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Unformatted text preview: Solution for Homework 19 Motion in a Circle I, Motion in a Plane problems Solution to Homework Problem 19.1() Problem: Explain why an object in two-dimensional frictionless projectile motion follows a parabolic trajectory. Solution In general, two-dimensional, frictionless projectile motion has both horizontal and vertical components (and if there isn’t any “cross-wind”, we can turn three-dimensional motion into two-dimensional motion by choosing our coordinate system appropriately). Gravity acts in the vertical direction only, so the horizontal component of the initial velocity is unaffected by gravity ( Δ v x = 0 ). The vertical motion is uniformly accelerated, and so is quadratic in time. And so x ( t ) = x + v ,x t and y ( t ) = y + v ,y t- 1 2 gt 2 and the vector position function is vector r ( t ) = parenleftbigg x ( t ) y ( t ) parenrightbigg The x- y graph of this function (parameterized by t ) is a parabola. Total Points for Problem: 2 Points Solution to Homework Problem 19.2() Problem: When is the maximum value for static friction obtained? Solution The maximum value of the static frictional force occurs the instant before the object actually moves (in which case the friction is then kinetic) when subject to a steadily increasing force. For example, suppose a block is sitting on a rough board, and you start trying to move it along the board. You slowly increase the force you apply to the block. Eventually the force you apply will be equal in magnitude to the maximum static frictional force, F pull = F s = μ s F N If the pulling force increases any more, the block will move, and the friction will then be kinetic. Total Points for Problem: 2 Points Solution to Homework Problem 19.3() Problem: What is the name given to the acceleration of an object in uniform circular motion? How is this acceleration oriented and what effect does it have? Solution An object in uniform circular motion experiences a “centripetal” acceleration. This is an acceleration directed toward the center of the circle along which the object travels. Since the object’s velocity and acceleration are always perpendicular to one another, the object’s speed does not change, only the direction of the velocity. An object must be subject to a centripetal acceleration to experience uniform circular motion. Total Points for Problem: 3 Points Solution to Homework Problem 19.4() Problem: A box leaves the edge of a table ( 1 . 00m above the ground) with an initial velocity of 1 . 00 m s in the horizontal direction. (a)How long does it take to hit ground? (b)How far does the box travel horizontally before it lands? (c)What is the velocity of the box when it lands? (d)Suppose the box is put on an incline (with angle of inclination θ ) on the table top so that it can gain the velocity it needs to leave the table. As it leaves the table, there is a frictionless curvey bit that changes the direction of the velocity to horizontal, without changing its size. What is the vertical height up thethe direction of the velocity to horizontal, without changing its size....
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