Exam (elaborations) TEST BANK FOR An Elementary Treatise on the Dynamics of A particle By S L Loney-Converted

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