Lecture Dependent Motion

Applications (2 of 2) Rope and pulley arrangements are often used to assist in lifting heavy objects. The total lifting force required from the truck depends on both the weight and the acceleration of the cabinet. How can we determine the acceleration and velocity of the cabinet if the acceleration of the truck is known? Section 12.9 Dependent Motion Dependent Motion (1 of 3) In many kinematics problems, the motion of one object will depend on the motion of another object. The blocks in this figure are connected by an inextensible cord wrapped around a pulley. If block A moves downward along the inclined plane, block B will move up the other incline. The motion of each block can be related mathematically by defining position coordinates, s and s .Each coordinate axis is A B defined from a fixed point or datum line, measured positive along each plane in the direction of motion of each block. Dependent Motion (2 of 3) In this example, position coordinates s A and sB can be defined from fixed datum lines extending from the center of the pulley along each incline to blocks A and B. If the cord has a fixed length, the position coordinates are related mathematically by the equation s A and sB s A  lCD  sB  lT Here lT is the total cord length and passing over the arc CD on the pulley. lCD is the length of cord Dependent Motion (3 of 3) The velocities of blocks A and B can be related by differentiating the position equation. Note that remain constant, so lCD and lt dlCD / dt  dlT / dt  0 ds A / dt  dsB / dt  0  vB   v A The negative sign indicates that as A moves down the incline positive s direction  , B moves up the incline A negative sB direction . Accelerations can be found by differentiating the velocity expression. Prove to yourself that aB   a A . Dependent Motion Example (1 of 3) Consider a more complicated example. Position coordinates  s and s  A B are defined from fixed datum lines, measured along the direction of motion of each block. Note that sB is only defined to the center of the pulley above block B, since this block moves with the pulley. Also, h is a constant. The red-colored segments of the cord remain constant in length during motion of the blocks. Dependent Motion Example (2 of 3) The position coordinates are related by the equation 2sB  h  s A  lT Where lT is the total cord length minus the lengths of the red segments. Since lT and h remain constant during the motion, the velocities and accelerations can be related by two successive time derivatives: 2vB   v A and 2a B   a A When block B moves downward   sB  block A moves to the left   s A  Remember to be consistent with your sign convention! Dependent Motion Example (3 of 3) This example can also be worked by defining the position coordinate for B  sB  from the bottom pulley instead of the top pulley. The position, velocity, and acceleration relations then become 2( h – sB )  h  s A  lT and 2vB  v A 2a B  a A Prove to yourself that the results are the same, even if the sign conventions are different than the previous formulation. Dependent Motion: Procedures These procedures can be used to relate the dependent motion of particles moving along rectilinear paths (only the magnitudes of velocity and acceleration change, not their line of direction). 1. Define position coordinates from fixed datum lines, along the path of each particle. Different datum lines can be used for each particle. 2. Relate the position coordinates to the cord length. Segments of cord that do not change in length during the motion may be left out. 3. If a system contains more than one cord, relate the position of a point on one cord to a point on another cord. Separate equations are written for each cord. 4. Differentiate the position coordinate equation(s) to relate velocities and accelerations. Keep track of signs! Example (1 of 3) Given: In the figure on the left, the cord at A is pulled down with a speed of 3 meter per second. Find: The speed of block D. Plan: There is only one cord involved in the motion, so only one position/length equation is required. Define position coordinates for block D and cable lengths that change, write the position relation and then differentiate it to find the relationship between the two velocities. Example (2 of 3) Solutions: 1) A datum line can be drawn through the upper, fixed pulleys. Two coordinates must be defined: one for block D and one for the changing cable length (s A ).  s can be defined to the point A. can be defined to the center of the pulley  sabove D. A D • All coordinates are defined as positive down and along the direction of motion of each point per object. (sD ) Example (3 of 3) 2) Write position/length equations for the cord. Define lT as the length of the cord, minus any segments of constant length. s A  3sD  lT 3) Differentiate to find the velocity relationship: v A  3vD  0 Since the cord at A is pulled down with a speed of 3 m/s, 3  3vD  0  vD   1 m/s  1 m/s  Practice Problems: Example Problems 12.21, 12,22, 12,23, 12.24 Exercise Problems: 12-196, 12-204, 12-207, 12-208