Chapter 20 - Engineering Mechanics - Dynamics Chapter 20...

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Unformatted text preview: Engineering Mechanics - Dynamics Chapter 20 Problem 20-1 The ladder of the fire truck rotates around the z axis with angular velocity 1 which is increasing at rate 1. At the same instant it is rotating upwards at the constant rate 2. Determine the velocity and acceleration of point A located at the top of the ladder at this instant. Given: 1 0.15 rad s rad s 2 1 0.8 2 0.6 rad s 30 deg L 40 ft Solution: 0 r L cos L sin 2 0 1 2 1 0 1 5.20 vA r vA 12.00 20.78 24.11 aA r r aA 13.25 7.20 ft s 2 ft s vA 24.6 ft s aA 28.4 ft s 2 666 Engineering Mechanics - Dynamics Chapter 20 Problem 20-2 The ladder of the fire truck rotates around the z axis with angular velocity 1 which is increasing at rate 1. At the same instant it is rotating upwards at rate 2 while increasing at rate 2. Determine the velocity and acceleration of point A located at the top of the ladder at this instant. Given: 1 0.15 rad s rad s 2 1 0.2 2 0.6 rad s rad s 2 2 0.4 30 deg L 40 ft Solution: 2 2 1 2 1 0 r L cos L sin 0 1 5.20 vA r vA 12.00 20.78 3.33 aA r r aA 21.25 6.66 ft s 2 ft s vA 24.6 ft s aA 22.5 ft s 2 667 Engineering Mechanics - Dynamics Chapter 20 Problem 20-3 The antenna is following the motion of a jet plane. At the instant shown , the constant angular rates of change are ' and '. Determine the velocity and acceleration of the signal horn A at this instant. The distance OA is d. Given: 25 deg ' 0.4 rad s 75 deg ' d 0.6 rad s 0.8 m Solution: ' 0 ' 0 ' ' 0 0 r d cos d sin 0.083 vA r vA 0.464 0.124 0.371 aA r r aA 0.108 0.278 m s 2 m s 668 Engineering Mechanics - Dynamics Chapter 20 *Problem 20-4 The propeller of an airplane is rotating at a constant speed si, while the plane is undergoing a turn at a constant rate t. Determine the angular acceleration of the propeller if (a) the turn is horizontal, i.e., tk, and (b) the turn is vertical, downward, i.e., tj . Solution: (a) tk si s tj (b) tj si s tk Problem 20-5 Gear A is fixed while gear B is free to rotate on the shaft S. If the shaft is turning about the z axis with angular velocity z, while increasing at rate z, determine the velocity and acceleration of point C at the instant shown. The face of gear B lies in a vertical plane. Given: z 5 rad s rad s 2 z 2 rA rB h Solution: z rA 160 mm 80 mm 80 mm B rB rA B z rB rA B 10 rad s z rA B rB B z rB B 4 rad s 2 669 Engineering Mechanics - Dynamics Chapter 20 0 B z 0 B z 0 0 z 0 r rA rB 1.6 vC r vC 0 0 0.64 aC r r aC 12 8 m s 2 m s vC 1.6 m s aC 14.436 m s 2 Problem 20-6 The conical spool rolls on the plane without slipping. If the axle has an angular velocity 1 and an angular acceleration 1 at the instant shown, determine the angular velocity and angular acceleration of the spool at this instant. Neglect the small vertical part of the rod at A. Given: 1 3 rad s rad s 2 1 2 20 deg Solution: L R 1m L tan 0 r L cos L sin R sin R cos 670 Engineering Mechanics - Dynamics Chapter 20 Guesses 2 1 rad s 2 1 rad s 2 ay 1 m s 2 az 1 m s 2 Given Enforce the no-slip constraint 0 2 cos 1 2 sin r 0 0 2 cos 1 2 sin 0 0 1 1 0 2 cos 2 sin 0 r ay az 2 2 Find ay az 2 2 ay az ay az 0 26.3 m s 2 2 8.77 rad s rad s 2 2 5.85 Now construct the angular velocity and angular acceleration. 