VECTOR MECH...,STAT.+DYNA.(LL)-W/ACCESS
VECTOR MECH...,STAT.+DYNA.(LL)-W/ACCESS
11th Edition
ISBN: 9781259633133
Author: BEER
Publisher: MCG
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Chapter 15.4, Problem 15.114P
To determine

The accelerations of points A, B, and C of the drums.

Expert Solution & Answer
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Answer to Problem 15.114P

The acceleration at point A (aA) is 48in./s2()_.

The acceleration at point B (aB) is 85.4in./s2_ with the angle of 69.4°(II-Quad)_.

The acceleration at point C (aC) is 82.8in./s2_ with the angle of 65°(III-Quad)_.

Explanation of Solution

Given information:

The radius of inner drum (r) is 3 in..

The radius of outer drum (R) is 5 in..

The velocity of the cord at D is 8in./s.

The acceleration of the cord at D is 30in./s2.

Calculation:

The velocity at point D and B is equal vB=vD.

Determine the angular velocity ω using the relation.

vA=lABω=(Rr)ω

Here, lAB is the length of the point AB.

Substitute 8 in. /s for vA, 5 in. for R, and 3 in. for r.

8=(53)ωω=82ω=4rad/s

The value of acceleration at point A is aA=[aA] for no slipping.

Write the expression for the acceleration at point A (aA).

aB=aA+(aA/B)t+(aA/B)n

Here, aA is the acceleration at point A, (aA/B)t is the tangential component of acceleration at point A with respect to B, and (aA/B)n is the normal component of acceleration at point A with respect to B.

Substitute [aA] for aA, [(aB)t+(aB)n] for aA, [(Rr)α] for (aB/A)t, and [(Rr)ω2] for (aB/A)n.

[aA]=[(aB)t+(aB)n]+[(Rr)α]+[(Rr)ω2] (1)

Substitute 30in./s2 for (aB)t, 5 in. for R, and 3 in. for r.

[aA]=[30+(aA)n]+[(53)α]+[(53)ω2] . (2)

Determine the angular acceleration α using the relation.

Equate the horizontal components in Equation (2).

0=30+2α30=2αα=302α=15rad/s2

Write the expression for the acceleration at point A (aA) with respect to G.

aA=aG+(aA/G)t+(aA/G)n

Substitute [aA] for aA, [aG] for aG, [rα] for (aA/G)t, and [rω2] for (aA/G)n.

[aA]=[aG]+[rα]+[rω2]

Substitute 3 in. for r.

[aA]=[aG]+[3α]+[3ω2] (3)

Equate the horizontal components in Equation (3).

0=aG3α

Substitute 15rad/s2 for α.

0=aG4×15aG=45in./s2

Equate the vertical components in Equation (3).

Determine the acceleration at point A (aA).

aA=3ω2

Substitute 4rad/s for ω.

aA=3×42=48in./s2()

Therefore, the acceleration at point A (aA) is 48in./s2()_.

Determine the acceleration at point B (aB).

aB=aG+(aB/G)t+(aB/G)n

Substitute [45in./s2] for aG, [Rα] for (aA/G)t, and [Rω2] for (aA/G)n.

aB=[45]+[Rα]+[Rω2]

Substitute 5 in. for R, 15rad/s2 for α, and 4rad/s for ω.

aB=[45]+[5×15]+[5×42]=[45]+[75]+[80]=[30in./s2]+[80in./s2]

Determine the magnitude of the acceleration at point B.

aB=(aB)x2+(aB)y2

Substitute 30in./s2 in (aB)x and 80in./s2 for (aB)y.

aB=302+802=85.4in./s2

Determine the direction of the acceleration at point B.

tanθB=(aB)y(aB)x

Substitute 80in./s2 for (aB)y and 30in./s2 in (aB)x.

tanθB=8030θB=tan1(2.667)θB=69.4°(IIQuad)

Therefore, the acceleration at point B (aB) is 85.4in./s2_ with the angle of 69.4°(II-Quad)_.

Determine the acceleration at point C (aC).

aC=aG+(aC/G)t+(aC/G)n

Substitute [45in./s2] for aG, [Rα] for (aC/G)t, and [Rω2] for (aC/G)n.

aC=[45]+[Rα]+[Rω2]

Substitute 5 in. for R, 15rad/s2 for α, and 4rad/s for ω.

aC=[45]+[5×15]+[5×42]=[45]+[75]+[80]=[35in./s2]+[75in./s2]

Determine the magnitude of the acceleration at point C.

aC=(aC)x2+(aC)y2

Substitute 35in./s2 in (aC)x and 75in./s2 for (aC)y.

aC=352+752=82.8in./s2

Determine the direction of the acceleration at point C.

tanθC=(aC)y(aC)x

Substitute 75in./s2 for (aC)y and 35in./s2 in (aC)x.

tanθC=7535θC=tan1(2.14)θC=65(IIIQuad)

Therefore, the acceleration at point C (aC) is 82.8in./s2_ with the angle of 65°(III-Quad)_.

