Suppose a block of mass m is placed on an inclined plane,

Chapter , Problem 7

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Suppose a block of mass m is placed on an inclined plane, as shown in the figure. The blocks descent down the plane is slowed by friction; if is not too large, friction will prevent the block from moving at all. The forces acting on the block are the weight W, where | W | mt (t is the acceleration due to gravity); the normal force N (the normal component of the reactionary force of the plane on the block), where | N | n; and the force F due to friction, which acts parallel to the inclined plane, opposing the direction of motion. If the block is at rest and is increased, | F | must also increase until ultimately | F | reaches its maximum, beyond which the block begins to slide. At this angle s, it has been observed that | F | is proportional to n. Thus, when | F | is maximal, we can say that | F | sn, where s is called the coefficient of static friction and depends on the materials that are in contact. (a) Observe that N 1 F 1 W 0 and deduce that s tanssd. (b) Suppose that, for . s, an additional outside force H is applied to the block, horizontally from the left, and let | H | h. If h is small, the block may still slide down the plane; if h is large enough, the block will move up the plane. Let hmin be the smallest value of h that allows the block to remain motionless (so that | F | is maximal). By choosing the coordinate axes so that F lies along the x-axis, resolve each force into components parallel and perpendicular to the inclined plane and show that hmin sin 1 mt cos n and hmin cos 1 sn mt sin (c) Show that hmin mt tans 2sd Does this equation seem reasonable? Does it make sense for s? Does it make sense as l 908? Explain. (d) Let hmax be the largest value of h that allows the block to remain motionless. (In which direction is F heading?) Show that hmax mt tans 1sd Does this equation seem reasonable? Explain.