* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download MOTION RELATIVE TO ROTATING AXES
Center of mass wikipedia , lookup
Photon polarization wikipedia , lookup
Theoretical and experimental justification for the Schrödinger equation wikipedia , lookup
Derivations of the Lorentz transformations wikipedia , lookup
Relativistic mechanics wikipedia , lookup
Classical mechanics wikipedia , lookup
Specific impulse wikipedia , lookup
Coriolis force wikipedia , lookup
Newton's theorem of revolving orbits wikipedia , lookup
Hunting oscillation wikipedia , lookup
Angular momentum operator wikipedia , lookup
Modified Newtonian dynamics wikipedia , lookup
Velocity-addition formula wikipedia , lookup
Newton's laws of motion wikipedia , lookup
Fictitious force wikipedia , lookup
Accretion disk wikipedia , lookup
Seismometer wikipedia , lookup
Equations of motion wikipedia , lookup
Relativistic angular momentum wikipedia , lookup
Jerk (physics) wikipedia , lookup
Rigid body dynamics wikipedia , lookup
Classical central-force problem wikipedia , lookup
PROBLEMS The gear has the angular motion shown. Determine the angular velocity and angular acceleration of the slotted link BC at this instant. The pin at A is fixed to the gear. C A w=2 rad/s 2m 0.5 m 0.7 m B O a=4 rad/s2 PROBLEMS Link 1, of the plane mechanism shown, rotates about the fixed point O with a constant angular speed of 5 rad/s in the cw direction while slider A, at the end of link 2, moves in the circular slot of link 1. Determine the angular velocity and the angular acceleration of link 2 at the instant represented where BO is perpendicular to OA. The radius of the slot is 10 cm. Take sin 37=06, cos 37=0.8 1 A 10 cm 2 20 cm 37o C 37o w1=5 rad/s B O 16 cm BO OA PROBLEMS For the instant shown, particle A has a velocity of 12.5 m/s towards point C relative to the disk and this velocity is decreasing at the rate of 7.5 m/s each second. The disk rotates about B with angular velocity w=9 rad/s and angular acceleration a=60 rad/s2 in the directions shown in the figure. The angle b remains constant during the motion. Telescopic link has a velocity of 5 m/s and an acceleration of -2.5 m/s. Determine the absolute velocity and acceleration of point A for the position shown. Problem 7 Velocity Analysis v A v B w r v rel 24 7 v B 5 i j 25 25 v B 4.8 i 1.4 j vB b 25 24 7 Velocity Analysis 2 yx 9 2 2 2 2 x 3 9 2 2 w r 9k i j 3 3 w r -6 i 6 j 2 x m 3 Velocity Analysis 2 2 2 2 yx x 9 3 9 dy 4 tan 2x dx x 2 / 3 3 2 vrel 3 5 4 3 4 v rel 12.5 i j 5 5 v rel 7.5 i 10 j v A 6.3 i 17.4 j 2 x m 3 Acceleration Analysis a A a B w w r a r 2w v rel a rel aB aB 24 7 -2.5 i j 25 25 -2.4 i - 0.7 j aB b 25 24 7 Acceleration Analysis 2 yx 9 2 2 2 2 x 3 9 2 x m 3 2 2 w w r 9k 9k i j 9k - 6 i 6 j -54 i - 54 j 3 3 2 2 a r -60k i j 40 i - 40 j 3 3 Acceleration Analysis 2w v rel 2 9k 7.5 i 10 j 2w v rel -180 i 135 j Acceleration Analysis +n +t (arel)n vrel 3 dy 4 2x dx x 2 / 3 3 d2y 2 dx 2 dy 1 dx d2y dx 2 2 3/ 2 v 2rel 12.52 a rel n 67.49 m / s 2 2.315 4 3 a rel n 67.49 - i j -53.992 i 40.494 j 5 5 3 4 a rel t -7.5 i j -4.5 i - 6 j 5 5 (arel)t 2.315 m a A -254.892 i 74.794 j 5 4 PROBLEMS The pin A in the bell crank AOD is guided by the flanges of the collar B, which slides with a constant velocity vB of 0.9 m/s along the fixed shaft for an interval of motion. For the position =30o determine the acceleration of the plunger CE, whose upper end is positioned by the radial slot in the bell crank. . Problem 8 Velocity Analysis vA vrel vA 30o vB=(vA)x 129.9 mm 60o vA 30o vB 0.9 1.039 m / s cos 30 cos 30 1.039 wAOD w 6.928 rad / s 0.15 vC -v C j (1) v C v O w rC / O v rel -6.928k 0.225 i 0.13 j v rel cos 30 i sin 30 j 0 v C 0.9 i - 1.56 j 0.866v rel i 0.5v rel j (2) vrel (1)=(2) vrel=-1.039 m/s vc=2.079 m/s Acceleration Analysis aA vrel VB=constant (aA)n o 30 aA So aA must be vertical. 