Ch15_Summary

# Ch15_Summary - bee29400_ch15_0914-1023.indd Page 1011...

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1011 REVIEW AND SUMMARY This chapter was devoted to the study of the kinematics of rigid bodies. We first considered the translation of a rigid body [Sec. 15.2] and observed that in such a motion, all points of the body have the same velocity and the same acceleration at any given instant. We next considered the rotation of a rigid body about a fixed axis [Sec. 15.3]. The position of the body is defined by the angle u that the line BP , drawn from the axis of rotation to a point P of the body, forms with a fixed plane (Fig. 15.39). We found that the magnitude of the velocity of P is v 5 ds dt 5 r u . sin f (15.4) where ˙ u is the time derivative of u . We then expressed the velocity of P as v 5 d r 5 V 3 r (15.5) where the vector V 5 v k 5 u ˙ k (15.6) is directed along the fixed axis of rotation and represents the angular velocity of the body. Denoting by A the derivative d V / dt of the angular velocity, we expressed the acceleration of P as a 5 A 3 r 1 V 3 ( V 3 r ) (15.8) Differentiating (15.6), and recalling that k is constant in magnitude and direction, we found that A 5 a k 5 v . k 5 ¨ u k (15.9) The vector A represents the angular acceleration of the body and is directed along the fixed axis of rotation. Rigid body in translation Rigid body in rotation about a fixed axis x z y O A ' A B P f r q Fig. 15.39

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Next we considered the motion of a representative slab located in a plane perpendicular to the axis of rotation of the body (Fig. 15.40). Since the angular velocity is perpendicular to the slab, the velocity of a point P of the slab was expressed as v 5 v k 3 r (15.10) where v is contained in the plane of the slab. Substituting V 5 v k and A 5 a k into (15.8), we found that the acceleration of P could be resolved into tangential and normal components (Fig. 15.41) respectively equal to a t 5 a k 3 r a t 5 r a a n 5 2 v 2 r a n 5 r v 2 (15.11 9 ) Recalling Eqs. (15.6) and (15.9), we obtained the following expres- sions for the angular velocity and the angular acceleration of the slab [Sec. 15.4]: v 5 d u dt (15.12) a 5 d v 5 d 2 u 2 (15.13) or a 5 v d v d u (15.14) We noted that these expressions are similar to those obtained in Chap. 11 for the rectilinear motion of a particle. Two particular cases of rotation are frequently encountered: uniform rotation and uniformly accelerated rotation. Problems involving either of these motions can be solved by using equations similar to those used in Secs. 11.4 and 11.5 for the uniform rectilin- ear motion and the uniformly accelerated rectilinear motion of a particle, but where x , v , and a are replaced by u , v , and a , respec- tively [Sample Prob. 15.1]. Rotation of a representative slab Tangential and normal components Angular velocity and angular acceleration of rotating slab x y O r P w k v = w k × r Fig. 15.40 x y O w = w k a = a k a t = a k × r a n = w 2 r P Fig. 15.41 1012 Kinematics of Rigid Bodies
1013 The most general plane motion of a rigid slab can be considered as the sum of a translation and a rotation [Sec. 15.5]. For example, the slab shown in Fig. 15.42 can be assumed to translate with point

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