hw2_sol - 180 Solutions Manual Problem 2.172 The...

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180 Solutions Manual Problem 2.172 The reciprocating rectilinear motion mechanism shown consists of a disk pinned at its center at A that rotates with a constant angular velocity ! AB , a slotted arm CD that is pinned at C , and bar that can oscillate within the guides at E and F . As the disk rotates, the peg at B moves within the slotted arm causing it to rock back and forth. As the arm rocks, it provides a slow advance and a quick return to the reciprocating bar due to the change in distance between C and B . Letting D 30 ı , ! AB D 50 rpm D constant , R D 0:3 ft , and h D 0:6 ft , determine P and R , i.e., the angular velocity and angular acceleration of the slotted arm CD , respectively. Solution Let R D AB and r D BC . The velocity of B in the . O u R ; O u / and . O u r ; O u / component systems are E v B D R! AB O u and E v B D P r O u r C r P O u : (1) To find P take advantage of the fact that for D 30 ı and R D 0:5h we have a right triangle ABC . Then O u r D O u and O u R D O u . Converting the O u component of E v B to O u R and equating components we find r P O u R D 0 ) P D 0: (2) The acceleration of B in the . O u R ; O u / and . O u r ; O u / component systems are E a B D R! 2 AB O u R and E a B D R r r P 2 O u r C r R C 2 P r P O u : (3) Convert the O u component of E a B to O u R , equate components of Eq. (3), plug in Eq. (2), and substitute r D p h 2 R 2 to find r R D R! 2 AB ) R D R! 2 AB p h 2 R 2 D 15:8 rad = s 2 .
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200 Solutions Manual Problem 2.190 An interesting application of the relative motion equations is the experimental determination of the speed at which rain falls. Say you perform an experiment in your car in which you park your car in the rain and measure the angle the falling rain makes on your side window. Let this angle be rest D 20 ı . Next, you drive forward at 25 mph and measure the new angle, motion D 70 ı , that the rain makes with the vertical. Determine the speed of the falling rain. Solution We will use a Cartesian Coordinate system where x and y represent the horizontal and vertical directions, respectively. The orientation of the angle allows us to write the velocity of the rain with respect to the stationary car as E v R D v R . sin rest O { C cos rest O |/ : (1) The second piece of information allows us to write the velocity of the rain with respect to the moving car as E v R=C D v R=C . sin motion O { C cos motion O |/: (2)
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