obtain y = 2x, which is true for all values of t. Hence the boat moves in a straight line (Corollary 1, page 57). Let C denote the position of the boat one hour after starting, and let OA and OB be the coördinates of C. The line OA represents the velocity of the water with reference to the earth, the direction of the line being that in which the water flows, and the length of the line being the number of miles per hour the water moves. Similarly, OB represents, in direction and magnitude, the velocity of the boat with reference to the water. (If the water were at rest, OB would represent the actual motion of the boat, in one hour, with reference to the earth.) With reference to the earth, the boat moves along the line OC, and in one hour moves from O to C. Hence OC represents, in direction and magnitude, the actual, or resultant, velocity of the boat. The resultant velocity is readily computed. Its magnitude is OC = √ÕA2 + AC2 = √√22 + 42 = 4.47 miles per hour. Its direction may be given by ZAOC. Since tan ZAOC AC/OA = 4/2 = 2, we have ZAOC = 63°.43. = This illustration exemplifies the law known as the Parallelogram of velocities. If a body is subjected to two different velocities, represented in direction and magnitude by two lines OA and OB, the actual or resultant velocity is represented by the diagonal OC of the parallelogram determined by OA and OB. If a body is moving through the air in any way, it has an acceleration of 32 feet per second per second directed toward the center of the earth (see Section 22, page 63). This means that its vertical velocity is increased or decreased each second by 32 feet per second, according as it is falling or rising. Its horizontal velocity is uniform, and is not affected by the action of gravity. No account is taken here of the resistance of the air, which would make the discussion very much more complicated. The acceleration due to gravity may be represented by a vertical line running downward whose length is 32. In like manner, any acceleration may be represented, in direction and magnitude, by a line. A force may also be represented, in direction and magnitude, by a line OA, O representing the point of application of the force. Accelerations and forces may also be combined according to the parallelogram law. Finding the resultant of two velocities is known as the composition of velocities, the two given velocities being called components of the resultant. The converse problem of determining two component velocities which have a given resultant is called the resolution of velocities. The same terms are also used with reference to accelerations and forces. If the components are at right angles, the parallelogram is a rectangle, and the composition or resolution may be effected by solving a right triangle. If the parallelogram is not a rectangle, it is necessary to use the methods to be developed in Sections 71-73. Problems of this sort will be found in the exercises following Section 73. EXAMPLE 2. A ball rolls down a plane whose inclination is 30°. Resolve the acceleration due to gravity into two components parallel and perpendicular to the plane. Let OC = 32 represent the acceleration due to gravity. Through O and C draw lines parallel and perpendicular to the plane DE, forming the rectangle OABC. Then OA and OB represent the required components, for the resultant of OA and OB is OC (parallelogram law). In the triangle OAC, OC E Α. 80 30 B F Hence = As there is no motion in a direction perpendicular to the plane, the component OB is neutralized by the plane. The component parallel to the plane, OA 16 feet per second per second, gives approximately the effective acceleration with which the ball rolls down the plane. If the ball starts to roll from rest, how fast will it be moving at the end of one second? At the end of two seconds? At the end of four seconds? If a ball is rolled up the plane, how will its velocity be affected during any second? If it is started up with a velocity of 50 feet per second, how long will it roll up the plane? 67. Conditions of Equilibrium of a Particle. In measuring a force we shall use the pound as the unit. If a number of forces act on a particle, which we take as the origin of a system of coördinates, each of the forces may be resolved into two components, one acting along each axis. If the particle is in equilibrium, there is no motion in the direction of the x-axis, and hence the algebraic sum of the components along that axis must be zero. Similarly, the sum of the components along the y-axis must be zero. And conversely, if each of these sums is zero the particle must be in equilibrium. Hence we have the Theorem. A particle is in equilibrium under the action of any number of forces if and only if the sum of the components in each of two perpendicular directions is zero. In applying this theorem, first determine all the forces acting on the particle, and then choose the perpendicular directions. If two of the forces are at right angles to each other, choose these directions. EXAMPLE 1. A particle weighing 4 ounces is supported on a smooth plane whose inclination is 30° by a cord parallel to the plane. Find the tension of the cord and the pressure on the plane. 