0 2 cos 1 2 sin 0.00 8.24 0.00 0 0 1 1 rad s 0 2 cos 1 2 sin 0 2 cos 2 sin 24.73 5.49 0.00 rad s 2 Problem 20-7 At a given instant, the antenna has an angular motion and ' about the z axis. At the same the angular motion about the x axis is and '2. Determine the velocity and instant = acceleration of the signal horn A at this instant. The distance from O to A is d. 671 Engineering Mechanics - Dynamics Chapter 20 Given: 1 3 rad s rad s 2 2 1.5 rad s 2 '1 2 '2 d 4 rad s 1 30 deg 3 ft Solution: 2 '2 0 '1 0 0 0 1 0 1 rA d cos 1 d sin 1 7.79 vA rA vA 2.25 3.90 8.30 aA rA rA aA 35.23 7.02 ft s 2 ft s vA 9 ft s aA 36.868 ft s 2 *Problem 20-8 The cone rolls without slipping such that at the instant shown z and 'z are as given. Determine the velocity and acceleration of point A at this instant. Given: z 4 rad s rad s 2 672 'z 3 Engineering Mechanics - Dynamics Chapter 20 20 deg a Solution: b z 2 ft a sin 2 sin 0 z 2 sin 'z sin 0 '2 cos 0 0 'z 2 11.695 rad s rad s 2 'z '2 sin 0 '2 '2 8.771 0 2 cos 2 sin z '2 sin 0 rA 1.532 ft 1.286 14.1 z 0 rA a 2b sin 2b cos vA rA vA 0.0 0.0 10.6 ft s vA 14.128 ft s aA rA rA aA 56.5 87.9 ft s 2 aA 105.052 ft s 2 Problem 20-9 The cone rolls without slipping such that at the instant shown z and 'z are given. Determine the velocity and acceleration of point B at this instant. Given: z 4 rad s rad s 2 'z 3 673 Engineering Mechanics - Dynamics Chapter 20 20 deg a Solution: b z 2 ft a sin 2 sin 0 z 2 sin 'z sin 0 '2 cos 2 11.695 rad s rad s 2 'z '2 sin 0 '2 '2 8.771 0 2 cos 2 sin z 0 0 'z z '2 sin b rB a b sin b cos rB 0.684 1.766 0.643 7.064 ft vB rB vB 0 7.518 77.319 ft s vB 10.316 ft s aB rB rB aB 28.257 5.638 ft s 2 aB 82.513 ft s 2 Problem 20-10 If the plate gears A and B are rotating with the angular velocities shown, determine the angular velocity of gear C about the shaft DE. What is the angular velocity of DE about the y axis? Given: A 5 rad s rad s 674 B 15 Engineering Mechanics - Dynamics Chapter 20 a b 100 mm 25 mm Solution: Guesses DE 1 rad s x 1 rad s rad s y 1 rad s z 1 Given 0 0 Aa 0 0 Ba x y z x y z 0 2b 0 0 DE 0 0 Ba 0 b 0 a 0 0 0 x x y z z DE 40 0 0 rad s DE Find x y z DE y 5 rad s Problem 20-11 Gear A is fixed to the crankshaft S, while gear C is fixed and gear B and the propeller are free to rotate. The crankshaft is turning with angular velocity s about its axis. Determine the magnitudes of the angular velocity of the propeller and the angular acceleration of gear B. Given: s 80 rad s 675 Engineering Mechanics - Dynamics Chapter 20 r2 r1 0.4 ft 0.1 ft Solution: vP s r2 s r2 B B 2r1 2r1 B 160 rad s vB B r1 vB 16 ft s 0 0.0 40.0 0.0 rad s vB prop r2 prop 0 0 0 B B 0 B prop 6400 0 0 rad s 2 0 *Problem 20-12 The right circular cone rotates about the z axis at a constant rate 1 without slipping on the horizontal plane. Determine the magnitudes of the velocity and acceleration of points B and C. Given: 1 4 rad s r 50 mm 45 deg Solution: Enforce no-slip condition Guess 2 1 rad s 676 Engineering Mechanics - Dynamics Chapter 20 0 Given 2 cos 2 sin 1 0 2r 0 0 2 Find 2 2 5.66 rad s Define terms 0 2 cos 2 sin 1 0 0 1 0 2 cos 2 sin 1 0 rB 2r 0 Find velocities and accelerations vB aB 0 vB 0 0 m s m s aB rB rB rB 0 0 1.131 m s m s 2 2 0 rC 0 2r vC aC rC rC 0.283 vC 0 0 m s m s aC rC 0 1.131 1.131 m s 2 m s 2 vB 0 aB 1.131 vC 0.283 aC 1.6 Problem 20-13 Shaft BD is connected to a ball-and-socket joint at B, and a beveled gear A is attached to its other end. The gear is in mesh with a fixed gear C. If the shaft and gear A are spinning with a constant angular velocity 1, determine the angular velocity and angular acceleration of gear A. Given: 1 8 rad s a 300 mm 677 Engineering Mechanics - Dynamics Chapter 20 rD rC Solution: Guesses Given a cos h 75 mm 100 mm 10 deg h 10 mm rD sin h h a sin rC rD cos Find h 0.293 m 32.904 deg 1 rD y a sin rD cos 1 sin y 6 rad s 4.346 y 1 cos 12.717 0 0.0 0.0 26.1 rad s 0 0 y rad s 2 0 Problem 20-14 The truncated cone rotates about the z axis at a constant rate z without slipping on the horizontal plane. Determine the velocity and acceleration of point A on the cone. Given: z 0.4 1 ft 2 ft 0.5 ft rad s a b c 678 Engineering Mechanics - Dynamics Chapter 20 Solution: asin c a z s z s sin 0 sin 0 s cos z s sin 0 0.693 