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**ABSOLUTE MOTION ANALYSIS (Section 16.4)**

Today’s Objective: Students will be able to determine the velocity and acceleration of a rigid body undergoing general plane motion using an absolute motion analysis. In-Class Activities: • Check homework, if any • Reading quiz • Applications • General Plane Motion • Concept quiz • Group problem solving • Attention quiz

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**1. A body subjected to general plane motion undergoes a/an **

READING QUIZ 1. A body subjected to general plane motion undergoes a/an A) translation. B) rotation. C) simultaneous translation and rotation. D) out of plane movement. Answers: 1. C 2. A 2. In general plane motion, if the rigid body is represented by a slab, the slab rotates A) about an axis perpendicular to the plane. B) about an axis parallel to the plane. C) about an axis lying in the plane. D) None of the above.

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APPLICATIONS The position of the piston, x, can be defined as a function of the angular position of the crank, q. By differentiating x with respect to time, the velocity of the piston can be related to the angular velocity, w, of the crank. The stroke of the piston is defined as the total distance moved by the piston as the crank angle varies from 0 to 180°. How does the length of crank AB affect the stroke?

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**APPLICATIONS (continued)**

The rolling of a cylinder is an example of general plane motion. During this motion, the cylinder rotates clockwise while it translates to the right. Yes. Remember the concept of instanteous centers? Where is the IC for this situation and what does it tell you about the velocity of G? The position of the center, G, is related to the angular position, q, by sG = r q if the cylinder rolls without slipping. Can you relate the translational velocity of G and the angular velocity of the cylinder?

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**PROCEDURE FOR ANALYSIS**

The absolute motion analysis method (also called the parametric method) relates the position of a point, P, on a rigid body undergoing rectilinear motion to the angular position, q (parameter), of a line contained in the body. (Often this line is a link in a machine.) Once a relationship in the form of sP = f(q) is established, the velocity and acceleration of point P are obtained in terms of the angular velocity, w, and angular acceleration, a, of the rigid body by taking the first and second time derivatives of the position function. Usually the chain rule must be used when taking the derivatives of the position coordinate equation.

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EXAMPLE 1 Given: Two slider blocks are connected by a rod of length 2 m. Also, vA = 8 m/s and aA = 0. Find: Angular velocity, w, and angular acceleration, a, of the rod when q = 60°. Plan: Choose a fixed reference point and define the position of the slider A in terms of the parameter q. Notice from the position vector of A, positive angular position q is measured clockwise.

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**By differentiating with respect to time, vA = -2 w sin q**

EXAMPLE 1 (continued) A reference sA q Solution: By geometry, sA = 2 cos q By differentiating with respect to time, vA = -2 w sin q Using q = 60° and vA = 8 m/s and solving for w: w = 8/(-2 sin 60°) = rad/s (The negative sign means the rod rotates counterclockwise as point A goes to the right.) Differentiating vA and solving for a, aA = -2a sin q – 2w2 cos q = 0 a = - w2/tan q = rad/s2

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EXAMPLE 2 Given: Crank AB rotates at a constant velocity of w = 150 rad/s Find: Velocity of P when q = 30° Plan: Define x as a function of q and differentiate with respect to time. xP = 0.2 cos q + (0.75)2 – (0.2 sin q)2 Solution: vP = -0.2w sin q + (0.5)[(0.75)2 – (0.2sin q)2]-0.5(-2)(0.2sin q)(0.2cos q)w vP = -0.2w sin q – [0.5(0.2)2sin2q w] / (0.75)2 – (0.2 sin q)2 At q = 30°, w = 150 rad/s and vP = ft/s = 18.5 ft/s

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CONCEPT QUIZ 1. If the position, s, is given as a function of angular position, q, by s = 10 sin 2q, the velocity, v, is A) 20 cos 2q B) 20 sin 2q C) 20 w cos 2q D) 20 w sin 2q 2. If s = 10 sin 2q, the acceleration, a, is A) 20 a sin 2q B) 20 a cos 2q - 40 w2 sin 2q C) 20 a cos 2q D) -40 a sin2 q

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GROUP PROBLEM SOLVING Given: The w and a of the disk and the dimensions as shown. Find: The velocity and acceleration of cylinder B in terms of q. Plan: Relate s, the length of cable between A and C, to the angular position, q. The velocity of cylinder B is equal to the time rate of change of s.

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**GROUP PROBLEM SOLVING (continued)**

Solution: Law of cosines: s = (3)2 + (5)2 – 2(3)(5) cos q vB = (0.5)[34 – 30 cosq]-0.5(30 sinq)w vB = [15 sin q w]/ – 30 cos q 30cosq)3/2 - (34 sinq) sinq)(30w (-0.5)(15w 30cosq 34 15a cosq (15w2 aB + = cosq)3/2 30 225w2sin2q cosq)0.5 a 15(w2cosq

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**2. If vA=10 m/s, determine the angular acceleration, a, when = 30. **

ATTENTION QUIZ 1. The sliders shown below are confined to move in the horizontal and vertical slots. If vA=10 m/s, determine the connecting bar’s angular velocity when = 30. A) 10 rad/s B) 10 rad/s C) 8.7 rad/s D) 8.7 rad/s Answers: 1. A 2. D (See example 1 for equations!) vA=10 m/s 2. If vA=10 m/s, determine the angular acceleration, a, when = 30. A) 0 rad/s2 B) rad/s2 C) rad/s D) -173 rad/s2

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Kinematics of Rigid Bodies

Kinematics of Rigid Bodies

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