TEST BANK FOR An Elementary Treatise on the Dynamics of A particle By S L Loney

FUNDAMENTAL DEFINITIONS
AND PRINCIPLES
1. The velocity of a point is the rate of its displacement, so that, if P
be its position at time t and Q that at time t +4t; the limiting value
of the quantity
PQ
4t
, as 4t is made very small, is its velocity.
Since a displacement has both magnitude and direction, the velocity
possesses both also; the latter can therefore be represented in
magnitude and direction by a straight line, and is hence called a vector
quantity.
2. A point may have two velocities in different directions at the same
instant; they may be compounded into one velocity by the following
theorem known as the Parallelogram of Velocities;
If a moving point possess simultaneously velocities which are represented
in magnitude and direction by the two sides of a parallelogram
drawn from a point, they are equivalent to a velocity which is
represented in magnitude and direction by the diagonal of the parallelogram
passing through the point.
Thus two component velocities AB, AC are equivalent to the resultant
velocity AD, where AD is the diagonal of the parallelogram of
which AB, AC are adjacent sides.
1
2 Chapter 1: Fundamental Definitions and Principles
If BAC be a right angle and BAD = q , then AB = ADcosq ,
AC = ADsinq , and a velocity v along AD is equivalent to the two
component velocities vcosq along AB and v sinq along AC.
Triangle of Velocities. If a point possess two velocities completely
represented (i.e. represented in magnitude, direction and sense) by
two straight lines AB and BC, their resultant is completely represented
by AC. For completing the parallelogram ABCD, the velocities
AB;BC are equivalent to AB;AD whose resultant is AC.
Parallelepiped of Velocities. If a point possess three velocities
completely represented by three straight lines OA;OB;OC their resultant
is, by successive applications of the parallelogram of velocities,
completely represented by OD, the diagonal of the parallelepiped
of which OA;OB;OC are conterminous edges.
Similarly OA;OB and OC are the component velocities of OD.
If OA;OB, and OC are mutually at right angles and u;v;w are the
velocities of the moving point along these directions, the resultant
velocity is
p
u2+v2+w2 along a line whose direction cosines are
proportional to w;v;w and are thus equal to
u
p
u2+v2+w2
;
v
p
u2+v2+w2
and
w
p
u2+v2+w2
Similarly, if OD be a straight line whose direction cosines referred
to three mutually perpendicular lines OA;OB;OC are l;m;n,
then a velocity V along OD is equivalent to component velocities
lV;mV;nV along OA;OB, and OC respectively.
3. Change of Velocity. Acceleration. If at any instant the velocity
of a moving point be represented by OA, and at any subsequent
instant by OB, and if the parallelogram OABC be completed whose
LONEY’S DYNAMICS OF A PARTICLE WITH SOLUTION MANUAL (Kindle edition) 3
diagonal is OB, then OC or AB represents the velocity which must be
compounded with OA to give OB, i.e. it is the change in the velocity
of the moving point.
Acceleration is the rate of change of velocity, i.e. if OA;OB represent
the velocities at times t and t+4t, then the limiting value of
BA
4t
(i.e. the limiting value of the ratio of the change in the velocity to the
change in the time), as 4t becomes indefinitely small, is the acceleration
of the moving point. As in the case of velocities, a moving
point may possess simultaneously accelerations in different directions,
and they may be compounded into one by a theorem known
as the Parallelogram of Accelerations similar to the Parallelogram of
Velocities.
As also in the case of velocities an acceleration may be resolved
into two component accelerations.
The results of Art. 2 are also true for accelerations as well as velocities.
4. Relative Velocity. When the distance between two moving points
is altering, either in direction or in magnitude or in both, each point
is said to have a velocity relative to the other.
P R Q
A B
Suppose the velocities of two moving points A and B to be represented
by the two lines AP and BQ (which are not necessarily in the
4 Chapter 1: Fundamental Definitions and Principles
same plane), so that in the unit of time the positions of the points
would change from A and B to P and Q.
Draw BR equal and parallel to AP. The velocity BQ is, by the Triangle
of Velocities, equivalent to the velocities BR, RQ, i.e. the velocity
of B is equivalent to the velocity of A together with a velocity
RQ.
The velocity of B relative to A is thus represented by RQ.
Now the velocity RQ is equivalent to velocities RB and BQ (by the
Triangle of Velocities), i:e: to velocities completely represented by
BQ and PA.
Hence the velocity of B relative