0 0 rad s 0 0 z 0.277 0 0 rad s 2 rA a b 2 b 2 b a a a 0 c sin rA 1.5 2.598 ft a c cos 1.8 vA rA vA 0 0 ft s 0.000 aA rA rA aA 0.720 0.831 ft s 2 Problem 20-15 The truncated cone rotates about the z axis at z without slipping on the horizontal plane. If at this same instant z is increasing at 'z, determine the velocity and acceleration of point A on the cone. Given: z 0.4 rad s rad s 2 a 1 ft 'z 0.5 b 2 ft 679 Engineering Mechanics - Dynamics Chapter 20 30 deg b a c a 0.5 ft 0 Solution: r c rA a b 2r sin 2r cos Guesses 1 rad s 1 rad s 2 2 2 ay 1 ft s 2 az 1 ft s 2 Given Enforce the no-slip constraints 0 2 cos z 2 sin 0 a 0 0 0 z z 0 0 2 cos 0 2 cos 2 sin 0 a 0 0 ay az 'z 2 2 2 sin Find ay az Define terms 2 2 ay az 2 0.8 rad s 2 1 rad s 2 0 2 cos z 2 sin 0 2 cos 0 0 z z 0 2 cos 2 sin 'z 2 sin Calculate velocity and acceleration. 1.80 vA rA vA 0.00 0.00 2.25 aA rA rA aA 0.72 0.831 ft s 2 ft s 680 Engineering Mechanics - Dynamics Chapter 20 *Problem 20-16 The bevel gear A rolls on the fixed gear B. If at the instant shown the shaft to which A is attached is rotating with angular velocity and has angular acceleration , determine the angular velocity and angular acceleration of gear A. Given: 1 2 rad s rad s 2 1 4 30 deg Solution: L R Guesses 2 1m L tan b L sec 1 rad s m s 2 2 1 rad s 2 ay Given 1 az 1 m s 2 Enforce the no-slip constraints. 0 2 cos 2 sin 1 0 b 0 0 0 1 1 0 0 2 cos 2 sin 0 2 cos 2 sin 1 0 b 0 0 ay az 2 2 Find ay az 2 2 ay az ay az 0 8 m s 2 2 4 rad s 2 8 rad s 2 681 Engineering Mechanics - Dynamics Chapter 20 Build the angular velocity and angular acceleration. 0 2 cos 2 sin 1 0.00 3.46 0.00 rad s 0 2 cos 2 sin 1 0 0 1 0 2 cos 2 sin 1 6.93 6.93 0.00 rad s 2 Problem 20-17 The differential of an automobile allows the two rear wheels to rotate at different speeds when the automobile travels along a curve. For operation, the rear axles are attached to the wheels at one end and have beveled gears A and B on their other ends. The differential case D is placed over the left axle but can rotate about C independent of the axle. The case supports a pinion gear E on a shaft, which meshes with gears A and B. Finally, a ring gear G is fixed to the differential case so that the case rotates with the ring gear when the latter is driven by the drive pinion H. This gear, like the differential case, is free to rotate about the left wheel axle. If the drive pinion is turning with angular velocity H and the pinion gear E is spinning about its shaft with angular velocity E, determine the angular velocity A and B of each axle. Given: H 100 rad s E 30 rad s rG rH rE rA 180 mm 50 mm 40 mm 60 mm Solution: H rH G rG 682 Engineering Mechanics - Dynamics Chapter 20 rH G H rG rad s vE 1.667 m s vE B E rE G 27.778 vE G rA vE E rE B rA rA vE E rE B 7.778 rad s vE E rE A rA A rA A 47.8 rad s Problem 20-18 Rod AB is attached to the rotating arm using ball-and-socket joints. If AC is rotating with constant angular velocity AC about the pin at C, determine the angular velocity of link BD at the instant shown. Given: a b c 1.5 ft 3 ft 6 ft d AC 2 ft 8 rad s Solution: Guesses BD 1 rad s rad s ABx 1 rad s rad s ABy 1 ABz 1 Given Note that AB is perpendicular to rAB. 683 Engineering Mechanics - Dynamics Chapter 20 0 0 AC a 0 0 ABx ABy ABz b d c BD 0 d 0 0 ft 0 s 0 ABx ABy ABz b d c 0 ft s 0 0 BD ABx ABx ABy ABz ABz BD 1.633 0.245 0.735 rad s rad s Find BD ABx ABy ABz ABy 2 Problem 20-19 Rod AB is attached to the rotating arm using ball-and-socket joints. If AC is rotating about the pin at C with angular velocity AC and angular acceleration AC, determine the angular velocity and angular acceleration of link BD at the instant shown. Given: a AC 1.5 ft 3 ft 6 ft 2 ft 8 rad s rad s 2 b c d AC 6 Solution: Guesses BD 1 rad s rad s rad s rad s BD 1 rad