To determine The angular accelerations of the bars AB and BC . Explanation Draw the schematic diagram of the system as shown in Figure (1). Write the expression for the acceleration of point B in vector form. a B = a A + a B / A = a A + ( a B / A ) t + ( a B / A ) n (I) Here, acceleration of point B in vector form is a B , acceleration of point A in vector form is a A , acceleration of point B with respect to point A in vector form is a B / A , tangential component of acceleration of point B with respect to point A in vector form is ( a B / A ) t and normal component of acceleration of point B with respect to point A in vector form is ( a B / A ) n . Write the expression for the acceleration of point B in vector form. a B = ( a B ) t + ( a B ) n (II) Here, tangential component of acceleration of point B in vector form is ( a B ) t and normal component of acceleration of point B in vector form is ( a B ) n . Conclusion: Calculate the position vector of point A with respect to point O . r O / A = ( O A ¯ ) ( cos 60 ° i + sin 60 ° j ) Here, distance between the points O and A is O A ¯ and unit vectors along x and y directions are i and j respectively. Substitute 80 mm for O A ¯ . r O / A = ( 80 mm ) ( cos 60 ° i + sin 60 ° j ) = ( 0.080 m ) ( cos 60 ° i + sin 60 ° j ) = 0.04 i + 0.069 j m Calculate the velocity of point A . v A = ω O A ( O A ¯ ) Here, angular velocity of the link OA is ω O A . Substitute 80 mm for O A ¯ and 10 rad / s for ω O A . v A = ( 10 rad / s ) ( 80 mm ) = ( 10 rad / s ) ( 0.080 m ) = 0.8 m / s From Figure (1), consider the triangle Δ I A B and then apply the law of sine rule. A I ¯ sin 75 ° = B I ¯ sin 75 ° = A B ¯ sin 30 ° Here, distance between the points A and I is A I ¯ , distance between the points B and I is B I ¯ and distance between the points A and B is A B ¯ . Calculate the distance between the points A and I . A I ¯ sin 75 ° = A B ¯ sin 30 ° Substitute 240 mm for A B ¯ . A I ¯ sin 75 ° = 240 mm sin 30 ° A I ¯ sin 75 ° = 0.240 m sin 30 ° A I ¯ = 0.4636 m Calculate the distance between the points B and I . B I ¯ sin 75 ° = A B ¯ sin 30 ° Substitute 240 mm for A B ¯ . B I ¯ sin 75 ° = 240 mm sin 30 ° B I ¯ sin 75 ° = 0.240 m sin 30 ° B I ¯ = 0.4636 m Calculate the angular velocity of the link AB . ω A B = v A A I ¯ Here, angular velocity of the link AB is ω A B . Substitute 0.4636 m for A I ¯ and 0.8 m / s for v A . ω A B = 0.8 m / s 0.4636 m = 1.7256 rad / s Calculate the velocity of point B . v B = ω A B ( B I ¯ ) Substitute 0.4636 m for B I ¯ and 1.7256 rad / s for ω A B . v B = ( 1.7256 rad / s ) ( 0.4636 m ) = 0.8 m / s Calculate the angular velocity of the link BC . ω B C = v B B C ¯ Here, angular velocity of the link BC is ω B C and distance between the points B and C is B C ¯ . Substitute 0.2 m for B C ¯ and 0.8 m / s for v B . ω B C = 0.8 m / s 0.2 m = 4 rad / s Write the expression for the angular acceleration of the link OA in vector form. α O A = α O A k = ( 5 rad / s 2 ) k = 5 k rad / s 2 Here, angular acceleration of the link OA in vector form is α O A , angular velocity of the link OA is α O A and unit vector along z direction is k . Calculate the tangential component of acceleration of point A in vector form. ( a A ) t = α O A × r O / A Here, tangential component of acceleration of point A in vector form is ( a A ) t . Substitute 5 k rad / s 2 for α O A and 0.04 i + 0.069 j m for r O / A . ( a A ) t = ( 5 k rad / s 2 ) × ( 0.04 i + 0.069 j m ) = − 0.345 i + 0.2 j m / s 2 Write the expression for the angular velocity of the link OA in vector form. ω O A = ω O A k = ( 10 rad / s ) k = 10 k rad / s Here, angular velocity of the link OA in vector form is ω O A and angular velocity of the link OA is ω O A . Calculate the normal component of acceleration of point A in vector form. ( a A ) n = ω O A × ( ω O A × r O / A ) Here, normal component of acceleration of point A in vector form is ( a A ) n . Substitute 10 k rad / s for ω O A and 0.04 i + 0.069 j m for r O / A . ( a A ) n = ( 10 k rad / s ) × [ ( 10 k rad / s ) × ( 0.04 i + 0.069 j m ) ] = − 4 i − 6.9 j m / s 2 Calculate the acceleration of point A in vector form. a A = ( a A ) t + ( a A ) n Substitute − 4 i − 6.9 j m / s 2 for ( a A ) n and − 0.345 i + 0.2 j m / s 2 for ( a A ) t . a A = ( − 4 i − 6.9 j m / s 2 ) + ( − 0.345 i + 0.2 j m / s 2 ) = − 4.345 i − 6.7 j m / s 2 Calculate the normal component of acceleration of point B with respect to point A . ( a B / A ) n = ω A B 2 ( A B ¯ ) Here, normal component of acceleration of point B with respect to point A is ( a B / A ) n

