A Second, the velocity of P. The linear velocity of Q will then be ao L per T in the direction of the tangent at Q. The velocity of P is the component in the direction OA; hence it is -aw sin(wt) L per T. A It has the negative sign because it is towards 0 (Fig. 16). Third, the acceleration of P. The acceleration of Q will be aw2 L per T per T in the direction towards the centre 0 (Art. 121). Hence the acceleration of P, being the component along AA', will be -au cos(wt) L per T per T. It has the negative sign because it is A towards 0 (Fig. 17). .. 980 (cm. along plane per sec.)2= cm. along plane, Ex. 2. A heavy body on a level plane has simultaneously communicated to it an upward vertical velocity of 48 feet per second, and a horizontal velocity of 25 feet per second. Find its greatest height, its range, and its whole time of flight. 2 x 32 (foot per sec.)2 deducted=foot rise, Ex. 3. A tooth in the blade of a reaper describes a simple harmonic motion of one and a half inch amplitude in a period of one seventh of a second. What is its maximum velocity and its maximum acceleration ? Now, the simple harmonic component is greatest when it is equal to the circular velocity; hence its greatest value is 21 inch per sec. Similarly, the maximum acceleration is 29472 inch per sec. per sec. EXERCISE XXI. 1. A heavy body starting from rest slides down a smooth plane inclined 30 degrees to the horizon. How many seconds will it occupy in sliding 240 feet down the plane, and what will be its velocity after traversing this distance? 2. From a point in a smooth inclined plane a ball is rolled up the plane with a velocity of 16'1 feet per second. How far will it roll before it comes to rest, the inclination of the plane to the horizon being 30 degrees? Also, how far will the ball be from the starting-point after five seconds from the beginning of motion? 3. Two bodies start together from rest, and move in directions at right angles to each other. One moves uniformly with a velocity of 3 feet per second, the other moves under the action of a constant force. Determine the acceleration due to this force, if the bodies at the end of 4 seconds are 20 feet apart. 4. A stone is let fall from the top of a railway carriage which is travelling at the rate of 30 miles an hour. Find what horizontal distance and what vertical distance the stone will have passed through in one tenth of a second. 5. The time of flight of a bullet on a horizontal rifle-range is observed to be 5 seconds; find the greatest elevation it attained. 6. A body is projected horizontally from the top of a tower with a velocity of 100 feet per second; find its distance from the point of projection at the end of 2 seconds. 7. From the top of a tower 169 feet high, a ball is projected horizontally with a velocity of 100 feet per second. When will it reach the ground, and at what distance from the foot of the tower? 8. If a body is projected in a direction inclined to the horizon by 45 degrees, and strikes the horizontal plane passing through the point of projection after 5 seconds, what is the velocity of projection? 9. A ball is projected with a velocity of 60 feet per second at an elevation of 15 degrees to the horizon. What will be its range on a horizontal plane as compared with the height ascended by a body projected vertically upwards with a velocity of 30 feet per second? 10. A rifle is pointed horizontally, with its barrel 5 feet above a lake. When discharged, the ball is found to strike the water 400 feet off. Find approximately the velocity of the ball. 11. A balloon is carried along at a height of 100 feet from the ground with a velocity of 40 miles an hour; a stone is dropped from it. Find the time before the stone reaches the ground, and the distance from the point where it reaches the ground to the point vertically below the point where it left the balloon. 12. What is the average velocity of a point executing a simple harmonic motion for the time occupied in moving from the one to the other extremity of its range, its maximum velocity being 5 feet per second? 13. A particle is describing simple harmonic motion in a period of 1/10 of a second, and with an amplitude of 4 centimetres. Find the acceleration of the particle when at the extremity of its range. Find also the velocity of the particle when passing through the middle of its range. CHAPTER FOURTH. DYNAMICAL. SECTION XXII.-MASS. ART. 124. Mass and Weight. By the mass of a body is meant the quantity of matter in it. Mass is the intrinsic property of a body; whereas weight is an accidental property depending on the presence of another body in the neighbourhood. These two ideas of mass and weight are confounded in the popular mind, and are not clearly discriminated in many text-books. A clear perception of the distinction greatly facilitates the application of arithmetic in the case of many problems. Mass is a fundamental idea, and the general unit of mass is appropriately denoted by M. ART. 125.-Imperial Standard of Mass. In the Imperial system the Standard of Mass is a cylinder of platinum, constructed in 1844, and now in the custody of the Warden of the Standards. It is denominated the "Imperial Pound Avoirdupois." The previous standard of mass, which was lost along with the standard of length, was a Pound Troy. The provision which had been made for its restoration was that one cubic inch of distilled water at 62° Fahr., the barometer standing at 30 inches, weighed 252-458 grains. This provision, however, was repealed on the recommendation of a scientific committee, and the new standard |