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Unformatted text preview: Solution for Practice Test 3 for Test 2 Solution to Practice Test Problem 3.1(Modified Atwood) Problem: In Activity 11, we analyzed the motion of an Atwood’s machine, idealized in figure (1). We varied the ratio of the masses, calculated the accelerations, and produced a graph of a versus a function of the masses, so that our graph had a theoretical slope of g . In other words, we graphed a line given by a = g ( f ( m 1 ,m 2 ) ) where f ( m 1 ,m 2 ) was a quantity we could calculate given m 1 and m 2 . Suppose we modify the machine as shown in figure (2), where mass 1 lies on a horizontal surface, connected by a string over a pulley to mass 2 (all of the usual assumptions apply; the string is massless and unstretchable, the pulley is massless and frictionless, the surface is frictionless). Derive the function f ( m 1 ,m 2 ) for this scenario. In other words, what quantity would we need to plot on the x-axis, versus a on the y-axis, so that the slope of our line would be g ? m 1 m 2 a 1 a 2 m 1 m 2 a 2 a 1 (1) (2) Solution We can use Newton II to solve this problem. Taking rightward to be positive for mass 1 and downward to be positive for mass 2, so that vectora 1 = vectora 2 ≡ a (since the string doesn’t stretch), the equations of motion for each of the two objects are Σ vector F 1 = m 1 vectora 1 = m 1 a Σ vector F 2 = m 2 vectora 2 = m 2 a T = m 1 a (Eq.1) m 2 g- T = m 2 a (Eq.2) Now we can solve each equation for T to yield T = m 1 a T = m 2 g- m 2 a m 1 a = m 2 g- m 2 a (Eq.3) a ( m 1 + m 2 ) = m 2 g 1 a = g parenleftbigg m 2 m 1 + m 2 parenrightbigg (Eq.4) so f ( m 1 ,m 2 ) = m 2 m 1 + m 2 1 point(s) (2 times) : Each force equation correct (Eqs 1 and 2) 2 point(s) : Simultaneous solve 1 point(s) : Algebra (can get from Eq 3 to Eq 4, or equivalent) 2 point(s) : Correct answer Total Points for Problem: 7 Points Solution to Practice Test Problem 3.2() Problem: A cart ( m = 4 . 5kg ) speeds along a frictionless track at some initial velocity toward a horizontal spring. (a)If the cart compresses the spring until it stops, creating an elastic potential of 10J , what is the initial velocity of the cart? Draw the initial and final frames, label all givens. Is this a conservative or nonconservative interaction? (b)Suppose the cart speeds toward the spring with an initial speed of 15m / s , compresses the spring until it stops, and the speeds away from the spring with a final speed of 10m / s . Draw the initial and final frames, label all givens. Is this a conservative or nonconservative interaction? If it is conservative, find the expression for the elastic potential energy of the spring. If it is nonconservative, find the dissipated energy, and explain where this energy goes....
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