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y. w 1. A. z. x. b. B. Problem 15.259. Rod AB of length 125 mm is attached to a vertical rod that rotates about the y axis at the constant rate w 1 = 5 rad/s. Knowing that the angle formed by rod AB and the vertical is increasing at the

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  1. y w1 A z x b B Problem 15.259 Rod AB of length 125 mm is attached to a vertical rod that rotates about the y axis at the constant rate w1 = 5 rad/s. Knowing that the angle formed by rod AB and the vertical is increasing at the constant rate db/dt = 3 rad/s, determine the velocity and acceleration of end B of the rod when b = 30o.

  2. Problem 15.259 y w1 A z x b B w w2 O w1 Solving Problems on Your Own Rod AB of length 125 mm is attached to a vertical rod that rotates about the y axis at the constant rate w1 = 5 rad/s. Knowing that the angle formed by rod AB and the vertical is increasing at the constant rate db/dt = 3 rad/s, determine the velocity and acceleration of end B of the rod when b = 30o. 1. To determine the velocity and acceleration of a point of a body rotating about a fixed point: 1a. Determine the angular velocity w of the body: The angular velocity w with respect to a fixed frame of reference is often obtained by adding two component angular velocities w1 and w2.

  3. Problem 15.259 y w1 A z x b B Solving Problems on Your Own Rod AB of length 125 mm is attached to a vertical rod that rotates about the y axis at the constant rate w1 = 5 rad/s. Knowing that the angle formed by rod AB and the vertical is increasing at the constant rate db/dt = 3 rad/s, determine the velocity and acceleration of end B of the rod when b = 30o. 1b. Compute the velocity of a point of the body: The velocity v of point B in the body is given by: v = wxr where r is the position vector connecting the fixed point O to point B. w B r O

  4. Problem 15.259 y w1 A z x b B Solving Problems on Your Own Rod AB of length 125 mm is attached to a vertical rod that rotates about the y axis at the constant rate w1 = 5 rad/s. Knowing that the angle formed by rod AB and the vertical is increasing at the constant rate db/dt = 3 rad/s, determine the velocity and acceleration of end B of the rod when b = 30o. 1c. Determine the angular acceleration a of the body: a = ( w )OXYZ = (w )oxyz + Wxw a is the rate of change (w)OXYZ of the vector w w.r.t. a fixed frame of reference OXYZ, (w)oxyz is the rate of change of w w.r.t. a rotating frame of reference oxyz, W is the angular velocity of the rotating frame . . . . Y y w a . W x o O X Z z

  5. Problem 15.259 y w1 A z x b B Y y w a r x o Z O X z Solving Problems on Your Own Rod AB of length 125 mm is attached to a vertical rod that rotates about the y axis at the constant rate w1 = 5 rad/s. Knowing that the angle formed by rod AB and the vertical is increasing at the constant rate db/dt = 3 rad/s, determine the velocity and acceleration of end B of the rod when b = 30o. 1d. Compute the acceleration of a point of a body: aB = axrB/A + wx(wxrB/A) Where aB is the acceleration of point B, aA is the (known) acceleration of point A, a and w are the angular acceleration and angular velocity of the body with respect to a fixed frame of reference, and rB/A is the position vector of B relative to A.

  6. y Problem 15.259 Solution ijk 0 5 3 0.0625 -0.1083 0 Determine the angular velocity of the body. w1 w1 = 5 j rad/s w2 = db/dt= 3 rad/s, w2 = 3 k rad/s w = w1 + w2 = 5 j + 3 k rad/s A z w2 x Compute the velocity of a point of the body. b B The velocity of end B: For b = 30o and rB/A = 0.125 rB/A = 0.125 sin30oi - 0.125 cos30oj = 0.0625 i - 0.1083 j vB = w x rB/A = vB = 0.325 i + 0.188 j - 0.313 k m/s

  7. y Problem 15.259 Solution Y w1 w2 o A Z O z x X b B Determine the angular acceleration a of the body. Frame OXYZ is fixed. Frame oxyz is attached to the vertical rod and rotates with constant angular velocity w1. . Consequently: w1= 0 and W = w1 . . . . a = w = w1 + w2 = 0 + w2 . . a = ( w2 )OXYZ = (w2)oxyz + Wxw2 a = 0 + a = 15 i rad/s2 ijk 0 5 0 0 0 3

  8. y Problem 15.259 Solution ij k 0 5 3 0.325 0.188 -0.313 ijk 15 0 0 0.0625 -0.108 0 aB = + aB = -2.13 i + 0.97 j - 3.25 k m/s2 Compute the acceleration of a point of a body. w1 The acceleration of end B: w = 5 j + 3 k rad/s Recall: A a = 15 i rad/s2 z rB/A = 0.0625 i - 0.1083 j w2 x b vB = wxrB/A = 0.325 i + 0.188 j - 0.313 k m/s B aB = axrB/A + w x(wxrB/A) aB = axrB/A + w xvB

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