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Chapter 15 Solutions

VECTOR MECH...,STAT.+DYNA.(LL)-W/ACCESS

Ch. 15.1 - The angular acceleration of a shaft is defined by...Ch. 15.1 - Prob. 15.10PCh. 15.1 - Prob. 15.11PCh. 15.1 - Prob. 15.12PCh. 15.1 - The rectangular block shown rotates about the...Ch. 15.1 - A circular plate of 120-mm radius is supported by...Ch. 15.1 - Prob. 15.15PCh. 15.1 - Prob. 15.16PCh. 15.1 - The earth makes one complete revolution on its...Ch. 15.1 - Prob. 15.18PCh. 15.1 - Prob. 15.19PCh. 15.1 - Prob. 15.20PCh. 15.1 - The rated speed of drum B of the belt sander shown...Ch. 15.1 - The two pulleys shown may be operated with the V...Ch. 15.1 - Prob. 15.23PCh. 15.1 - A gear reduction system consists of three gears A,...Ch. 15.1 - A belt is pulled to the right between cylinders A...Ch. 15.1 - Prob. 15.26PCh. 15.1 - Prob. 15.27PCh. 15.1 - A plastic film moves over two drums. During a 4-s...Ch. 15.1 - Cylinder A is moving downward with a velocity of 3...Ch. 15.1 - The system shown is held at rest by the...Ch. 15.1 - A load is to be raised 20 ft by the hoisting...Ch. 15.1 - A simple friction drive consists of two disks A...Ch. 15.1 - Prob. 15.33PCh. 15.1 - Two friction disks A and B are to be brought into...Ch. 15.1 - Two friction disks A and B are brought into...Ch. 15.1 - Steel tape is being wound onto a spool that...Ch. 15.1 - In a continuous printing process, paper is drawn...Ch. 15.2 - The ball rolls without slipping on the fixed...Ch. 15.2 - Three uniform rodsABC, DCE, and FGHare connected...Ch. 15.2 - Prob. 15.38PCh. 15.2 - Prob. 15.39PCh. 15.2 - A painter is halfway up a 10-m ladder when the...Ch. 15.2 - Rod AB can slide freely along the floor and the...Ch. 15.2 - Rod AB can slide freely along the floor and the...Ch. 15.2 - Rod AB moves over a small wheel at C while end A...Ch. 15.2 - The disk shown moves in the xy plane. 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Knowing that...Ch. 15.2 - Prob. 15.46PCh. 15.2 - Velocity sensors are placed on a satellite that is...Ch. 15.2 - In the planetary gear system shown, the radius of...Ch. 15.2 - Prob. 15.49PCh. 15.2 - 15.50 Arm AB rotates with an angular velocity of...Ch. 15.2 - Prob. 15.51PCh. 15.2 - A simplified gear system for a mechanical watch is...Ch. 15.2 - 15.53 and 15.54Arm ACB rotates about point C with...Ch. 15.2 - 15.53 and 15.54Arm ACB rotates about point C with...Ch. 15.2 - 15.55 Knowing that at the instant shown the...Ch. 15.2 - Prob. 15.56PCh. 15.2 - Knowing that the disk has a constant angular...Ch. 15.2 - The disk has a constant angular velocity of 20...Ch. 15.2 - The test rig shown was developed to perform...Ch. 15.2 - Prob. 15.60PCh. 15.2 - In the engine system shown, l = 160 mm and b = 60...Ch. 15.2 - In the engine system shown, l = 160 mm and b = 60...Ch. 15.2 - Prob. 15.63PCh. 15.2 - Prob. 15.64PCh. 15.2 - Prob. 15.65PCh. 15.2 - Prob. 15.66PCh. 15.2 - Prob. 15.67PCh. 15.2 - Prob. 15.68PCh. 15.2 - 15.69 In the position shown, bar DE has a constant...Ch. 15.2 - Both 6-in.-radius wheels roll without slipping on...Ch. 15.2 - The 80-mm-radius wheel shown rolls to the left...Ch. 15.2 - For the gearing shown, derive an expression for...Ch. 15.3 - The disk rolls without sliding on the fixed...Ch. 15.3 - Prob. 15.6CQCh. 15.3 - A juggling club is thrown vertically into the air....Ch. 15.3 - At the instant shown during deceleration, the...Ch. 15.3 - A helicopter moves horizontally in the x direction...Ch. 15.3 - Prob. 15.76PCh. 15.3 - Prob. 15.77PCh. 15.3 - Prob. 15.78PCh. 15.3 - Prob. 15.79PCh. 15.3 - The arm ABC rotates with an angular velocity of 4...Ch. 15.3 - The double gear rolls on the stationary left rack...Ch. 