Solved: Suppose a block of mass is placed on an inclined

Chapter 17, Problem 17.181

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Suppose a block of mass 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 , where ( is the acceleration due to gravity); the normal force (the normal component of the reactionary force of the plane on the block), where ; 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, must also increase until ultimately reaches its maximum, beyond which the block begins to slide. At this angle , it has been observed that is proportional to . Thus, when is maximal, we can say that , where is called the coefficient of static friction and depends on the materials that are in contact. (a) Observe that N F W 0 and deduce that . (b) Suppose that, for , an additional outside force is applied to the block, horizontally from the left, and let . If is small, the block may still slide down the plane; if is large enough, the block will move up the plane. Let be the smallest value of that allows the block to remain motionless (so that is maximal). By choosing the coordinate axes so that lies along the -axis, resolve each force into components parallel and perpendicular to the inclined plane and show that and (c) Show that Does this equation seem reasonable? Does it make sense for ? As ? Explain. (d) Let be the largest value of that allows the block to remain motionless. (In which direction is heading?) Show that Does this equation seem reasonable? Explain.