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MOTION AND FORCE 17 
20. Composition and Resolution of Velocities and Accelerations.— 
Bs Two or more velocities may be reduced to a single velocity, and two or 
_ more accelerations may be reduced to a single acceleration by composition, 
_ exactly as for forces. Also, conversely, a single velocity may be resolved 
into two or more velocities, and a single acceleration may be resolved 
_ into two or more accelerations, exactly as for forces. For the com posi- 
tion and resolution of forces, see Chapter IV. 
21. Radial Acceleration of a Point moving in a Circle with 
Uniform Velocity.—Let a point A (Fig. 14) be moving with a uniform 
velocity v along the circumference of the circle, 
whose centre is C and radius CA=r. Let Ade- P : 
scribe the small arc AB in the time ¢. The 
velocity of the point when at A is in the direction @ 
of the tangent to the circle at A or perpendicular 
to CA, and the direction of the velocity of the 
point when at B is in the direction of the tangent 
to the circle at B or perpendicular to CB. Draw 
OP perpendicular to CA and equal to v ; also draw c r 
OQ perpendicular to CB and equal to », and join Fic. 14 
PQ. The change in the velocity of the point in i 
moving from A to B is represented by PQ=wu. If ACB isa very small 
angle, the difference between the chord AB and the arc AB may be 
‘neglected, and ACB and POQ are then similar triangles. 
PQ _ AB - &@ vt uw ¥ u 
OP = Ga’ that is, Phar or Vanes But > =f 
Hence 
is the rate of change of velocity of the point moving in the circle, 
therefore. f= eS Also, when the angle ACB is indefinitely small the direc- 
tion of u is perpendicular to that of v, and is therefore at any instant in 
the direction of the radius of the circle from the moving point at that 
instant. Therefore if a point moves witha uniform velocity v in a circle of 
radius 7, there is a constant acceleration f -2 towards the centre of the 
circle. If v is in feet per second, and / in feet per second per second, then 
- » must be in feet. If » is the angular velocity of A about C in radians 
| per second, then w= and f= wr. 
22. Instantaneous or Virtual Centre.—Let A and B (Fig. 15) be 
two definite points in a rigid body which has plane motion, the plane of 
the paper being the plane of motion of. the points 
A and B. Suppose that at the instant that the b 
body is in the position shown the point A is 
moving in the direction Aa, and that the point 
B is moving in the direction Bb. Draw AC and 
BD perpendicular to Aa and Bé respectively, and 
let AC and BD meet at O. Just for an instant /¥ a 
the point A can be made to revolve about any 
_ point in AC without altering the direction of its Fic. 15. 
motion. Also, just for an instant the point B 
can be made to revolve about any point in BD without altering the 
B 