60o (aA)t 30o 129.9 mm a A n w2 OA 6.9282 0.15 7.195 m / s 2 a A n 2 aA 8.308 m / s cos 30 a A t a A sin 30 4.154 m / s 2 a A t a AOD AO a AOD a 27.695 rad / s 2 aC aC j (3) a C a O w w r a r 2w v rel a rel 0 a C -6.928k - 6.928k 0.225 i 0.13 j 27.695k 0.225 i 0.13 j 2 - 6.928k - 1.039 cos 30 i - 1.039 sin 30 j arel cos 30 i sin 30 j a C -21.58 i 12.464 j 0.866a rel i 0.5a rel j (4) (3)=(4) arel=24.92 m/s2 aC=27.92 m/s2 PROBLEMS 1. The uniform 30-kg bar OB is secured to the accelerating frame in the 30o position from the horizontal by the hinge at O and roller at A. If the horizontal acceleration of the frame is a=20 m/s2, compute the force FA on the roller and the x- and y-components of the force supported by the pin at O. PROBLEMS 2. The block A and attached rod have a combined mass of 60 kg and are confined to move along the 60o guide under the action of the 800 N applied force. The uniform horizontal rod has a mass of 20 kg and is welded to the block at B. Friction in the guide is negligible. Compute the bending moment M exerted by the weld on the rod at B. SOLUTION FBD Kinetic Diagram mTax=60ax x x N 60o W=60(9.81) N Fx ext. forces max 800 - 60(9.81) sin 60 60a x a x 4.84 m / s 2 By FBD of rod KD of rod m1ax=20ax Bx M W1=20(9.81) N M B ma x d M 196 m / s 2 M - 20 (9.81)0.7 (20 )( 4.94 )( 0.7 sin 60 ) PROBLEMS 3. The parallelogram linkage shown moves in the vertical plane with the uniform 8 kg bar EF attached to the plate at E by a pin which is welded both to the plate and to the bar. A torque (not shown) is applied to link AB through its lower pin to drive the links in a clockwise direction. When reaches 60o, the links have an angular acceleration an angular velocity of 6 rad/s2 and 3 rad/s, respectively. For this instant calculate the magnitudes of the force F and torque M supported by the pin at E. PROBLEMS 4. The uniform 100 kg log is supported by the two cables and used as a battering ram. If the log is released from rest in the position shown, calculate the initial tension induced in each cable immediately after release and the corresponding angular acceleration a of the cables. SOLUTION +n FBD KD +n TA TB ma n +t mat +t W=100(9.81) N When it starts to move, v=0, w=0 but a≠0 Fn ext. forces 0 Ft ext. forces mat at ar a an w 2 r 0 TA TB - mg cos 30 0 mg sin 30 mat TA TB 849.57 at 4.905 m / s 2 3TA TB 4.905 2.45 rad / s 2 2 Length of the cables The motion of the log is curvilinear translation. M G d .k . 0 TA 212 .39 N TA sin 60 (1.5) - TB sin 60 (0.5) 0 TB 637 .17 N * * PROBLEMS 5. An 18 kg triangular plate is supported by cables AB and CD. When the plate is in the position shown, the angular velocity of the cables is 4 rad/s ccw. At this instant, calculate the acceleration of the mass center of the plate and the tension in each of the cables. C A 60° 24 cm B 10 cm 60° D G Answer: 20 cm 20 cm a 6.23 m / s 2 TAB 143 .11 N TCD 78 .93 N PROBLEMS 6. The uniform 8 kg slender bar is hinged about a horizontal axis through O and released from rest in the horizontal position. Determine the distance b from the mass center to O which will result in an initial angular acceleration of 16 rad/s2, and find the force R on the bar at O just after release. PROBLEMS 7. The spring is uncompressed when the uniform slender bar is in the vertical position shown. Determine the initial angular acceleration a of the bar when it is released from rest in a position where the bar has been rotated 30o clockwise from the position shown. Neglect any sag of the spring, whose mass is negligible. SOLUTION Unstrecthed length of the spring: When =30o , length of the spring: When =30o lo (2l / 4) 2 l 2 l spring 5 l 2 3 l 2 5 5 3 3 , spring force: Fspring k ll kl 2 2 2 2 (in compression) M O Ia mat 30o W +t l l 1 l - mg cos 60 Fspring ml 2 a m at 4 4 2 12 l . 60o Ot +n O lspring G 60o 30o l On a Fspring +t mat +n G Ia man mw 2 a 0.864 l 0 4 k g - 0.857 m l 4