EFFORT, ACCELERATION, AND VELOCITY DIAGRAMS 299 
For example, take the case of the simple pendulum (Fig. 464), where 
a small body of weight w swings in a vertical plane in a Nr 
small arc of a circle of radius/, T=w-O,N=wlsin@. ; | 
Since @ is a small angle, sin @ may be taken equal to 6, ae] 
also I = wi?, hence ear? ' 
gS a6 
T = 10 4°00 > ' ‘a \ 
gt an A 
therefore ‘i it 
Ar] 1 “" yi! 
?—=—— ort=2r ~ Pi ‘ i, 
il J g’ £ So 
a well-known result, which may be proved in other 
ways. Fic. 464. 
Exercises XVIIb. 
1. A body of 60 lb. has a simple vibration, the total length of a swing being 
3 feet ; there are 200 complete vibrations (or double swings) per minute; calculate 
the forces which act on the body at the ends of a swing, and show on a diagram 
to scale what force acts upon the body in every position. [B.E.] 
2. A weight of 5 lbs, is supported by a spring. ‘The stiffness of the spring is 
such that putting on or taking off a weight of 1 lb. produces a downward or 
upward motion of 0°04 foot. What is the time of a complete oscillation, neglect- 
ing the mass of the spring ? [B.E.] 
8. A weight of 10 lbs. suspended by a spiral spring makes 107 complete 
vertical oscillations in 1 minute. What weight applied gradually will lengthen 
the spring 1 inch? 
4. A U tube (Fig. 465) of uniform bore contains a liquid which fills a length 
of 2 feet of the tube. Find the time of a complete 
oscillation of the liquid in the tube. 
5. A steel wire 0°16 inch diameter fixed at its upper 
end and guided at the lower end has a wheel weighing 
12°3 lbs. fixed to it near its free end and 40 inches 
from its fixed end, as shown in Fig. 466. The wheel is 
turned through a small angle and then liberated, and it 
is then found to make 4 complete oscillations in 6 
seconds, Taking the modulus of rigidity of the wire as 
13,000,000 Ibs, per square inch, calculate the radius of 
gyration of the wheel about its axis. 
6. A fly-wheel weighs 5 tons, and its radius of gyra- Fra. 465. Fia. 466. 
tion is 6°30 feet. It is at the end of a shaft 40 feet 
long, 5 inches diameter, modulus of rigidity of material 12,000,000 lbs. per square 
inch, what is the natural time of torsional vibration of the system, neglecting the 
inertia of the shaft itself ? . [B.E.] 
7. A fly-wheel weighs 5 tons, and has a radius of gyration of 6 feet. It is at 
one end of a shaft, the other end of which is fixed. It is found that a torque of 
200,000 1b.-feet is sufficient to turn the wheel through 1°. If the wheel is twisted 
slightly and then released, how many vibrations per minute will it make? [B.E.] 
8. A uniform circular plate, 1 foot in diameter, and weighing 4 lbs., is hung 
in a horizontal plane by three fine parallel cords from the ceiling, and when set 
into small torsional vibrations about a vertical axis is found to have a period of 
8 seconds. A body, whose moment of inertia is required, is laid horizontally 
across it, and the period is then found to be 5 seconds, the weight of the body 
being 6 lbs. Find the moment of inertia of the body about the axis of oscilla- 
tion, [Inst.C.E.] 
