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Unformatted text preview: 7########,##################o ###### C##########################d##q##d##q##h####h####h#,##h#<##i#####i#####i##P## ic####iy####i####i#L##i#x##h#,##jY# ##jy####mo#*##m#3##h########l#;##j#S##m##:##mY####n#=##mo####mo####mo####mo# ###mo####mo####Name 1. [25 points total] Check the Fundamental Theorem for Gradients for the function: t = 3x3 + 4xy2 + 6yz3 over the two paths sketched below from (0,0,0) to (1,1,1). Show your work.#A. Path #1: (0,0,0) (1,0,0) (1,1,0) (1,1,1)B. Path #2: (0,0,0) (1,1,1) along the curve z = x2, x = y #4. [30 points total] A cylindrical room is rotating fast enough that two blocks stacked against the wall do not drop. The mass of block A is 500 g and the mass of block B is 300 g. A. [4 pts] On the diagram to the right, indicate the direction of the acceleration of block A. If the acceleration is zero, state that explicitly. B. [8 pts] Draw separate freebody diagrams for blocks A and B and label the forces as instructed in tutorial.# C. [6 pts] Identify any Third Law companion forces on your diagrams above with tick marks as was done in tutorial. D. [6 pts] Rank the magnitudes of all the horizontal forces that you identified above in order from largest to smallest. Explain your reasoning. Nw,B > NA,B by Newtons 2nd Law. Acceleration of B is toward center of room, so net force must be directed toward center of room. NA,B = NB,A by Newtons 3rd Law. E. [6 pts] Determine the magnitude (if possible) of each of the vertical forces on block A that you identified above. Show your work and explain your reasoning. (Use the approximation g 10 m/s2.) WE,A = (0.5 kg)(10 m/s2) = 5 N Since ay = 0, the forces acting on block A in the ydirection must cancel. Hence, #fB,A = WE,A = 5 N. 3. [25 points total] The Colossus Rollercoaster involves a very exciting loopdeloop at the bottom of a long straight incline, as shown below. Suppose a cart (with you in it) were released from rest at point A at time to = 0.6s, and at time t = 5.6s it had reached point B, 50m down the track.#A. [5 points] On the diagram above, sketch vectors to show the direction of the instantaneous velocity of the cart at each of the labeled points. If #o(v,) = 0 at any point, indicate that explicitly.B. [6 points] Find the magnitude of the instantaneous velocity of the cart at point B. Show your work.C. [5 points] On the diagram below, sketch arrows to show the approximate direction of the acceleration of the cart at each of the labeled points. Do not try to draw the arrows to scale. If #o(a,) = 0 at any point, indicate that explicitly.#D. [4 points] Explain how you knew to draw the acceleration vector (if any) as you did: i. at point D. ii. at point E.E. [5 points] Find the magnitude of the instantaneous acceleration of the cart at point B. Show your work. 4. [25 points total] A cannon ball is fired at an angle of 37o above a level field. The initial velocity has a magnitude of 100 m/s. In the questions that follow, you may neglect air resistance and assume that g 10 m/s2.#A. [6 pts] follow, you may neglect air resistance and assume that g 10 m/s2....
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 Fall '99
 GregoryFrancis

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