8th Edition

ISBN: 9781118885840

Author: James L. Meriam, L. G. Kraige, J. N. Bolton

Publisher: WILEY

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Chapter 5.6, Problem 153P

To determine

The angular accelerations of the bars AB and BC.

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Engineering Mechanics: Dynamics

Ch. 5.2 - Prob. 1P Ch. 5.2 - The circular sector rotates about a fixed axis...Ch. 5.2 - Prob. 3P Ch. 5.2 - Prob. 4P Ch. 5.2 - When switched on, the grinding machine accelerates...Ch. 5.2 - The small cart is released from rest in position 1...Ch. 5.2 - The flywheel has a diameter of 600 mm and rotates...Ch. 5.2 - Prob. 8P Ch. 5.2 - Prob. 9P Ch. 5.2 - The angular acceleration of a body which is...

Ch. 5.2 - The device shown rotates about the fixed z-axis...Ch. 5.2 - Prob. 12P Ch. 5.2 - The T-shaped body rotates about a horizontal axis...Ch. 5.2 - Prob. 14P Ch. 5.2 - Prob. 15P Ch. 5.2 - Prob. 16P Ch. 5.2 - The bent flat bar rotates about a fixed axis...Ch. 5.2 - At time t = 0, the arm is rotating about the fixed...Ch. 5.2 - A variable torque is applied to a rotating wheel...Ch. 5.2 - Prob. 20P Ch. 5.2 - Prob. 21P Ch. 5.2 - Prob. 22P Ch. 5.2 - Prob. 23P Ch. 5.2 - Prob. 24P Ch. 5.2 - Prob. 25P Ch. 5.2 - During its final spin cycle, a front-loading...Ch. 5.2 - Prob. 27P Ch. 5.2 - Prob. 28P Ch. 5.3 - Slider A moves in the horizontal slot with a...Ch. 5.3 - The fixed hydraulic cylinder C imparts a constant...Ch. 5.3 - Prob. 31P Ch. 5.3 - At the instant under consideration, the hydraulic...Ch. 5.3 - The hydraulic cylinder D is causing the distance...Ch. 5.3 - The Scotch-yoke mechanism converts rotational...Ch. 5.3 - Prob. 35P Ch. 5.3 - The wheel of radius r rolls without slipping, and...Ch. 5.3 - Link OA rotates with a clockwise angular velocity...Ch. 5.3 - Determine the acceleration of the shaft B for θ =...Ch. 5.3 - Prob. 39P Ch. 5.3 - Prob. 40P Ch. 5.3 - Boom OA is being elevated by the rope-and-pulley...Ch. 5.3 - The hydraulic cylinder imparts a constant upward...Ch. 5.3 - Prob. 43P Ch. 5.3 - The rod OB slides through the collar pivoted to...Ch. 5.3 - Prob. 45P Ch. 5.3 - Prob. 46P Ch. 5.3 - Link OA is given a clockwise angular velocity ω =...Ch. 5.3 - Prob. 48P Ch. 5.3 - Derive an expression for the upward velocity v of...Ch. 5.3 - Prob. 50P Ch. 5.3 - Show that the expressions v = rω and at = rα hold...Ch. 5.3 - Prob. 52P Ch. 5.3 - Prob. 53P Ch. 5.3 - Prob. 54P Ch. 5.3 - Prob. 55P Ch. 5.3 - Prob. 56P Ch. 5.3 - Prob. 57P Ch. 5.3 - The punch is operated by a simple harmonic...Ch. 5.4 - The right-angle link AB has a clockwise angular...Ch. 5.4 - The uniform rectangular plate moves on the...Ch. 5.4 - The cart has a velocity of 4 ft/sec to the right....Ch. 5.4 - Prob. 62P Ch. 5.4 - The speed of the center of the earth as it orbits...Ch. 5.4 - Prob. 64P Ch. 5.4 - The