q5s - Physics 2211 Spring 2008 Quiz #5 Solutions Unless...

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Physics 2211 Quiz #5 Solutions Spring 2008 Unless otherwise directed, all springs, cords, and pulleys are ideal, and drag should be neglected. I . (16 points) An amusement park sled and rider, with total mass m , is to be sent around a loop-the-loop of radius R . The sled and rider will accomplish this by compressing a large spring with Hooke’s Law constant k , and being released from rest. The loop is frictionless, but the level ground has a coefficient of kinetic friction μ k with the sled. After the spring has been compressed a distance s , the sled is a distance d from the loop. At what maximum distance, d , from the loop can the sled be released if the loop is to be completed? Express your answer in terms of any or all of m , R , k , μ k , and physical or mathematical constants? ( On Earth. ) . . . . . . . . . . . . . . . . . . . . . . . Announced during quiz: Your answer may depend on s . Use the Work-Energy Theorem. Consider the Earth-spring-car sys- tem. Elastic (spring) and gravitational potential energies change within the system. Friction is a non-conservative force doing work within the system. There are no external forces doing work on the system. W ext + W nc = Δ K + Δ U = ( K f - K i ) + ( U s f - U s i ) + ( U g f - U g i ) 0 + Z ~ f · d ~ d = ( 1 2 mv 2 f - 1 2 mv 2 i ) + 1 2 k s f ) 2 - 1 2 k s i ) 2 · + ( mgh f - mgh i ) The frictional force, ~ f , has constant magnitude μ k n , and is opposite the displacement. Let the zero of height be at ground level (the initial position). The positive direction must be up and the zero of spring compression be at the relaxed (final) position to use the expressions above. The initial compression of the spring is s , and the initial speed of the car is zero. The height of the car at the top of the loop is 2 R . μ k nd cos180 = ( 1 2 mv 2 f - 0 ) + ( 0 - 1 2 ks 2 ) + ( mg 2 R - 0) The normal force while the car is on the ground and friction is acting can be found from Newton’s Second Law. The car has no vertical ( y ) acceleration at that point. X
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q5s - Physics 2211 Spring 2008 Quiz #5 Solutions Unless...

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