s rad s 2 2 ABx 1 ABx 1 ABy 1 ABy 1 rad s 2 ABz 1 ABz 1 rad s 2 684 Engineering Mechanics - Dynamics Chapter 20 Given 0 0 AC 2 Note that a 0 0 AB and AB are perpendicular to rAB. BD ABx ABy ABz ABx ABy ABz b d c 0 d 0 ABx ABy ABz 0 0 0 0 d BD d BD 2 0 0 ABx ABy ABy ABx ft s a AC a AC 0 ABx ABy ABz BD ABx ABy ABz BD ABx ABy ABz ABx ABy ABz b d c b d c b d c 0 b d c 0 ABy ABz Find BD ABx ABy ABz BD ABx ABy ABz 1.633 0.245 0.735 rad s ABx ABy ABz 0.495 14.18 4.479 rad s 2 BD 2 rad s rad s 2 BD 31.6 *Problem 20-20 If the rod is attached with ball-and-socket joints to smooth collars A and B at its end points, determine the speed of B at the instant shown if A is moving downward at constant speed vA. Also, determine the angular velocity of the rod if it is directed perpendicular to the axis of the rod. 685 Engineering Mechanics - Dynamics Chapter 20 Given: vA a Solution: Guesses vB 1 ft s rad s rad s rad s 8 ft s b 6 ft 3 ft c 2 ft x 1 y 1 z 1 Given 0 0 vA vB x x y z z x y z c b a vB 0 0 x y z c b a 0 ft s 0.980 1.061 1.469 rad s vB 12 ft s Find vB x y z y Problem 20-21 If the collar at A is moving downward with an acceleration aA, at the instant its speed is vA, determine the acceleration of the collar at B at this instant. Given: vA 8 ft s a 3 ft c 2 ft 686 Engineering Mechanics - Dynamics Chapter 20 aA 5 ft s 2 b 6 ft Solution: Guesses vB 1 ft s rad s ft s y 2 x x 1 rad s rad s rad s 2 y 1 z 1 aB 1 1 1 rad s 2 z 1 rad s 2 Given 0 0 vA 0 0 aA x y z x x y z z x y z x y z c b a vB 0 0 x y z x y z x y z c b a aB 0 0 x y z 0 c b a c b a c b a 0 0.98 1.06 1.47 rad s Find x y z x y z vB aB y vB aB x y z 687 0.61 5.70 11.82 rad s 2 Engineering Mechanics - Dynamics Chapter 20 vB 12 ft s ft s 2 aB 96.5 Problem 20-22 Rod AB is attached to a disk and a collar by ball-and-socket joints. If the disk is rotating at a constant angular velocity , determine the velocity and acceleration of the collar at A at the instant shown. Assume the angular velocity is directed perpendicular to the rod. Given: 2 r b rad s 1 ft 3 ft Solution: Guesses x 1 rad s rad s rad s ft s x 1 rad s 2 y 1 y 1 rad s 2 z 1 z 1 rad s 2 vA 1 aA 1 ft s 2 Given 0 r 0 x y z b r r vA 0 0 x y z b r r 0 x y z b r r 0 688 Engineering Mechanics - Dynamics Chapter 20 0 0 2 x y b r r x y z x y z b r r aA 0 0 r z x y z x y z Find x y z x y z vA aA vA 0.667 ft s aA 0.148 ft s 2 vA aA x y z 0.182 0.061 0.606 rad s x y z 0.364 1.077 0.013 rad s 2 Problem 20-23 Rod AB is attached to a disk and a collar by ball and-socket joints. If the disk is rotating with an angular acceleration , and at the instant shown has an angular velocity , determine the velocity and acceleration of the collar at A at the instant shown. Given: 2 rad s rad s Solution: Guesses x 2 r 1 ft 4 b 3 ft 1 rad s rad s y 1 rad s ft s z 1 vA 1 689 Engineering Mechanics - Dynamics Chapter 20 x 1 rad s 2 y 1 rad s 2 z 1 rad s 2 aA 1 ft s 2 Given 0 r 0 0 r 2 x y z x y z b r r vA 0 0 b r r x y z x y z x y z b r r aA 0 0 0 x y z b r r 0 b r r r x y z x y z Find x y z x y z vA aA vA 0.667 ft s aA 1.185 ft s 2 vA aA x y z 0.182 0.061 0.606 rad s x y z 3.636 10 7 rad s 2 1.199 1.199 *Problem 20-24 The rod BC is attached to collars at its ends by ball-and-socket joints. If disk A has angular velocity A, determine the angular velocity of the rod and the velocity of collar B at the instant shown. Assume the angular velocity of the rod is directed perpendicular to the rod. Given: a 200 mm 690 Engineering Mechanics - Dynamics Chapter 20 b c Solution: Guesses vB 100 mm 500 mm d A 10 rad s 300 mm 1 m s rad s x 1 rad s rad s y 1 z 1 Given Ab x y z a b d c 0 vB 0 x y z a b d c 0 m s 0 0 vB x x y z z 0.204 0.612 1.361 rad s vB 0.333 m s Find vB x y z y Problem 20-25 The rod BC is attached to collars at its ends. There is a ball-and-socket