circular disk of radius 8 in. is released very...Ch. 5.4 - For a short interval, collars A and B are sliding...Ch. 5.4 - Prob. 67P Ch. 5.4 - The magnitude of the absolute velocity of point A...Ch. 5.4 - Prob. 69P Ch. 5.4 - Prob. 70P Ch. 5.4 - Determine the angular velocity of bar AB just...Ch. 5.4 - For the instant represented, point B crosses the...Ch. 5.4 - Prob. 73P Ch. 5.4 - For a short interval, collars A and B are sliding...Ch. 5.4 - Determine the angular velocity of link BC for the...Ch. 5.4 - The elements of a switching device are shown. If...Ch. 5.4 - Determine the angular velocity ωAB of link AB and...Ch. 5.4 - Determine the angular velocity ωAB of link AB and...Ch. 5.4 - The rotation of the gear is controlled by the...Ch. 5.4 - Prob. 80P Ch. 5.4 - Prob. 81P Ch. 5.4 - The ends of the 0.4-m slender bar remain in...Ch. 5.4 - Prob. 83P Ch. 5.4 - Prob. 84P Ch. 5.4 - Pin P on the end of the horizontal rod slides...Ch. 5.4 - A four-bar linkage is shown in the figure (the...Ch. 5.4 - The mechanism is part of a latching device where...Ch. 5.4 - The elements of the mechanism for deployment of a...Ch. 5.4 - Prob. 89P Ch. 5.4 - Prob. 90P Ch. 5.5 - The slender bar is moving in general plane motion...Ch. 5.5 - Prob. 92P Ch. 5.5 - Prob. 93P Ch. 5.5 - Roller B of the quarter-circular link has a...Ch. 5.5 - Prob. 95P Ch. 5.5 - Prob. 96P Ch. 5.5 - Prob. 97P Ch. 5.5 - At a certain instant vertex B of the...Ch. 5.5 - Prob. 99P Ch. 5.5 - Prob. 100P Ch. 5.5 - The mechanism of Prob. 5/100 is now shown in a...Ch. 5.5 - Prob. 102P Ch. 5.5 - Prob. 103P Ch. 5.5 - The switching device of Prob. 5/76 is repeated...Ch. 5.5 - The shaft of the wheel unit rolls without slipping...Ch. 5.5 - Prob. 106P Ch. 5.5 - The attached wheels roll without slipping on the...Ch. 5.5 - The mechanism of Prob. 5/77 is repeated here. By...Ch. 5.5 - Prob. 109P Ch. 5.5 - Prob. 110P Ch. 5.5 - Prob. 111P Ch. 5.5 - Prob. 112P Ch. 5.5 - Prob. 113P Ch. 5.5 - Solve for the speed of point D in Prob. 5/64 by...Ch. 5.5 - Link OA has a counterclockwise angular velocity =...Ch. 5.5 - Vertical oscillation of the spring-loaded plunger...Ch. 5.5 - A device which tests the resistance to wear of two...Ch. 5.5 - Motion of the roller A against its restraining...Ch. 5.5 - In the design of the mechanism shown, collar A is...Ch. 5.5 - Determine the angular velocity ω of the ram head...Ch. 5.6 - For the instant represented, corner C of the...Ch. 5.6 - The two rotor blades of 800-mm radius rotate...Ch. 5.6 - Prob. 123P Ch. 5.6 - Determine the angular velocity and angular...Ch. 5.6 - The wheel of radius R rolls without slipping, and...Ch. 5.6 - The 9-m steel beam is being hoisted from its...Ch. 5.6 - The bar of Prob. 5/82 is repeated here. The ends...Ch. 5.6 - Prob. 128P Ch. 5.6 - Prob. 129P Ch. 5.6 - Prob. 130P Ch. 5.6 - Prob. 131P Ch. 5.6 - Prob. 132P