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q4s - Physics 2211 Spring 2008 Quiz#4 Solutions Unless...

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Physics 2211 Quiz #4 Solutions Spring 2008 Unless otherwise directed, all springs, cords, and pulleys are ideal, and drag should be neglected. I . (16 points) Two blocks of mass m 1 and m 2 are tied together and pulled so that it travels up a frictionless incline that makes an angle θ with the horizontal, as illustrated. The force pulling them is a tension of magnitude T 1 , and their resulting acceleration is not zero. What is the tension magnitude T 2 in rope 2, which ties the blocks together, in terms of any or all of m 1 , m 2 , θ , T 1 , and physical or mathematical constants? ( On Earth. ) . . . . . . . . . . . . . . . . . . . . . . . There is more than one approach to solving this Newton’s Second Law problem. The easiest is probably to look first at the two blocks m 1 and m 2 along with rope 2 as a single object being pulled up the incline by rope 1. A Free-Body Diagram has been sketched for this object. A coordinate system has been chosen so the acceleration on one axis is zero, and the weight force has been resolved into components. Applying Newton’s Second Law to the x direction: X F x = T 1 - w tot x = m tot a x T 1 - ( m 1 + m 2 ) g sin θ = ( m 1 + m 2 ) a x a x = T 1 - ( m 1 + m 2 ) g sin θ m 1 + m 2 Looking next at just the block with mass m 2 , which is pulled up the ramp by rope 2, a Free-Body Diagram has been sketched for this object. The same coordinate system is used, so the acceleration of this block will be represented by the same symbol as the acceleration of the two blocks together. The weight force has been resolved into components. Again, applying Newton’s Second Law to the x direction: X F x = T 2 - w 2 x = m tot a x T 2 - m 2 g sin θ = m 2 a x Substituting the expression for the acceleration and solving for T 2 : T 2 - m 2 g sin θ = m 2 T 1 - ( m 1 + m 2 ) g sin θ m 1 + m 2 T 2 = m 2 T 1 - ( m 1 + m 2 ) g sin θ m 1 + m 2 + m 2 g sin θ That is sufficient to answer the question. A bit more algebra, though, yields an interesting result: T 2 = m 2 T 1 - ( m 1 + m 2 ) g sin θ m 1 + m 2 + m 1 + m 2 m 1 + m 2 m 2 g sin θ = m 2 T 1 - ( m 1 + m 2 ) g sin θ + ( m 1 + m 2 ) g sin θ m 1 + m 2 = m 2 m 1 + m 2 T 1 Quiz #4 Solutions Page 1 of 5
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II . (16 points) A physics book of mass M is connected by a string to a coffee cup of mass m . The book is given a push up a slope that makes an angle θ with the horizontal, and released at initial speed v i , as illustrated. The coefficient of static friction between the book and the slope is μ s , while the coefficient of kinetic friction is μ k . How much time is required for the book to come to a stop after it is released, in terms of any or all of M , m , θ , μ s , μ k , and physical or mathematical constants? (
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