at C. The connection at B now consists of a pin as shown in the bottom figure. If disk A has angular velocity A, determine the angular velocity of the rod and the velocity of collar B at the instant shown. Hint: The constraint allows rotation of the rod both along the bar DE (j direction) and along the axis of the pin ( n direction). Since there is no rotational component in the u direction, i.e., perpendicular to n and j u 0. The vector n is where u = j n, an additional equation for solution can be obtained from in the same direction as rBC Given: A rDC. 10 rad s a b 200 mm 100 mm 691 Engineering Mechanics - Dynamics Chapter 20 c d 500 mm 300 mm Solution: a rBC b d a rDC 0 d rBC rBC rDC rDC 0.728 n 0.485 0.485 0.485 n u 0 0.728 m s rad s rad s rad s c n 0 u 1 0 Guesses Given Ab vB 1 x 1 y 1 z 1 x y z a b d c 0 vB 0 x y u z 0 0 vB 0 rad s x x y z z 0.769 2.308 0.513 rad s vB 0.333 m s Find vB x y z y Problem 20-26 The rod AB is attached to collars at its ends by ball-and-socket joints. If collar A has a speed vA, determine the speed of collar B at the instant shown. 692 Engineering Mechanics - Dynamics Chapter 20 Given: vA a b 20 2 ft 6 ft 45 deg ft s Solution: Guesses Given 0 vA 0 x x y z x y z x 1 rad s y 1 rad s z 1 rad s vB 1 ft s b a 0 vB sin 0 cos x y z b a 0 0 0.333 1 3.333 rad s vB 9.43 ft s Find x y z vB y z vB Problem 20-27 The rod is attached to smooth collars A and B at its ends using ball-and-socket joints. Determine the speed of B at the instant shown if A is moving with speed vA. Also, determine the angular velocity of the rod if it is directed perpendicular to the axis of the rod. Given: vA a 6 m s b c 1m 1m 0.5 m 693 Engineering Mechanics - Dynamics Chapter 20 Solution: Guesses x 1 rad s y 1 rad s z 1 rad s vB 1 m s Given vA 0 0 x x y z x y z a b c 0 vB 0 x y z a b c 0 1.33 2.67 3.33 rad s vB 3.00 m s Find x y z vB y z vB *Problem 20-28 The rod is attached to smooth collars A and B at its ends using ball-and-socket joints. At the instant shown, A is moving with speed vA and is decelerating at the rate aA. Determine the acceleration of collar B at this instant. Given: vA aA 6 m s m s 2 a 0.5 m 5 b c 1m 1m Solution: Guesses x 1 rad s rad s 2 y 1 rad s rad s 2 z 1 rad s rad s 2 vB 1 m s m s 2 x 1 y 1 z 1 aB 1 694 Engineering Mechanics - Dynamics Chapter 20 Given vA 0 0 aA 0 0 x y z x y z x y z x y z a b c a b c 0 vB 0 x y z x y z x y z a b c a b c 0 aB 0 0 x y z a b c 0 Find x y z x y z vB aB vB aB x y z 1.33 2.67 3.33 rad s x y z 21.11 2.22 12.78 rad s 2 vB 3.00 m s aB 47.5 m s 2 Problem 20-29 Rod AB is attached to collars at its ends by using ball-and-socket joints. If collar A moves along the fixed rod with speed vA, determine the angular velocity of the rod and the velocity of collar B at the instant shown. Assume that the rod's angular velocity is directed perpendicular to the axis of the rod. Given: vA a 8 8 ft ft s c d 6 ft 8 ft 695 Engineering Mechanics - Dynamics Chapter 20 b Solution: 5 ft e 6 ft atan d c x Guesses Given 0 vA 0 x y z x y z 1 rad s y 1 rad s z 1 rad s vB 1 ft s c e b cos b sin a vB cos sin 0 x y z c e b cos b sin a 0 cos Find x y z vB 2.82 vBv 3.76 0.00 x y z vBv vB sin 0 ft s vB 0.440 0.293 0.238 rad s Problem 20-30 Rod AB is attached to collars at its ends by using ball-and-socket joints. If collar A moves along the fixed rod with a velocity vA and has an acceleration aA at the instant shown, determine the angular acceleration of the rod and the acceleration of collar B at this instant. Assume that the rod' s angular velocity and angular acceleration are directed perpendicular to the axis of the rod. Given: vA a 8 ft s aA b 4 ft s 5 ft 2 8 ft c 6 ft d 696 8 ft e 6 ft Engineering Mechanics - Dynamics Chapter 20 Solution: atan d c x Guesses 1 rad s rad s 2 y 1 rad s rad s 