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THE PARALLELOGRAM OF ACCELERATIONS.
DEF. The resultant of two simultaneous accelerations is an acceleration, such that the distance due to it after any interval is the resultant of the distances due to the two accelerations.
103. The parallelogram of accelerations. When two simultaneous accelerations are represented (in direction and magnitude) by two straight lines OA, OB, their resultant is represented (in direction and magnitude) by OR, the diagonal of the parallelogram AOB.
Let a point have the two simultaneous accelerations represented by OA and OB, and other motions besides.
Let OA represent a celos;
Let O'O” represent the distance due to all the other motions of the moving point for the interval t seconds.
Draw O"Q parallel to 0 A to represent fa't ft.,
Then P is the position of the point after t seconds; O'Q is the distance due to the acceleration OA; QP is the.
distance due to the acceleration OB; and therefore O'P is the distance due to their resultant acceleration.
Ο"Q α' ОА 0A
QP OB AR'
That is, the distance due to the accelerations OA and OB together, viz. O'P, is the distance due to the acceleration OR.
Therefore, the acceleration represented by OR is the resultant of the accelerations represented by OA, OB.
104 Art. 103, proves that if we have two points having exactly the same motion except that one of them has the two simultaneous accelerations represented by OA, OB and the other the acceleration represented by OR, their motions will be exactly the same.
in fact that the resultant of two accelerations is an acceleration.
105. When a moving point is moving with given variable simultaneous accelerations we can find its actual acceleration at any instant by the parallelogram of acceleration.
For, if we consider another point Q moving with uniform accelerat such that at that instant it has the same velocities and accelerations as the first point P, then will, at that instant, be moving with the same actual velocity and with the same actual acceleration as the first point P.
106. The resultant of any number of simultaneous accelerations
may be found in exactly the same way as the resultant of any number of simultaneous distances or velocities.
107. When the resultant of all the accelerations which a point has is zero, the point is moving with uniform velocity.
For the motion is that of a point having no acceleration.
108. When a point has simultaneous acceleration, each acceleration adds its own velocity to the initial velocity of the point independently of any other velocities which the point may have.
109. The parallelogram of accelerations may be deduced from the parallelogram of velocities in precisely the same words as the parallelogram of velocities was deduced from that of distances by reading velocity for distance and acceleration for velocity in Art. 95.
The parallelogram of acceleration may also be stated as the Triangle of Accelerations in the words of
EXAMPLES. XXVIII. 1. A man is starting to leave a railway carriage walking directly towards the door with an acceleration 3 celos, and the train itself is stopping with retardation 4 celos; what is the direction of the resultant acceleration of the man?
2. A man in a train which is getting up its speed with an acceleration 5 celos, gives a ball an acceleration 12 celos, in the direction at right angles to the motion of the train ; find the resultant acceleration of the ball.
3. Supposing in Question 2. the man commences to throw the ball just as the train is starting and he keeps up the pressure on the ball with his hand (thus producing 12 celos) for 1 second, find the initial velocity of the ball on leaving his hand.
4. Supposing in Question 2. the train is moving at the rate of 30 miles an hour, (and with 5 celos) when the man commences his throw, and that he keeps up the pressure for 1 sec., find the initial velocity of the ball.
5. A man runs across a railway carriage which is 7 ft. across with an acceleration of 5 celos; the carriage is moving with a uniform velocity of 30 miles an hour; find the velocity with which he leaves the carriage.
6. A train is pulling up with a retardation of 10 celos; I wish to walk straight across the carriage with an acceleration of 10 celos; in what direction shall I have to apply a horizontal force to my body in order to do so ?
THE PARALLELOGRAM OF FORCES.
DEF. The resultant of two forces is that force which acting on any mass will produce in that mass the same acceleration as the two forces would produce when acting together on the same mass.
The two forces which together are equivalent to a resultant are called its components. When components are mutually at right angles they are each called the resolved parts of their resultant in their own direction.
112. PROP. The parallelogram of Forces. When two forces acting on a particle are represented (in direction and magnitude) by the straight lines OA, OB, their resultant is represented (in direction and magnitude) by OR the diagonal of the parallelogram AOB.
For, by Art. 33, when two forces act on a particle each force produces in the particle its own acceleration in its own direction; and since the forces are acting on the same mass, the accelerations produced are proportional to the forces producing them.
The resultant of the two forces must therefore be such that it produces in the same particle the resultant of these two accelerations.
Therefore the resultant force is parallel and proportional to the resultant acceleration.
That is, the two forces and their resultant are proportional to and in the same direction as the two accelerations and their resultant.
Therefore, if we represent the two forces by the two straight lines OA, OB, then the diameter OR of the parallelogram AOB represents their resultant. 'Q. E. D.
113. In considering the motion of a mass acted on by several forces either
I. We may consider that each force produces its own acceleration in the mass, and then find the acceleration of the mass by finding the resultant of all these accelerations, or II.
We may first find the resultant of all the forces, and then consider that the forces are equivalent to this resultant only, which resultant produces the actual acceleration of the mass.
The second method is the one usually adopted.
114. It should be noticed that the three propositions called the Parallelogram of Distances, the Parallelogram of Velocities, and the Parallelogram of Acceleration are geometrical facts; while the proposition known as the Parallelogram of Forces is deduced from the Parallelograin of Accelerations, by means of the definition of Force as that which produces acceleration in mass, each force producing its own acceleration exactly as if the others did not exist.
115. When the resultant of any number of forces is zero, they produce no resultant acceleration; in other words, they have no effect on the motion of any mass on which they simultaneously act.
Forces whose resultant is zero are said to be in equilibrium.
As a particular case we may notice, that when one of three forces is equal and opposite to the resultant of the other two, the three forces are in equilibrium.