motion of a Rigid Body about a Fixed Axia
The Compound Pendulum
XV. Motion of a rigid body in two dimensions
.
.
D'Alenibort's principle
..
..
Toivuonal vibrations
. .
.
Ballistic
Pendulum
XVI. Further Examples of motion of
a rigid body in two dimensions
Reaction on the Axis of Rotation
Impulsive Forces
Centre of Permission
ANSWERS
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...
.
. .
305
305
...
307
307
...
314
.
32 2
. . .
vw
The
following approximate results
numerical calculations :
V2
-V/5
V?
=1-414
-2-236
=2-646
may be
V3
V6
referred to for
=1-732
=2-449
V10=3'162
V13-3-606
n=3'1416
=0-31831
1C
log10e=0-4343
Tt=9'870
Weight of one gallon of water
Weight of one cubic foot of water
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= 10 Ibs.
=62'5
Ibs.
Elementary
Analytical Dynamics
CHAPTER
I
VELOCITY, ACCELERATION
ri.
Dynamics is the science which treats of bodies in motion.
a body moves all parts of the body have not necessarily
the same motion hut if the body is a very small one, the differences
When
between the motions of
its different
parts are unimportant.
In fact,
we may regard the body as coinciding with a geometrical point and
then consider its motion. Such a body, the position of which is defined by that of a geometrical point at each instant, is called a
particle. Our main object in this book is to deal with motion of a
particle.
If a body moves in such a way that all its points move
i*ii.
through equal and parallel distances in all equal intervals of time,
however small, the body is said to have a Motion of Translation. In
such cases the body moves without rotation and between any
two 0C
%
its points there is no relative motion in space.
Its motion is completely determined when the motion of any one of its points is
known.
The body is then taken as replaced by a material partiole
coincident with this point and having its mass equal to that of the
whole body. Jt is usual to take the centre of gravity of the body at
the point whose motion determines the motion of the whole body.
When a point changes its position relative to any object
1*9.
and occupies different positions at different times, it is said to be in
motion relative to that object. The curve drawn through the successive positions occupied by the point, is called its path.
1*3.
The path of a Ynoving point may be
a straight line or
a
curve.
In earlier chapters of this book we shall deal with the motion
a particle along a straight line and we now discuss some concept!
necessary to the investigation of such a motion.
of!
In order to
1*31. Displacement in a
straight line.
determine the position of a point P in a straight line, a fixed point
must be marked on the line. The position of P is then defined, by
the length OP. If at any instant the position of a moving partiole
be P, and at any subsequent instant it be Q, then PQ. is the change
in position or the
displacement of the particle in the intervening
time.
Whenever a relation betweeft-the distance QP^x ancTtime
known, the displacement or the distance travelled dttrirur A
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t i
ELEMENTABY ANALYTICAL DYNAMICS
we put f=2, we get
ar^W andfor t=3 we get a:=U4; so that the displacement in tho
If,
for example, in the. relation ar=16f*,
third second
1*32.
is
144-64=80
ft.
Average Velocity during an Interval.
If a particle
x
jnoveu a distance x along a straight line in time
t,
then
is
said to
be the average velocity during the interval.
For example, if a train travels from Amritear to Delhi, a
distance of 300 miles in 12 hours, then $T
miles per hour is
its average velocity.
It means that if the train were supposed to
have bee n ntoving with the same velocity of 25 miles per hour during
the whole interval, the train would have covered the distance of 300
miles in 12 hours. But as a matter of fact this is not so. We know
from experit nco that the train starts from rest, gathers speed and
stops at the next station and continues doing so between every two
^=25
stopping stations.
It is evident, therefore, that the average velocity is related to
the intervals as a whole and is simply a matter of calculation. It is
distinct from the actual velocity which a moving body may have at
any particular time.
With a view to understand the motion in
duce the notion of velocity at a point.
1*33.
to
a
f
x
littlr
Then
is
o*
As
is
intro-
Suppose that a particle begin*
the
along
straight line OX and comes in tht*
after descoibing a
position
point
$x
P
(I
distance 8# in a
8#
we
Velocity at a Point.
move from a
U
all details,
JL
little
t
Further let it muvo
time $t and come in the position Q.
