. BALANCING 415 
A, B, C, ete., are W,0r,/9, W.w*r,/9, W,w"r,/9, ete., respectively, and the 
- eentrifugal force of the mass x 
_ the forces, it wil be 
sufficient to consider the 
is Ww*r/g. Since w?/y is common to all 
forces as represented in 
magnitude by W,?,, 
Were, Wr, ete., and 
Wr. The problem evi- 
dently reduces to the 
simple one of finding a 
foree Wr which will 
balance the forces W,7,, 
Wry, W,rs, etc., all the Fa. 679. 
forces acting in the same plane and at the same point. ‘This is easily 
done by drawing the polygon of forces shown to the right in Fig. 679. 
The closing line 2 gives the direction and magnitude of the foree Wr. 
The radius 7 may be chosen, and then W =2/r. 
347. Variation in the Pressure on the Road of an Unbalanced 
Rolling Wheel.—If the want of balance of a wheel carrying a load W 
(including the weight of the wheel) rolling on a road is equivalent to 
a weight w at a distance 7 from its axis, and if the angular velocity of the 
wheel is , there will be at every instant a radial force F equal to ww?r/g 
acting from the centre of the wheel, and the vertical component of this 
force will cause a variation in the pressure of, the wheel on the road. 
The greatest pressure on the road will be W+F when the centrifugal 
force is acting vertically downwards, and the least pressure will be W — F 
when the centrifugal force is acting vertically upwards. When the line 
of action of the centrifugal force F makes an angle 6 with its position 
when acting verti- 
cally downwards 
(Fig. 680) the ver- 
tical component of 
F is F cos 0, and the 
ressure on the road 
is then W + F cos 6. 
_ Fig. 681 shows 
the obvious con- 
struction for draw- 
ing the curve whose ry. 680. FIG. 681. 
ordinates represent 
the term F cos @ on a base CA, representing the distance travelled by 
the centre of the wheel during half a revolution. 
If F is greater than W, then once during each revolution the wheel 
will rise off the road and return with a blow. 
If D is the diameter of the wheel in feet, V the speed of the centre 
of the wheel in miles per hour, then the number of revolutions made by 
5280V _ 44V 
7™Dx60x60 307rD 
velocity of the wheel in radians per second, is aa x 44V  88V. 
the wheel in one second is , and », the angular 
307D 30D° 
