Chapter 19 - Engineering Mechanics Dynamics Chapter 19...

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Engineering Mechanics - Dynamics Chapter 19 Problem 19-1 The rigid body (slab) has a mass m and is rotating with an angular velocity Z about an axis passing through the fixed point O. Show that the momenta of all the particles composing the body can be represented by a single vector having a magnitude mv G and acting through point P , called the center of percussion , which lies at a distance r PG = k 2 G / r GO from the mass center G. Here k G is the radius of gyration of the body, computed about an axis perpendicular to the plane of motion and passing through G. Solution: H o r GO r PG . + , mv G r GO mv G I G Z . Where I G mk G 2 r GO mv G r PG mv G . r GO mv G mk G 2 Z . r PG k G 2 Z v G k G 2 v G v G r GO § ¨ © · ¸ ¹ k G 2 r GO Q.E.D Problem 19-2 At a given instant, the body has a linear momentum L = mv G and an angular momentum H G = I G Z computed about its mass center. Show that the angular momentum of the body computed about the instantaneous center of zero velocity IC can be expressed as H IC = I IC Z where I IC represents the body’s moment of inertia computed about the instantaneous axis of zero velocity. As shown, the IC is located at a distance r GIC away from the mass center G . Solution: H IC r GIC mv G I G Z . Where v G Z r GIC 632
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Engineering Mechanics - Dynamics Chapter 19 H IC r GIC m Z r GIC I G Z . H IC I G mr GIC 2 . + , Z H IC I IC Z Q.E.D. Problem 19-3 Show that if a slab is rotating about a fixed axis perpendicular to the slab and passing through its mass center G, the angular momentum is the same when computed about any other point P on the slab. Solution: Since v G = 0, the linear momentum L = mv G = 0. Hence the angular momentum about any point P is H P I G Z Since Z is a free vector , so is H P . Q.E.D. *Problem 19-4 Gear A rotates along the inside of the circular gear rack R . If A has weight W and radius of gyration k B , determine its angular momentum about point C when (a) Z R = 0, (b) Z R = Z . Given: W 4 lbf r 0.75 ft Z CB 30 rad s a 1.5 ft k B 0.5 ft Z 20 rad s g 32.2 ft s 2 Solution: a ( ) Z R 0 rad s v B a Z CB Z A Z R a r . ( ) Z CB a 0 r H c W g § ¨ © · ¸ ¹ v B a W g § ¨ © · ¸ ¹ k B 2 Z A . H c 6.52 slug ft 2 s ¡ 633
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Engineering Mechanics - Dynamics Chapter 19 b ( ) Z R Z v B a Z CB Z A Z R a r . ( ) Z CB a 0 r H c W g § ¨ © · ¸ ¹ v B a W g § ¨ © · ¸ ¹ k B 2 Z A . H c 8.39 slug ft 2 s ¡ Problem 19-5 The fan blade has mass m b and a moment of inertia I 0 about an axis passing through its center O . If it is subjected to moment M = A (1 0# e bt ) determine its angular velocity when t = t 1 starting from rest. Given: m b 2 kg A 3 N m ¡ t 1 4 s I O 0.18 kg m 2 ¡ b 0.2 0 s 1 0 Solution: 0 0 t 1 t A 1 e b t 0 + , ´ µ d . I O Z 1 Z 1 1 I O 0 t 1 t A 1 e b t 0 + , ´ µ d Z 1 20.8 rad s Problem 19-6 The wheel of mass m w has a radius of gyration k A . If the wheel is subjected to a moment M = bt , determine its angular velocity at time t 1 starting from rest. Also, compute the reactions which the fixed pin A exerts on the wheel during the motion.
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