2 z 1 rad s rad s 2 vB 1 ft s ft s 2 x 1 y 1 z 1 aB 1 Given 0 vA 0 0 aA 0 x y z x y z x y z x y z x y z c e b cos b sin a vB cos sin 0 x y z x x y z c e b cos b sin a c e b cos b sin a aB cos sin 0 c e b cos b sin a 0 c e b cos b sin a 0 y z Find x y z x y z vB aB vB aB cos vBv vB sin 0 aBv aB cos sin 0 aBv 5.98 7.98 0.00 ft s 2 697 Engineering Mechanics - Dynamics Chapter 20 x y z 0.440 0.293 0.238 rad s x y z 0.413 0.622 0.000 rad s 2 2.824 vBv 3.765 0 ft s Problem 20-31 Consider again Example 20.5. The pendulum consists of two rods: AB is pin supported at A and swings only in the y-z plane, whereas a bearing at B allows the attached rod BD to spin about rod AB. At a given instant, the rods have the angular motions shown. Also, a collar C has velocity vC and acceleration aC along the rod. Determine the velocity and acceleration of the collar at this instant. Solve such that the x, y, z axes move with curvilinear translation, = 0, in which case the collar appears to have both an angular velocity xyz = 1 + 2 and radial motion. Given: 1 4 rad s rad s rad s 2 vCB aCB a 3 m s m s 2 2 5 2 '1 1.5 0.5 m '2 6 rad s 2 b 0.2 m Solution: 1 0 0 a vB 0 2 0 m s vB 0 0 '1 0 0 a 0 1 1 1 0 0 a 1.00 vC 5.00 0.80 m s aB 0 0.75 8 m s 2 aB 0 0 0 0 0 b 0 0 0 vC vB vCB 0 0 2 698 Engineering Mechanics - Dynamics Chapter 20 0 aC aB aCB 0 1 '1 0 '2 0 vCB 0 1 1 0 b 0 1 1 0 b 0 0 0 0 2 0 2 0 2 2 0 2 28.8 aC 5.45 32.3 m s 2 Problem 20-32 Consider again Example 20.5. The pendulum consists of two rods: AB is pin supported at A and swings only in the y-z plane, whereas a bearing at B allows the attached rod BD to spin about rod AB. At a given instant, the rods have the angular motions shown. Also, a collar C has velocity vC and acceleration aC along the rod. Determine the velocity and acceleration of the collar at this instant. Solve by fixing the x, y, z axes to rod BD in which case the collar appears only to have radial motion. Given: 1 4 rad s rad s b '1 1.5 rad s 2 2 5 '2 6 rad s 2 a 0.5 m m s 0.2 m m s 2 vCB Solution: 3 aCB 2 1 0 0 a 0 0 a 1 1 vB 0 0 '1 0 0 a 699 aB 0 0 0 0 0 0 Engineering Mechanics - Dynamics Chapter 20 0 vC vB vCB 0 0 aC aB aCB 0 1 1 0 b 0 1 1 1.00 vC 5.00 0.80 0 b 0 1 1 0 2 m s '1 0 '2 0 vCB 0 0 b 0 0 0 0 2 0 2 0 2 2 0 2 28.8 aC 5.45 32.3 m s 2 Problem 20-33 At a given instant, rod BD is rotating about the y axis with angular velocity BD and angular acceleration 'BD. Also, when = 1 , link AC is rotating downward such that ' = 2 and '' = Determine the velocity and acceleration of point A on the link at this instant. Given: BD 2. 2 rad s rad s 2 1 60 deg rad s rad s 2 'BD L 3 ft 5 2 2 2 8 Solution: 0 BD 2 BD 2 0 rA L cos 1 'BD 0 0 0 L sin 1 700 Engineering Mechanics - Dynamics Chapter 20 5.196 vA rA vA 5.196 3 24.99 aA rA rA aA 26.785 8.785 ft s 2 ft s Problem 20-34 During the instant shown the frame of the X-ray camera is rotating about the vertical axis at z and 'z. Relative to the frame the arm is rotating at rel and 'rel . Determine the velocity and acceleration of the center of the camera C at this instant. Given: z 5 rad s rad s 2 a 1.25 m 'z 2 b 1.75 m rel 2 rad s rad s 2 c 1m 'rel Solution: 1 0 rel z 0 'rel 'z 0 vC 0 z 0 0 z a 0 0 0 b c 701 6.75 vC 6.25 0 m s Engineering Mechanics - Dynamics Chapter 20 a z aC 2 0 b c 0 b c aC 28.75 26.25 4 m s 2 a 'z 0 Problem 20-35 At the instant shown, the frame of the brush cutter is traveling forward in the x direction with a constant velocity v, and the cab is rotating about the vertical axis with a constant angular velocity 1. At the same instant the boom AB has a constant angular velocity ', in the direction shown. Determine the velocity and acceleration of point B at the connection to the mower at this instant. Given: 1 0.5 rad s rad s ' 0.8 m s v a b 1 1m 8m Solution: v vB 0 