FRICTION AND LUBRICATION 279 
6 A piece slides on a bar of square section by the action of a force P, as 
‘shown in Fig. 434. If Q is the force pressing the sliding 
on the bar, show that P=yQ,/2. la 
_ 9. An inclined plane has a base 90 feet long, and is. ae = 
20 feet high; the coeflicient of friction between it and <» === 
a body weighing 800 Ibs. placed on it is 0°3: how many ; 
foot-pounds of work are done in drawing the body up the 
whole length of the plane, and how many in drawing it Fia. 434. 
down the plane, the pulling force being parallel to the plane ? 
_ 8. What must be the effective sgpen) ancl of a locomotive which moves at 
the steady speed of 45 miles per hour on level rails, the resistances being 15 Ibs. 
per ton, and the weight of the engine and train 220 tons? If the rails were laid 
at a gradient of 1 in 130, what additional horse-power would be required ? 
9. If the engine of the preceding exercise exerts the same power on the 
incline as on the horizontal, at what speed, in miles per hour, would it ascend 
an incline of 1 in 180 with the same train, assuming the frictional resistances to 
be unaltered ? 
10. Calculate the horse-power required to drive a motor car weighing 1 ton 
up an incline of 1 in 14 at 24 miles an hour, supposing it to reach the same 
velocity when running freely down the incline. (U.L.] 
11. A window sash (Fig. 435), of height h, is counterbalanced by weights ; 
show that it can be raised by a vertical force, if its point of 
application is not further than 44 cot¢ from the centre, where ¢ 
is the angle of friction. [B.E.] 
12. A square threaded screw, whose mean diameter is 1} 
inches, and pitch } inch, has its axis vertical, and carries at its 
upper end a weight W, which is raised by the application of a 
torque T to the screw. It was found by experiment that the Fia. 435. 
relation between T and W was given by the equation T=) +8, where T is in 
inch-lbs. and W in lbs. Determine the values of » for the screw and nut when, 
(1) W=50 lIbs., (2) W=100 Ibs., (3) W =200 lbs. 
13. Particulars are given in the following table of certain standard Whitworth 
Outside diameter (d), inches 4 1 1} 2 24 3 34 
Diameter at bottom of 
thread, inches ;: - | 0°393 | 0-840 | 1:287 | 1-715 | 2-180 | 2°634 | 3-106 
Number of threads per inch 12 8 6 | 43 4 34 34 
screws in which the angle of the V-thread is 55°. Calculate 
the efficiencies of these screws, taking ~=0°15, and plot 
the results, taking efficiencies for ordinates, and d for 
abscissze. 
14. A simple screw-jack (Fig. 436) has a square threaded 
screw whose mean diameter is 1-8 inches and pitch 0:4 inch, 
If the coefficient of friction between the screw and nut is 0°12, 
what force at the end of a lever 24 inches long, measured 
from the axis of the screw, will raise a load of 2 tons? 
Assume that the load rotates with the screw, thus eliminat- 
ing collar friction. Calculate also the efficiency. What 
force at the end of the lever will be necessary to lower the 
load of 2 tons? 
15. A weight W is carried by a square threaded screw 
and nut, as shown in Fig. 437. Outside diameter of screw, 
15 inches; pitch, 0-4 inch; thickness and depth of thread, 
0-2 inch. Outside diameter of collar on nut, 3 inches; in- 
side ditto, 1-5 inches. The nut is rotated by a force of 80 lbs. 
at the end of a spanner 18 inches long. Find the load W, in 
Ibs., (1) when friction is neglected, (2) when yw for the collar 
and for the screw is 0°2. 
16. A flat pivot has to carry a load of 5000 Ibs., and the 
intensity of the pressure (assamed to be uniform) is to be 
