This preview shows pages 1–4. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Engineering Mechanics  Dynamics Chapter 19 Problem 191 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 . + , m v G r GO m v G I G Z . Where I G m k G 2 r GO m v G r PG m v G . r GO m v G m k 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 192 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 m v G I G Z . Where v G Z r GIC 632 Engineering Mechanics  Dynamics Chapter 19 H IC r GIC m Z r GIC I G Z . H IC I G m r GIC 2 . + , Z H IC I IC Z Q.E.D. Problem 193 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 194 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 rad s v B a Z CB Z A Z R a r . ( ) Z CB a 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 Engineering Mechanics  Dynamics Chapter 19 b ( ) Z R Z v B a Z CB Z A Z R a r . ( ) Z CB a 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 195 The fan blade has mass m b and a moment of inertia I 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 s 1 Solution: t 1 t A 1 e b t + , ´ µ ¶ d . I O Z 1 Z 1 1 I O t 1 t A 1 e b t + , ´ µ ¶ d Z 1 20.8 rad s Problem 196 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...
View
Full
Document
This note was uploaded on 04/13/2008 for the course ME 2580 taught by Professor Vandenbrink during the Spring '08 term at Western Michigan.
 Spring '08
 Vandenbrink

Click to edit the document details