Unc6 x jn time
the average velocity of the particle during the interval
J.
and -consequently $x becomes smaller and smaller, the point Q
8#
taken nearer and nearer to the point P so that
chaiu<>$t
ot
terizca
more and more accurately the nature of the motion at P.
Thus
it is
natural to take
Lt
~~-T
dt
8*
M
as the velocity at
P,
Thou the derivative of x with respect to
t
expresses the rate,
of
Or
the
v
at
the
time t,
change
displacement
velocity
dx
*-ar
\
1*34.
Speed. The velocity may be positive or negative ac
ding as the particle is moving fn one or the other direction aloirj;
straight line. The magnitude of the velocity without any contwl* m
tion of direction i usually called tpttd, A man walking north at
re.tc of 4 miles per hour and another walking east at 4 miles per
!
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t
-^
i
VELOCITY, ACCELERATION
have the same speed but different
motion are different.
Uniform
1*35.
moving along a
velocities because the directions of
Velocity. In case the velocity of a particle
is the same at every instant
during an
straight line
remains constant throughout the interval, the
*f-jrdt
interval,
i.
motion
said to be uniform.
is
3
For measuring velocity one second is r usumlly taken as the
unit of time and either one foot or one centimetre as the unit of
length.
A train which travels uniformly 60 miles in one hour will travel
88 feet in one second and is said to have a velocity of 88 feet per
second or 60 miles per hour,
Velocity of
1*36. Relative
straight line.*
and
two bodies moving in
Let the velocities of two bodies A and B at any instant be u>
and let x> x' be their distances from a fixed point in the line.
ti'
The
velocity of
B relative to A
=rate of change of the distance
d
^ di
,
(X
Thus the velocity of
the velocity of
JS,
AB
~ X) = dx^ dx __ U ~~ U
~~
'
~dt
B relative to A
is
obtained by adding to
the reversed velocity of A.
A
B
are moving on parallel
and
for example, two trains
same direction with equal velocities, their relative
appears to be
velocity is -zero, so that to a person in A, the train
If the trains are moving with unequal velocities of 20
stationary.
and 25 miles per hour respectively in the same direction, the motion
If,
tracks in the
B
of
B relative
a*ains are
A
is
25
to
A
is
25
20,
i.e.,
5 miles per hour.
(20),
Beparating from
Again,
if
the
B
relative to
directions the velocity of
In this case a person in
i*
i.e., 45 miles per hour.
at the rate of 45 miles per hour.
a person in
moving in opposite
B
A
We
have seen that the
Acceleration at a Point.
1*37.
rate of change of displacement at the particular instant is the velocity at that instant, similarly the rate of change of the velocity
with respect to time
is
called acceleration
and
is
generally denoted
by/
t;
be the velocity at any instant, of a particle moving
variable
velocity and S^> the^change in the velocity in a little
yith
Let
time
gt.
*f_-
is
then the average rate of change of the velocity in the short
8^
i
interval
*
*For
relative. motion in
a plane, tec Chapter
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XL
ELEMENTARY ANALYTICAL DYNAMICS
4
The
/ is
rate of change of velocity,
therefore
i.e.,
acceleration at
any instant
^-^o *r ~~dt'
-
dv
d
.
77 =2 ~j
'
dt
d
.
dx
Thus general expressions
~j7~
/
~"*~"
dx \
"~j
dx
dt
for acceleration of a
moving
particle
are
dv
dv
d*x
or
-j- or -7^
eft
dt 2
t;
-y-
dx
The student should clearly bear in inind these three forms for
as to which one is best to employ in a given case will become
and
/,
clear from the examples given in the next article.
The word
city,
acceleration is used to denote any change in the velowhether that change be an increase, or a decrease. In other
words the acceleration
may be in the direction of motion, or in the
contrary direction, i.e., the acceleration/ may have the same sign as
that of the velocity v or may have the opposite sign.
In case the sign of/ is opposite to that of v, the acceleration
So that retardation implies
sometimes known as retardation.
decrease in the magnitude of velocity.