0 0 aB 1 ' ' 0 1 a b 0 a b 0 ' 0 1 5 vB 0.5 6.4 ' 0 1 m s a b 0 0.25 0 aB 7.12 0.00 702 m s 2 Engineering Mechanics - Dynamics Chapter 20 *Problem 20-36 At the instant shown, the frame of the brush cutter is traveling forward in the x direction with a constant velocity v, and the cab is rotating about the vertical axis with an angular velocity 1, which is increasing at '1. At the same instant the boom AB has an angular velocity ', which is increasing at ''. Determine the velocity and acceleration of point B at the connection to the mower at this instant. Given: 1 0.5 rad s rad s 2 '1 0.4 ' 0.8 rad s rad s 2 '' 0.9 m s v a b Solution: 1 1m 8m v vB 0 0 '' aB 1 ' ' 0 1 a b 0 ' 0 1 5 vB 0.5 6.4 ' 0 1 m s a b 0 a b 0 aB 2.95 7.52 7.20 m s 2 '1 Problem 20-37 At the instant shown, rod BD is rotating about the vertical axis with an angular velocity BD and an angular acceleration BD. Link AC is rotating downward. Determine the velocity and acceleration of point A on the link at this instant. 703 Engineering Mechanics - Dynamics Chapter 20 Given: BD 7 rad s rad s 2 60 deg rad s rad s 2 BD 4 ' 2 l Solution: 0.8 m '' 3 ' 0 BD 0 r l sin cos 0 0 BD z '' 0 BD y 4.85 vA r vA 0.80 1.39 13.97 aA r r aA 35.52 3.68 m s 2 m s Problem 20-38 The boom AB of the locomotive crane is rotating about the Z axis with angular velocity 1 which is increasing at '1. At this same instant, = 1 and the boom is rotating upward at a constant rate of '= Given: 1 2. Determine the velocity and acceleration of the tip B of the boom at this instant. 0.5 rad s 2 '1 3 rad s 1 30 deg 704 Engineering Mechanics - Dynamics Chapter 20 2 3 rad s L r 20 m 3m Solution: 0 vB 0 1 0 r 0 '1 r 0 2 0 L cos 1 0 1 L sin 1 0 L cos 1 2 2 0 L cos 1 aB 2 1 r 1 2 0 1 0 1 0 10.2 vB 30.0 52.0 m s '1 L sin 1 31.0 aB 161.0 90.0 m s 2 L sin 1 Problem 20-39 The locomotive crane is traveling to the right with speed v and acceleration a. The boom AB is rotating about the Z axis with angular velocity 1 which is increasing at '1. At this same instant, = 1 and the boom is rotating upward at a constant rate of acceleration of the tip B of the boom at this instant. Given: v 2 m s 705 '= 2. Determine the velocity and Engineering Mechanics - Dynamics Chapter 20 a 1.5 m s 2 1 0.5 rad s 2 '1 3 rad s 1 30 deg rad s 2 3 L r Solution: 20 m 3m 0 vB v 0 0 0 1 0 r 0 '1 r 2 1 r 2 0 L cos 1 0 1 L sin 1 0 L cos 1 2 2 0 aB a 0 10.2 vB 28.0 52.0 m s 0 1 2 0 L cos 1 0 1 0 1 0 '1 31.0 aB L sin 1 L sin 1 159.5 90.0 m s 2 *Problem 20-40 At a given instant, the rod has the angular motions shown, while the collar C is moving down relative to the rod with a velocity v and an acceleration a. Determine the collar's velocity and acceleration at this instant. Given: v 6 ft s 706 Engineering Mechanics - Dynamics Chapter 20 1 8 rad s rad s 2 '1 12 30 deg a 2 ft s b 2 0.8 ft Solution: 0 vC v cos sin 0 aC a cos sin 0 2 0 1 0 0 1 0 b cos b sin 0 b cos b sin 0 0 1 0 0 '1 0 v cos v sin 0 0 1 0 b cos b sin 5.54 vC 5.20 3.00 ft s aC 91.45 42.61 1.00 ft s 2 Problem 20-41 At the instant shown, the arm OA of the conveyor belt is rotating about the z axis with a constant angular velocity 1, while at the same instant the arm is rotating upward at a constant rate 2. If the conveyor is running at a constant rate r' = v, determine the velocity and acceleration of the package P at the instant shown. Neglect the size of the package. Given: 1 6 rad s v 5 ft s 707 Engineering Mechanics - Dynamics Chapter 20 2 4 rad s r 6 ft 45 deg Solution: 2 0 0 1 0 1 0 rP r cos r sin vP vrel 25.5 vP 13.4 20.5 ft s aP rP aP vrel 0 v cos v sin rP 161.2 248.9 39.6 ft s 2 rP 2 vrel Problem 20-42 At the instant shown, the arm OA of the conveyor belt is rotating about the z axis with a constant angular velocity 1, while at the same instant the arm is rotating