Ch. 5.6 - Prob. 133P Ch. 5.6 - The switching device of Prob. 5/76 is repeated...Ch. 5.6 - Prob. 135P Ch. 5.6 - Prob. 136P Ch. 5.6 - If the wheel in each case rolls on the circular...Ch. 5.6 - Prob. 138P Ch. 5.6 - The system of Prob. 5/101 is repeated here. Crank...Ch. 5.6 - Prob. 140P Ch. 5.6 - The mechanism of Prob. 5/77 is repeated here. The...Ch. 5.6 - The system of Prob. 5/84 is repeated here. If the...Ch. 5.6 - The shaft of the wheel unit rolls without slipping...Ch. 5.6 - Plane motion of the triangular plate ABC is...Ch. 5.6 - The system of Prob. 5/110 is repeated here. At the...Ch. 5.6 - The velocity of roller A is vA = 0.5 m/s to the...Ch. 5.6 - In the design of this linkage, motion of the...Ch. 5.6 - The mechanism of Prob. 5/112 is repeated here. If...Ch. 5.6 - The bar AB from Prob. 5/74 is repeated here. If...Ch. 5.6 - If the piston rod of the hydraulic cylinder C has...Ch. 5.6 - Prob. 151P Ch. 5.6 - Prob. 152P Ch. 5.6 - The four-bar linkage of Prob. 5/86 is repeated...Ch. 5.6 - Prob. 154P Ch. 5.6 - Prob. 155P Ch. 5.6 - Prob. 156P Ch. 5.7 - The disk rotates about a fixed axis through O with...Ch. 5.7 - The sector rotates with the indicated angular...Ch. 5.7 - The slotted wheel rolls to the right without...Ch. 5.7 - The disk rolls without slipping on the horizontal...Ch. 5.7 - Prob. 161P Ch. 5.7 - An experimental vehicle A travels with constant...Ch. 5.7 - Prob. 163P Ch. 5.7 - Prob. 164P Ch. 5.7 - The small collar A is sliding on the bent bar with...Ch. 5.7 - Prob. 167P Ch. 5.7 - Vehicle A travels west at high speed on a...Ch. 5.7 - Prob. 169P Ch. 5.7 - Prob. 170P Ch. 5.7 - Prob. 171P Ch. 5.7 - Prob. 172P Ch. 5.7 - Prob. 173P Ch. 5.7 - Prob. 174P Ch. 5.7 - Prob. 175P Ch. 5.7 - Prob. 176P Ch. 5.7 - Prob. 177P Ch. 5.7 - Refer to the figure for Prob. 5/177. Car A is...Ch. 5.7 - For the instant represented, link CB is rotating...Ch. 5.7 - The disk rotates about a fixed axis through point...Ch. 5.7 - All conditions of the previous problem remain the...Ch. 5.7 - Prob. 182P Ch. 5.7 - Prob. 183P Ch. 5.7 - One wheel of an experimental vehicle F, which has...Ch. 5.8 - Prob. 185RP Ch. 5.8 - Prob. 186RP Ch. 5.8 - Prob. 187RP Ch. 5.8 - Prob. 188RP Ch. 5.8 - Prob. 189RP Ch. 5.8 - Roller B of the linkage has a velocity of 0.75 m/s...Ch. 5.8 - Prob. 191RP Ch. 5.8 - Prob. 192RP Ch. 5.8 - Prob. 193RP Ch. 5.8 - Prob. 194RP Ch. 5.8 - Prob. 195RP Ch. 5.8 - Prob. 196RP Ch. 5.8 - The isosceles triangular plate is guided by the...Ch. 5.8 - Prob. 198RP Ch. 5.8 - The hydraulic cylinder C imparts a velocity υ to...Ch. 5.8 - Prob. 200RP Ch. 5.8 - The figure illustrates a commonly used...Ch. 5.8 - Prob. 202RP Ch. 5.8 - Prob. 203RP Ch. 5.8 - Prob. 204RP Ch. 5.8 - Prob. 206RP Ch. 5.8 - For the slider-crank configuration shown, derive...Ch. 5.8 - Prob. 212RP

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