Just like velocity acceleration may be uniform t>r variable.
is
As acceleration is the change of velocity per unit of time,
1*38.
a point is said to be moving with n units of acceleration when a
change of n units of velocity takes place every unit of time.
Thus a point is moving with 5 foot-second units of acceleration
change of velocity of 5 ft. per second takes place every second;
it is usual to express it by saying that the point possesses an acceleration of 6 feet per second per second or 5 ft./sec. 2
Similarly 5 centimetre-second units of acceleration are written as 5 cm./sec.* and 10
miles-hour units of acceleration as 10 mi./hr. a
.
A
particle moves in a straight line in such a way
a fixed point O in the line at the end of t sees., is
3
Find (i) its distance from O, (ii) its speed, (Hi) its accelell-f-Jj-f-l /*.
ration at the end of 4 sees.
Example
i
that the distancd froi*
Here
*=< 3 45H~11==:95
ft.
when *=4,
ds
v=-=:3t*+5=63
and
/=^4a80f<&=:'24
ft.
ft>
per
per sec. when <=4,
3ec.
seconds.
ft
when t=4.
A body rnoves so that the distance x feet which it
ihe
relation x*=t*+- 6f+U where t is (he time in
by
given
Find the velocity and acceleration 4 sees, after the body
Example ?.
travels
per sec.
begin* to move.
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VELOCITY, ACCELERATION
x
Here
The
acceleration
v==0 or f=
3,
The body begins
constant.
is
to
move when
3 seconds before the time* was begun to be
i.e.,
observed.
To find the velocity 4 sees, after the body
(=1
which gives v=S ft. per sec.
put
Example
where x
we
// a particle is moving according to the law
3.
v*
= 2(x sin x -\-coa x),
is the distance described,
Here
begins to move,
v*=2(x
#+cos
sin
its
find
a,).
Differentiating with respect to
=3 2 [sin
2v
acceleration.
or,
x+x cos x
we
get
sin x].
dv
*
/=v
-=- ==# cos
ar.
4. The speed of a particle moving along the axis of x
v*~4x~x?~
Find its range of motion and if f is its acceleraby
Example
is given
tion,
show
ilwt
27*=8(2-f) (4+ff.
Here
t;
The L.H.S. of
which requires
2
=4x x*=x(t-x*)=*x$-x)
(2+ar)
.
.(1)
being positive, the R.H.S. must also be positive,
(1)
:
x^2, when x is positive, and
2, when x is negative.
(ii) x^
Hence the particle moves within the range defined by
(i)
of x lying between
and 2 and
Now differentiating
(1)
for values (of
with respect to
x
less
3**=4 2/
squaring (1), we get
t^a 1 ^
/.
from
(2)
and
(3),
the values
2.
we have
x,
or
And
than
..(2)
1
..(3)
a*)
we have
Examples
I
If a body be moving in a straight line and its distance in foot from a
+ 2J-M8, find
given point hi the line after t seconds is given by
of
at
the
end
the
2} sees,
(t)
apeed
(it) the acceleration at the end of 3} seconds,
I,
*5
(Hi) the average speed during the 4th second.
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,
**
""J&I^MENTARY ANALYTICAL DYNAMICS
a- "A particle moves along a
straight line so that after t seconds its difr
tance x from a fixed point
on the line is given by
Find
<*(*
1) .
(t) the velocity and acceleration on each occasion when it passe*
through
;
(ii)
its
distance from
O when
(tit)
its
distance from
when
its
velocity
is
zero
;
its acceleration is zero.
Describe the nature of motion.
A particle moves
3.
along a straight
8=*A cos
(nt
line,
the law of motion being
+ k) t
and varies as the distance.
particle moving along a straight line is given by
4.
i>***aa* + 26* + c.
the relation
Prove that the acceleration varies as the distance from a fixed point in
show that the acceleration
is
directed to the origin
The velocity of a
the line.
A point moves in a straight line so that its distance a from a fixed
5.
n If v be the
velocity and/ the accelerapoint at any time t is proportional to t
tion at any time t, show that v 2 a=sn//(n
1).
