the area of the rectangle BC will represent to scale 
WORK AND ENERGY 25 
Work done = ,(k, — h,) + wo(Ig — hy) + w4(Ig — hy) + ete. 
oR pak ~~ ag leat — (wyh, + thy + wyhy + ete...) 
= - by a property of the centre of gravity 
= W(K-H). 
41. Diagram of Work.—If a straight line OC (Fig. 22) represents 
to seale the distance 8 through which a body moves under the action of 
a force, and if OB drawn at right angles to OC 
represents to scale the magnitude P of the force, then 8 
the work done by P in acting through the distance 9 
8. For, let the linear scale be 1 inch to m feet, and 
the force scale 1 inch to m Ibs.; also let OC bez P12 
inches, and let OB be h inches long. Then the magnitude P of the force 
is hn lbs., and the distance S is /m feet. Work done=PS =hnlm=hlmn 
=Aman, where A is the area of the rectangle BC in square inches. 
If the body moves along the horizontal path represented to scale by 
OC (Fig. 23) under the action of a force which varies in magnitude, and 
if the magnitude of the force at each point of the path is represented to 
scale by the height of the diagram BC at that point, then the area of the 
diagram will still represent to scale the work done. Consider the work 
done from E to F, two points near to one another, and Jet the dimensions 
of the diagram be in inches, and let the linear and 
force scales bé the same as before. At E the magnitude D 
of the force is ED x  lbs., and at F the magnitude of 
the force is FH x lbs., and since E and F are near to 
one another DH may be considered to be a straight line. QE F c 
and the mean magnitude of the force between E and F Fic. 23. 
is }(ED+FH)xxn lbs. The work done between E and 
F is }(ED+FH) xxx EF xm foot-pounds. But the area of EDHF 
is }(ED+FH) x EF square inches, therefore the work done from E to 
F is equal to the area of the vertical strip DF in square inches multiplied 
by m and by n. Hence dividing the whole diagram BC into vertical 
narrow strips, it follows that the work done in moving the body through 
the distance represented by OC is equal to Amn, where A is the area 
of the diagram BC in square inches. 
42. Turning Moment— Work in Turning.—When a force P acting 
on a body causes that body to rotate about a fixed axis, the line of action 
of the force being in a plane perpendicular to that axis, the product of P, 
the magnitude of the force, and the perpendicular distance R of its line 
of action from the axis is called the turning moment or torque of the driving 
force P. If P is in pounds and R is in feet, the turning moment PR is 
_in pound-feet or foot-pounds ; but if P is in pounds and R is in inches, PR 
is in pound-inches or inch-pounds. If the line of action of P is not in a 
plane perpendicular to the axis of rotation, but makes an angle @ with 
that plane, then the turning moment is PR cos @. 
tf R, the leverage of P, remains constant during the rotation of the 
body, and if the magnitude of P is also constant, then if is the angle in 
radians through which the body turns, the distance through which P acts 
is oR, and the work done by P is PoR, or Tw, where T is the turning 
moment. If the leverage R or the force P, or both, should vary, then if 
T is the mean turning moment the work done is Tw. 