upward at a constant rate 2. If the conveyor is running at the rate r' = v which is increasing at the rate r'' = a, determine the velocity and acceleration of the package P at the instant shown. Neglect the size of the package. Given: 1 6 ft s rad s a 2 4 ft s rad s v 5 8 2 r 6 ft 45 deg 708 Engineering Mechanics - Dynamics Chapter 20 Solution: 2 0 0 1 0 rP r cos r sin 0 0 1 0 vrel v cos v sin vP vrel 25.5 vP 13.4 20.5 ft s aP rP aP arel a cos a sin arel 161.2 243.2 33.9 ft s 2 rP rP 2 vrel Problem 20-43 At the given instant, the rod is spinning about the z axis with an angular velocity 1 and angular acceleration '1. At this same instant, the disk is spinning, with 2 and '2 both measured relative to the rod. Determine the velocity and acceleration of point P on the disk at this instant. Given: 1 3 4 '1 rad s rad 2 a b 2 ft 3 ft 2 '2 Solution: s rad 2 s rad 1 2 s c r 4 ft 0.5 ft 0 vP 0 1 b c 0 ft s 2 0 r 0 0 1 10.50 vP 9.00 1.00 709 Engineering Mechanics - Dynamics Chapter 20 0 aP 0 '1 '2 1 2 b c 0 0 r 0 0 0 1 2 0 0 1 2 b c 0 0 r 0 aP 41.00 17.50 0.50 ft s 2 0 1 0 1 '1 *Problem 20-44 At a given instant, the crane is moving along the track with a velocity vCD and acceleration aCD. Simultaneously, it has the angular motions shown. If the trolley T is moving outwards along the boom AB with a relative speed vr and relative acceleration ar, determine the velocity and acceleration of the trolley. Given: 1 0.5 rad s rad s m s '1 0.8 rad s 2 2 0.4 '2 0.6 rad s 2 vCD 8 m s aCD 9 m s 2 m s 2 vr 3 ar 5 l 3m Solution: vCD vA 0 0 0 r l 0 vrel aA aCD 0 0 0 vr 0 710 2 '2 1 2 0 1 '1 0 arel ar 0 Engineering Mechanics - Dynamics Chapter 20 9.50 vT vA vrel r vT 3.00 1.20 14.40 aT aA arel r r 2 vrel aT 3.77 4.20 m s 2 m s Problem 20-45 At the instant shown, the base of the robotic arm is turning about the z axis with angular velocity 1, which is increasing at '1. Also, the boom segment BC is rotating at constant rate BC. Determine the velocity and acceleration of the part C held in its grip at this instant. Given: 1 4 rad s rad s 2 a 0.5 m '1 3 b 0.7 m BC 8 rad s Solution: 0 BC 1 1 BC b r 0 0 0 0 '1 vC r vC 2.8 5.6 56 m s aC r r aC 2.1 0 m s 2 711 Engineering Mechanics - Dynamics Chapter 20 Problem 20-46 At the instant shown, the base of the robotic arm is turning about the z axis with angular velocity 1, which is increasing at '1. Also, the boom segment BC is rotating with angular velociy BC which is incrasing at 'BC. Determine the velocity and acceleration of the part C held in its grip at this instant. Given: 1 4 rad s rad s b '1 3 rad s 2 BC 8 'BC 2 rad s 2 a Solution: 0.5 m 0.7 m 0 BC 1 1 BC 'BC '1 b r 0 0 0 vC r vC 2.8 5.6 aC r 56 aC 2.1 1.4 m s 2 m s r Problem 20-47 The load is being lifted upward at a constant rate v relative to the crane boom AB. At the instant shown, the boom is rotating about the vertical axis at a constant rate 1, and the trolley T is moving outward along the boom at a constant rate vt. Furthermore, at this same instant the rectractable arm supporting the load is vertical and is swinging in the y-z plane at an angular rate 2, with an increase in the rate of swing instant. 2. Determine the velocity and acceleration of the center G of the load at this 712 Engineering Mechanics - Dynamics Chapter 20 Given: 1 4 rad s rad s m s h 2 7 rad s 2 2 5 3m vt 2 s1 4m v 9 m s Solution: 0 vT vt 0 0 vG vT 0 v 2 0 0 1 0 s1 0 2 0 aT 0 1 0 0 1 0 s1 0 2 0 0 1 0 vt 0 0 0 h 0 0 h 2 2 0 1 0 0 h 2 2 0 0 v aG aT 1 2 0 1 0 1 0 1 0 16.0 vG 17.0 9.0 m s 136.0 aG 133.0 75.0 m s 2 713 ...
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This homework help was uploaded on 04/13/2008 for the course ME 2580 taught by Professor Vandenbrink during the Spring '08 term at Western Michigan.

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