.
A
6.
particle
moves along a
straight lino according to the law
Prove that the acceleration varies as -~~.
The
x is given by the formula
range of motion and if /is its acceleration, show that
27*4=4(4 -/)(8+/)2.
P and Q are two inverse points with respect to a circle of radius a
8.
and centre 0. Show that if P begins to move along the line PQ in one direction,
Q always moves in the opposite direction.
Show also that the magnitude of their velocities are always unequal
except at the point where they meet.
7.
t;2=8a;
2s3
.
spei'd of a particle along the axis of
Find
[Hint.
its
OP.OQa*].
A man runs
100 yards in 19 seconds. What is his average speed in
9.
miles per hour ?
If a velocity of 16 miles an hour be the unit of velocity and a mile,
ip.
the unit of space, find the number which represents a velocity of 32 ft. per sec.
'
A passenger in a railway carriage observes another train moving on
ii.
a parallel line in the opposite direction to take two seconds in passing him but
if the other train have been proceeding in the same direction as the observer, it
would have appeared to pass him in 30 seconds. Compare the rates of motion
of the two trains.
The law of motion of a body moving along a straight
ia.
prove that
its
acceleration
[Hint.
13.
distance
Show
is
that
line is
*}*
;
constant.
~~ = 0,
..
at*
-^-=const.]
J
at
Prove that a point cannot move so that
has travelled from rest.
its
velocity shall vary as the
it
[Hint.
A shell
Show that the
initial acceleration ia zero.]
at a target 2200 yards away and explodes at the ins14.
tant of hitting. At points 1650 yards from the gun and 440 yards from tho
target, the sounds of firing and exploding of the shell arrive simultaneously.
Taking the velocity of sound to be 1100 /t. per sec. and assuming the path of
the shell straight find its average velocity.
line of men are running along a road at 8 m. p. h. behind one
15.
line of cyclists are riding in the same
another at equal intervals of 20 yds.
direction at 15 m.p.h. at equal intervals of 30 yds. At what speed must an
observer travel along tho road in the opposite direction so that whenever he
meets a runner, he also meets a cyclist.
(Roorke 1952)
A
is fired
'
A
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Distance-Time Graph* If a particle moves
1*4*
straight line, the distance moved is a function, of time.
If a graph is drawn such that
v
the time is represented on some scale
along a
i.e., by a length drawn
and the corresponding distance, by the ordinate on the same
by the
along
abscissa,
OX
,
scale, the graph is called the distancetime graph, and the velocity which
i
s
is
ctt
evidently given by the gradient
of the tangent to the curve at a particular point determined by an assigned value of
T
*
I
t.
1*5.
Velocity-Time Graph. In exactly the same way, we
can draw a graph representing the velocity in terms of time. The
at a particular
acceleration
instant
is
then represented by
the
gradient of the tangent at the corresponding point on the graph.
When the acceleration is uniform the velocity-tima graph will
be evidently a straight line.
Examples
For the velocity of a tram car it is given that
4
6
'0
1
5
7
t
3
s
2
14
27
20
35
9
Plot the space-tiraa curve and find the velocity at tha end of each.
second, honce find the acceleration at the end of the 4th, second.
1.
(Roortee)
that the above figures are connected by the relation
2a=ta -f3J and hence verify the results obtained in Example 1.
Show th&t in the velocity space curve (in which abscissae
3.
denote spaces covered and ordinates represent velocities) the sub2.
Show
normal represents acceleration,
Hint.
Subnormal =t>X
(B.U.)
-T-
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CHAPTER
MOTION
IN
II
A STRAIGHT LINEI
In the preceding chapter we have shown how by
differentiation, we di'duce the velocity from a knowledge of the
relation between the distance travelled and the time taken and how
we deduce the acceleration from the relation between the velocity and
the time.
2*i
We now consider; the more important problem of deducing the
velocity from a knowledge of the acceleration, and the distance
travelled from a knowledge of the velocity.
This is the inverse problem and hence can be solved by integration.
