310 APPLIED MECHANICS 
the inner dead centre radius, « the distance of the piston from the outer end of 
its ae and v the velocity of the piston. 
When the length of the connecting-rod is equal to that of the crank, show 
that ‘the stroke of the piston is four times the length of the crank. Also, if the 
crank has uniform velocity, show that the piston has simple harmonic motion, 
aud that the maximum velocity of the piston is twice the velocity, of the 
crank pin, 
4. In a direct-acting engine the connecting-rod is 50 inches, and the crank 
10 inches long. If the crank makes 120 revolutions per minute, calculate the 
mean velocity of the piston, in feet per minute, also the velocity of the piston, 
in feet per minute, when the crank and connecting-rod are at right angles to 
one another. 
5. Estimate the greatest and least forward velocity of the piston of a loco- 
motive engine, relative to the rails, when the train is running at 50 miles per 
hour, the diameter of the driving wheels being 66 inches, the length of stroke 
27 inches, and the length of the engine connecting-rod 54 inches. [Inst.C.E.] 
6. Construct the piston acceleration diagrams, as shown in Fig. 474, p. 303, 
for the following cases :—(1) l/r=~, (2) d/r=4°5, (3) l/r=2, where /=length of 
connecting-rod, and »=radius of crank. The three sets of diagrams to be 
drawn on the same corresponding bases, or, in the case of the polar diagrams, 
from the same pole, in order to show the differences due to variations in the 
value of Z/r. Take r=10 inches, V=10 feet per second, and linear scale 4. 
- Construct on the diagrams the acceleration scale, showing feet per second per 
second. ‘Take from the diagrams the values of f, the piston acceleration, in feet 
per second per second, when @=30°, and state the results. 
7. Same as preceding exercise, “but for the following cases:—(1) J/r=1- 1, 
(2) Ye 1°144, (3) U/r=1°2. 
. If tke acceleration of a piston is 350 feet per second per second when it 
has haven 4 inches from one end of its stroke, which is 24 inches, at what 
speed is the crank shaft running, in revolutions per minute? Assume an 
infinite connecting-rod. 
9. In a direct-acting engine, 7=length of connecting-rod, r=radius of Coes 
n=l/r, x=distance of piston from outer end of stroke, V= velocity of crank pin, 
v=velocity of piston, and f=acceleration of piston. Show that— 
(1) when the crank is perpendicular to the line of stroke, 
ss v3 1 
=n+1-,/n?-1, v=V, and f= F = West 
n _— 
(2) when the crank is at right angles to the a 
ee ee 24 . nt+] 
~=nt1— jn +l, gesett , and f= * NOE 
v_nv/4n?-1 —V?_ n(4n*— 6n? + 1) 
(3) when z=7, ‘ve, and f= poet (Qt 
The upper sign in the value of f in each case applying to the ‘‘ in” stroke, and 
the lower sign to the ‘‘ out »” stroke. 
10. If w is the angular velocity, and a the angular acceleration of the con- 
necting-rod, then, using the notation of Exercise 9, show that (1) when the 
crank and connecting-rod are in a straight line w= V/nr,anda=0; (2) when the 
2 
crank is perpendicular to the line of stroke, w=0, and a= Vata . 2 and 
n? — 
(3) when the crank is perpendicular to the connecting-rod, 
~1,V?, 
n> 2 
11. Construct the connecting-rod angular velocity diagrams, as shown in 
Fig. 477, p. 306, for the following cases :—(1) l/r=4'5, (2) l/r=2, (3) Z/r=1, 
where /=length of connecting-rod, and r=radius of crank, Take r=10 inches, 
velocity of crank pin 10 feet per second, and linear scale 3. Construct the 
angular velocity scale, showing radians per second, Take from the diagrams the 
‘values of w in radians per second when @=30°, and state the results. 
12. In an ordinary steam-engine the stroke is 18 inches, the length of the 
connecting-rod is 36 inches, and the revolutions are 400 per minute. The 
