Ex. 4. What is the amount of work done per stroke by an engine when the average pressure of the steam during the stroke is 38 lb. weight per square inch, the length of the stroke being 3 feet, and the diameter of the piston 14 inches;

1 foot-pound = lb. wt. per sq. in. by sq. in. by ft. 38 lb. wt. per sq. in. by 72 sq. in. by 3 ft. per stroke, 38 × 72 × 3′′ foot-pounds per stroke,

1. Reduce a kilogrammetre to foot-pounds, and reciprocally.

2. The resistance to traction on a level road is 150 lbs. per ton moved; how many foot-pounds of work are expended in drawing 6 tons through a distance of 150 yards?

3. A hole is punched through a plate of wrought-iron one half inch in thickness, and the pressure actuating the punch is estimated at 36 tons. Assuming that the resistance to the punch is uniform, find the number of foot-pounds of work done.

4. It is found, neglecting friction, that a force acting horizontally will move 10 lbs. up 5 feet of an incline rising 1 in 4. Find the work done, and find also the force parallel to the plane which will just support the weight of 10 lbs. 5. Calculate the amount of work done, independently of that lost through friction, in drawing a car of two tons weight, laden with 30 passengers averaging 11 stones each in weight, up a slope, the ends of which differ in level by 50 feet.

6. The inclination of a mountain path to the horizon is 30°; how much work is done against gravity by a man of 12 stones weight in walking a mile along the path? 7. How much work is done by means of a crane in raising from the ground the material required to build a stone wall 100 feet long, 36 feet high, and 2 feet thick, the density of the stone being 153 pounds per cubic foot.

S. A fly-wheel weighing 7 tons turns on a horizontal axle 1 foot in diameter. If the coefficient of friction between the axle and its bearing is 0075, what number of foot-pounds of work must be done against friction while the wheel makes 10 turns?

9. The resistance of friction along an inclined plane is taken at 150 lbs. for each ton of weight moved. Find the work done in drawing 2 tons up 100 feet of an incline which rises 1 foot in height for 25 in length.

10. Weights of 10 lbs. and 8 lbs. are connected by a very fine thread which rests on a rough fixed pulley, so that they hang suspended; the heavier weight is found to be just not heavy enough to fall and draw the lighter weight up; if now we suppose the weights to move uniformly, so that one goes up and the other down through 12 feet, how many foot-pounds of work are done against friction during the motion?

11. The plunger of a force-pump is 8 inches in diameter, the length of the stroke is 2 feet 6 inches, and the pressure of the water is 50 lbs. per square inch; find the number of units of work done in one stroke.

12. Determine the unit of mass in order that the absolute unit of work may be equal to the foot-pound, the second and foot being the units of time and length, and 32-2 feet per sec. per sec. being the implied intensity of gravity.

SECTION XXXIV.-KINETIC ENERGY.

ART. 159.-Kinetic Energy. By kinetic energy is meant the equivalent of the work spent in increasing the speed of a body. The work spent is proportional to the mass of the body, to its initial speed, and to the increment of speed. It is evident that the increment of speed must be taken small, in order that the initial speed may not vary sensibly. Hence

1 W = M by (L per T) initial by (L per T) increment. To find the entire amount of kinetic energy in a body, we have to 'consider that the total increment is the final speed, and that the average initial speed is half the final speed; hence

As particular cases

WM by (L per T final).

foot-poundal = lb. by (foot per second)",

erg=gm. by (em. per second).

Observe that the dimensions of each of the equivalents of W are the same.

ART. 160.-Kinetic Energy of Rotation; Moment of Inertia. We have

W = M by (L per T)2.

Suppose that a rigid body revolves round an axis, and that the whole of the mass of the body is at nearly the same distance from the axis; then since

it follows that

1 L per T = L radius by (radian per T),

W=M by (L radius)2 by (radian per T)2,

or W per (radian per T)2= M by (L radius).

The idea M by (L radius)2 depends on the body only; it is called the moment of inertia of the body. From the above the unit is seen to be equivalent to half unit of work per square of unit of angular velocity.

ART. 161.-Radius of Gyration. When the several portions of the body are at different distances from the axis, an equivalent radius can be found such that, were the whole mass situated at its end, the value of M by L2 would be the same as the sum of the actual values of M by L for the different portions. This equivalent radius is called the radius of gyration. Compare Arts. 31 and 136.

EXAMPLES.

Ex. 1. A locomotive weighing 30 tons is moving with a speed of 60 miles an hour. Express its kinetic energy in foot-pounds, and also in foot-poundals.

Ex. 2. A body, whose mass is 100 gms., is thrown vertically upwards with a velocity of 980 cm. per sec.

What is the kinetic

energy of the body, first, at the moment of propulsion; second, after half a second; third, after one second?

erg=gm. by (cm. per sec.),

100 gm.

50 erg= (cm. per sec.)2.

First, at the moment of propulsion the speed is 980 cm. per sec., ... the energy is

50 × 9802 ergs, i.e., 4·802 × 107 ergs, or 48 megalergs.

X

Second, after half a second the speed is 490 cm. per sec., for gravity in that time has deducted 490 cm. per sec.; hence the energy is

50 x 4902 ergs, i.e., 1·2005 × 107 ergs.

Third, after one second the speed has been reduced to zero; .. the energy has also been reduced to zero.

Ex. 3. Find the number of units of work required to start on a level a laden tram car of 4 tons so as to give it a speed of 6 miles an hour; and also the average force which must be applied in order that the start may be made in two seconds.

Ex. 4. What average force will bring to rest in 100 ft. a train of 30 tons which has a speed of 10 miles an hour. Also, what average force will bring it to rest in 5 seconds?

Ex. 5. What is the moment of inertia of a grindstone, 18 inches in diameter, and 4 inches in thickness, the density being 09 lb. per cubic inch ?