Motion with constant acceleration*
A
particle movex
initial velocity u and
constant acceleration /, in its direction of motion. To find the velocity
v and the distance s travelled after any time t.
2*2,
along a
straight line
from a fixed point in
it witfi
The
acceleration of the particle is given constant and equal to
distance at any time t from the point at which it starts,
/,
then its equation of motion is
and
s
is its
""'
Integrating with respect to
at
It
is
given that initially
'
t,
we have
==/lt4-c,
where
when J=0,
c is
a constant.
ds
at
-=w,
i.e.,
w=0+c
c=u.
or
--(I)
which giVes
v,
the velocity at any time.
Integrating again with respect to
f,
we
get
s=ut-\-%ft*-\-k (a constant),
when
t*0, *=*0, so that fc=0
..(2)
(v
v -T- being another expression for acceleration, the equation of motion can b$ written as v -r-=/.
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MOTION IN A STRAIGHT LINE
I
t?
Integrating with respect to
It is given that initially
p
$,
2t
when s=0, v=v,
i.e.,
1
we have -
~-p
so th.^o
The
(1)
and
last equation
(2) as follows
From
can also be obtained by eliminating
t
from
:
&=
(1)
If the particle starts from rest, so that
take the simpler forms
u=0,
these equations
2
and
*=i/*
r*2/j.
three
The
equations obtained above are of fundamental importance. They are the equations of rnoi ion of a particle moving along a
t;=/J,
straight line with constant acceleration.
Each of these equations involves four quantities and any one of
them can be obtained whenever the other throe are known.
In solving problems we select the equation* \\hich contain the
quantities that are given and the one \vhicli is required.
A partith starts with a velocity of 20 ft. per sec.
i.
100 yards in 30 sees, along a straight line. Find (i) the
acceleration supposed uniform (ii) the times cU which it is 166\ ft. from
the starting point, and (Hi) tlie time when it comes to rest.
(i) Here v =-20 ft. per sec., *=100 yds. =300 ft. and t =30 ace*.
Example
and
travels
%
SiiUttihitiii;!
these in 8~nt-^-\JP,
we
get
300 == 20 X 30 + |/X 30 x 30
/
i.e.,
or
ft.
600=^(50i
(Hi)
sec.
-
-
In the
t
f
per see. per sec.
or
(t
10)(
50)=0.
2=10 or 50 sees.
formula r=w-f/f, we have v=0, i/=20and/
020 -It or <=30 sees.
f
Thus the particle moves with a retardation of ft. per sec. per
and it comes to rest after 30 sees., then u returns back and is
again at a distance cf 166|
obtained above.
ft.
at
t~50
sees.,
the second value of*
A body waving with uniform acceleration in a
2.
has a velocity 12 ft. per sec. at a distance of JO ft. from a
and 16 ft. per #tc. at 24 ft. f/'Ofti O. Find the actdefition and
Example
straight line
point
O
velocity
'
tit
Let
O.
t/
be the velocity at O, then using the formula
f8
= !iM-2/*
f
we have
144w+2/X 10 and 2u6=w+2/x24
which give/B>4
ft.
per sec. per sec. and i/=8
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ft.
per sec.
ELEMENTARY ANALYTICAL DYNAMICS
10
7Vo cars start off to race with velocities
3.
in a straight line with 'uniform accelerations a and j3
ends in a dead htat, prove that the length of the course is
Example
trarel
2(v
u and v and
;
if the race
ro)
~*
r) (?//?
(-P) 2
Let the length of the course be s units. As the race ends in
dead heat, the cars reach the destination at the tame time, i.e., each
of them describes the distance s in the same time, say t.
Then by the question, we have
which gives
or
<=
rejecting the value J=0, which corresponds to the initial position.
2
Substituting the value oft in $=?/$-{- a( we now get
,
w
r
?/
v
r
,
x
,
,
-2(-t;)(^-~t;a)/(a-/3)
Other result^*
2*21.
