298 APPLIED MECHANICS 
represents NS Oa or x will represent fon that scale. This also follows 
r 
from Art. 253, p. 287, because x is the sub-normal of the velocity curve 
“on a space base. ' 
Fig. 461 shows the velocity v and acceleration f/ plotted on a time 
base. The constructions are obvious, and clearly shown in the figure. 
259. Forces giving a Body Simple Harmonic Motion.—If a body 
weighing W lbs. has simple harmonic motion along the line BOC (Fig. 
462) under the action of a force P, then since acceleration is proportional 
ft , and using the notation and results of the 
’ WwW 9g 
2 S P 
preceding Article, = ie ; and when “= Tr; a = x = ne 
The force P must always act towards the centre O, so that while the 
body is moving towards O, P is an effort, rf 
but when the body is moving away from O, 
P becomes a resistance. 
The force diagram on a space base is 
evidently a straight line one, like the ac- B 
celeration diagram. In moving from B to 
O the work done by the effort is represented 
by the area of the triangle B&O, and is stored 
up in the body as kinetic energy, to be given 
out again in overcoming the resistance in 
moving from O to C, the work done on the resistance from O to C being 
represented. by the area of the triangle OcC. 
The foregoing results may be applied to the case of a body which has 
angular harmonic motion, Let O,A (Fig. 463) a 
to the force producing it 
Fia@. 462. 
be a bar upon which is mounted a mass M, ee 
the weight of the rod O,A and the mass M Nae 
being W, and let the whole body oscillate with Ps Se ae 
harmonic motion about an axis O, perpen- og@—m— 
dicular to the plane of the paper. Let O,A be ‘arse 
the central position, and O,B, making an angle o7-4s re 
@.with O,A, any other position. Let & be the Nae wer, 
radius of gyration of the whole body about aA Fie 
the axis O,, and let P be a force acting on the 
body at a distance from O, equal to & and 
in the direction of motion, which will give the harmonic motion. 
Fic. 463. 
2 
Applying the formula z = a to this case, r=are OB=k6, Hence 
P = ala and Pk = An? Wi0 = 4n710 , where I is the moment of inertia 
Ww gf ge? gt 
of the body about the axis O,. The product Pé is the turning moment 
of the force P about the axis O,. If T denotes this turning moment, 
2 
then T = vee: . If the force which gives harmonic motion to the body 
g 
be a force Q acting as shown, R being the perpendicular distance of its 
2 
line of action from O,, then T=QR= ae : : 
