HW3 - Homework Set 3 Solutions Ch 5 16 A block is sliding...

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Homework Set 3 Solutions Ch 5 16 A block is sliding on a frictionless surface along a loop-the-loop, as in Figure 5- 59. The block is moving fast enough so that it never loses contact with the track. Match the points along the track to the appropriate free-body diagrams in the figure. Determine the Concept The only forces acting on the block are its weight and the force the surface exerts on it. Because the loop-the-loop surface is frictionless, the force it exerts on the block must be perpendicular to its surface. At point A the weight is downward and the normal force is to the right. The normal force is the centripetal force. Free-body diagram 3 matches these forces. At point B the weight is downward, the normal force is upward, and the normal force is greater than the weight so that their difference is the centripetal force. Free-body diagram 4 matches these forces. At point C the weight is downward and the normal force is to the left. The normal force is the centripetal force. Free-body diagram 5 matches these forces. At point D both the weight and the normal forces are downward. Their sum is the centripetal force. Free-body diagram 2 matches these forces. 51 •• Two blocks attached by a string (Figure 5-64) slide down a 10º incline. Block 1 has mass m 1 = 0.80 kg and block 2 has mass m 1 = 0.25 kg. In addition, the kinetic coefficients of friction between the blocks and the incline are 0.30 for block 1 and 0.20 for block 2. Find ( a ) the magnitude of the acceleration of the blocks, and ( b ) the tension in the string. Picture the Problem Assume that the string is massless and does not stretch. Then the blocks have a common acceleration and the tension in the string acts on both blocks in accordance with Newton’s third law of motion. Let down the incline be the positive x direction. Draw the free-body diagrams for each block and apply Newton’s second law of motion and the definition of the kinetic friction force to each block to obtain simultaneous equations in a x and T .
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Draw the free-body diagram for the block whose mass is m 1 : Apply to the upper block: = a F r r m x x a m g m T f F , 1 1 1 1 k,1 sin = + + = θ (1) and 0 cos 1 n,1 = = g m F F y (2) The relationship between f k,1 and F n,1 is: n,1 k,1 k,1 F f μ = (3) Eliminate f k,1 and F n,1 between (1), (2), and (3) to obtain: x a m g m T g m , 1 1 1 1 1 k,1 sin cos = + + (4) Draw the free-body diagram for the block whose mass is m 2 : Apply to the block: = a F r r m x x a m g m T f F , 2 2 2 2 k,2 sin = + = (5) and 0 cos 2 n,2 = = g m F F y (6) x y k,1 f r g m r 1 n,1 F r 1 T r 1 m x y 2 T r n,2 F r g m r 2 k,2 f r 2 m
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The relationship between f k,2 and F n,2 is: n,2 k,2 k,2 F f μ = (7) Eliminate f k,2 and F n,2 between (5), (6), and (7) to obtain: x a m g m T g m , 1 1 2 2 2 k,2 sin cos = + θ (8) Noting that T 2 = T 1 = T, add equations (4) and (8) to eliminate T, and solve for x a , 1 : g m m m m a x + + = μμ cos sin 2 1 2 k,2 1 k,1 , 1 Substitute numerical values and evaluate a 1, x : () ( )( )( ) 2 2 , 1 m/s 96 . 0
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HW3 - Homework Set 3 Solutions Ch 5 16 A block is sliding...

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