For a bod}7 moving along a
2
.
st raiglit
line
with uniform acceleration, we have
(1) the average velocity is the mean of the initial and final velociis equal to the velocity at the middle of the interval.
and
ties
Lot v be the
mi
The
initial velocity
i
The moan
=
-^
average velocity
velocity
and
v,
the final velocity, then
t
t
==
5-= --^gr-^-
^2
The
velocity at the middle of the interval asti+/-o-
Thus
and
all
the three are equal.
(2) 7'Atf distance travelled
the mean velocity.
m
t
seconds
is
equal to the product oft
*
For,
mean
If the particle starts from rest
=o
(3)
and
velocity
x time.
and acquires a velocity v
in time
*=}<.
Distance travelled in the nth second
Distance described in n sees.
that described in
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(
1) sees.
MOTION IK A STRAIGHT LINE
1
I
Thus the distance described in the first, second, thin!
w/
seconds of the motion aro w
/, u
$f, u+%f
A .P. with common difference/.
+
+
Examples
f
1
rm an
II
[N.B. In the following questions, bodies arc supp'wd to be moving
along a straight line and their acceleration is taken to be uniform.]
The speed of a train increases from 30 m.p.h. to 45 m.p.h. in 11
I.
seconds. What is the acceleration of the train and what distance does it describe during this time ?
Find also tho velocity of the train whon
tance of 847
it
has travelled a further dis-
ft.
If a body, moving with Uniform acceleration passes over 300 feet
velocity increases from 50 to 70 ft. per second, find the acceleration
and tho time of motion.
The velocity of a particle, which is moving in a straight line with
3.
constant retardation, decreases 10 ft./sec. while the particle travels 10 fcvst; and
15 ft./.sec. while it travels 12J ft. from the starting point. Find the distance
the particle travels from the starting point until it comes to rest.
A car is moving at 30 m.p.h. when passing one lamp-post, and at 15
4.
m.p.h. when passing tho other. If tho lamp-posts are 110 yards apart, how far
will tlio car travel before it comes to rest if this retardation is maintained ?
How long after passing tho first iainp-post will tho car bo moving ?
whilo
^(^
its
A train is moving with a speed of 45 m.p.h. and the brakes produce
5.
a retardation of 4 ft. sec.* At what distance from a station should the brakes
be applied so that the train may stop at the station ?
If the brakes aro pu on at half this distance, with what spocd will the
train pass tho station
?
engine driver whos<* train is travelling at tho rato of 30 miles an
hour sees a danger signal at a distance of 220 yards and does his best to stop
the tram. Supposing that ho ean stop the train whon travelling 30 miles an
hour in 440 yards, show that hi* train will roach tho .signal with a velocity of
(f-^M
21y miles per hour nearly.
A body start* with velocity u and rmvos in a straight line with
7.
Constant acceleration/. When tho velocity ha^ increased to 5w the acceleration
is rovi rsed, its magnitude being unaltered.
Find tho velocity with which the
body again reaehes its storting point.
A train moving with constant acceleration takes t sees, to pass a
8.
certain point and th" ends of tho train pass tho point with velocities u, v respect ively. Find (i) the, acceleration, () the length of tho train, (Hi) tho velo(L.U)
city with which tho centre of tho train passes tho point.
A body starting with initial velocity and moving with uniform accelg.
eration acquires a velocity of 20 ft. por sec. after moving through 10 ft. and a
volouKy of 30 ft. per see. after moving through a further 15 i't. When and
where will its velocity be 40 ft. per sec.
[Hint. Write down one equation for the first distance and the second
equation for the sum of the two distances.]
6.
An
In a certain interval of 10 seconds, a point passes over 220 ft., in
10.
tho next interval of 5 seconds it passes ovor 330 ft. ; if the poinf is moving
with uniform acceleration, find its velocity at tho beginning of each of the two
,
intervals.
(/*
")
A particle moving in a straight lino with constant acceleration
11.
and BO
parses in succession through throe points A, B, C, tho distances
each being equal to 12 ft. Tho particlo takes 1 soc. to travel from A to B,
-md 2 fcocs. to travel from B to C. Dotermin" tho point at which the particle
to rest and the point at which its velocity ix 8 ft. por soo.
(Born. U.)
AB
12.
A
particle
moving with uniform
acceleration in a straight line
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