300 R 4 sin 30 FIG. 108. W (1) Its weight, W = 4, acting vertically. (2) The tension of the cord, T, acting parallel to the plane. (3) The resistance of the plane, R, acting perpendicular to the plane. Since R and T act at right angles, we proceed to resolve all the forces into components parallel and perpendicular to the plane. Choosing these directions facilitates the work, because W is then the only force whose components must be determined. By the parallelogram law and the solution of a right triangle, the component of W parallel to the plane is found to be 4 sin 30°, and that perpendicular to the plane is 4 cos 30°, both acting downward. Then by the theorem, taking the directions of R and T as positive, we have In pointing out the part played by mathematics in the illustration of the scientific method in Example 1, page 79 (see bottom page 80), it was stated that a more satisfactory verification of the law obtained would be indicated later, by deducing from the law some fact that may be verified by an experiment of a different nature. This deduction will be made in Example 2 below. Friction between an object and a plane acts parallel to the plane, and in the direction opposite to that in which the object moves or tends to move. When motion is about to take place, the force of friction is equal to the coefficient of friction for the surfaces in contact (see page 81) multiplied by the pressure of the object on the plane in the direction perpendicular to the plane. It is this law that we wish to verify. EXAMPLE 2. A block of wood weighing 20 grams rests on a horizontal board. If the coefficient of friction is 0.29, the result obtained in Example 1, page 79, at what angle may the board be tipped before the block are: (1) The resistance of the plane, R, (2) The friction, F, and (3) The weight W = 20. It is known that Ꭱ since the pressure on the plane is numerically equal to the resistance of the plane. Resolving W into components parallel and perpendicular to the plane, as in Example 1 above, we get where is the angle at which the block is on the point of sliding. Substituting in (1) the values of F and R given by (2), we get, 20 sin 0 = 0.29 × 20 cos 0. Dividing both sides by 20 cos 0, we obtain sin (2) This result may be readily tested by experiment. If it is found that when one end of the board is gradually raised the block begins to slide when the inclination is a little over 16°, then we have a verification of the correctness of the deduction by which was found, and also a verification of the law (1) on which the deduction was based. That is, we have a verification of the law obtained in Example 1, page 79, which was re-stated on page 81. EXERCISES 1. A boy kicks a football so that it would roll across a street at the rate of 15 feet per second, and simultaneously a second boy kicks it so that it would roll along the street with a velocity of 12 feet per second. Find the actual velocity of the ball (magnitude and direction). 2. A man walks across a canal boat. If his velocity with reference to the earth is 7 feet per second in a direction inclined at 30° to the bank of the canal, find the velocity of the boat and that at which he walks. 3. A ball is thrown into the air at an angle of 40° with a velocity of 30 feet per second. Find the horizontal and vertical components of the velocity. 4. If a ball is placed on a plane inclined at 20°, find the acceleration with which it rolls down the plane, and how fast it will be moving at the end of 3 seconds. 5. A ball is rolled up a plane inclined at 10° with an initial velocity of 20 feet per second. How long will it roll up the plane? 6. What is the inclination of a sidewalk if a ball rolls down it with an acceleration of 8 feet per second per second? 7. A cake of ice weighing 200 pounds is held on an ice slide at an ice house by a rope parallel to the slide. If the inclination of the slide is 45°, find the pull on the rope and the pressure on the slide. 8. An automobile weighing 2500 pounds stands on a pavement inclined at 12°. What force do the brakes exert? 9. Will a box slide down a board inclined at 15° if the friction is 0.3 of the pressure of the box on the plane? 10. A bar of iron weighing 5 pounds rests on a rough board. As one end of the board is gradually raised, it is found that the bar is just ready to slide when the inclination is 20°. Find the force of friction and the coefficient of friction (Definition, page 81). |