15.3 - Prob. 15.82PCh. 15.3 - Rod ABD is guided by wheels at A and B that roll...Ch. 15.3 - 15.84 Rod BDE is partially guided by a roller at D...Ch. 15.3 - Prob. 15.85PCh. 15.3 - Prob. 15.86PCh. 15.3 - Prob. 15.88PCh. 15.3 - Small wheels have been attached to the ends of bar...Ch. 15.3 - Prob. 15.90PCh. 15.3 - The disk is released from rest and rolls down the...Ch. 15.3 - Prob. 15.92PCh. 15.3 - Two identical rods ABF and DBE are connected by a...Ch. 15.3 - Arm ABD is connected by pins to a collar at B and...Ch. 15.3 - 15.95 Two 25-in. rods are pin-connected at D as...Ch. 15.3 - Prob. 15.96PCh. 15.3 - At the instant shown, the velocity of collar A is...Ch. 15.3 - Prob. 15.98PCh. 15.3 - Describe the space centrode and the body centrode...Ch. 15.3 - Describe the space centrode and the body centrode...Ch. 15.3 - Prob. 15.101PCh. 15.3 - Using the method of Sec. 15.3, solve Prob. 15.64....Ch. 15.3 - Using the method of Sec. 15.3, solve Prob. 15.65....Ch. 15.3 - Using the method of Sec. 15.3, solve Prob. 15.38....Ch. 15.4 - A rear-wheel-drive car starts from rest and...Ch. 15.4 - Fig. P15.105 and P15.106 15.105A 5-m steel beam is...Ch. 15.4 - For a 5-m steel beam AE, the acceleration of point...Ch. 15.4 - A 900-mm rod rests on a horizontal table. 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the motion...Ch. 15.5 - Prob. 15.162PCh. 15.5 - Prob. 15.163PCh. 15.5 - At the instant shown, the length of the boom AB is...Ch. 15.5 - At the instant shown, the length of the boom AB is...Ch. 15.5 - Prob. 15.166PCh. 15.5 - Prob. 15.167PCh. 15.5 - Prob. 15.168PCh. 15.5 - 15.168 and 15.169A chain is looped around two...Ch. 15.5 - Prob. 15.170PCh. 15.5 - Prob. 15.171PCh. 15.5 - The collar P slides outward at a constant relative...Ch. 15.5 - Pin P slides in a circular slot cut in the plate...Ch. 15.5 - Prob. 15.174PCh. 15.5 - Prob. 15.175PCh. 15.5 - Knowing that at the instant shown the rod attached...Ch. 15.5 - Prob. 15.177PCh. 15.5 - In Prob. 15.177, determine the angular velocity...Ch. 15.5 - At the instant shown, bar BC has an angular...Ch. 15.5 - Prob. 15.180PCh. 15.5 - Rod AB passes through a collar that is welded to...Ch. 15.5 - Prob. 15.182PCh. 15.5 - Prob. 15.183PCh. 15.6 - The bowling ball shown rolls without slipping on...Ch. 15.6 - Prob. 15.185PCh. 15.6 - Prob. 15.186PCh. 15.6 - Prob. 15.187PCh. 15.6 - The rotor of an electric motor rotates at the...Ch. 15.6 - Prob. 15.189PCh. 15.6 - Prob. 15.190PCh. 15.6 - In the system shown, disk A is free to rotate...Ch. 15.6 - Prob. 15.192PCh. 15.6 - Prob. 15.193PCh. 15.6 - Prob. 15.194PCh. 15.6 - A 3-in.-radius disk spins at the constant rate 2 =...Ch. 15.6 - Prob. 15.196PCh. 15.6 - The cone shown rolls on the zx plane with its apex...Ch. 15.6 - At the instant shown, the robotic arm ABC is being...Ch. 15.6 - Prob. 15.199PCh. 15.6 - Prob. 15.200PCh. 15.6 - Several rods are brazed together to form the...Ch. 15.6 - In Prob. 15.201, the speed of point B is known to...Ch. 15.6 - Prob. 15.203PCh. 15.6 - Prob. 15.204PCh. 15.6 - Rod BC and BD are each 840 mm long and are...Ch. 15.6 - Rod AB is connected by ball-and-socket joints to...Ch. 15.6 - Prob. 15.207PCh. 15.6 - Prob. 15.208PCh. 15.6 - Prob. 15.209PCh. 15.6 - Prob. 15.210PCh. 15.6 - Prob. 15.211PCh. 15.6 - Prob. 15.212PCh. 15.6 - Prob. 15.213PCh. 15.6 - Prob. 15.214PCh. 15.6 - In Prob. 15.205, determine the acceleration of...Ch. 15.6 - In Prob. 15.206, determine the acceleration of...Ch. 15.6 - In Prob. 15.207, determine the acceleration of...Ch. 15.6 - Prob. 15.218PCh. 15.6 - Prob. 15.219PCh. 15.7 - A flight simulator is used to train pilots on how...Ch. 15.7 - A flight simulator is used to train pilots on how...Ch. 15.7 - Prob. 15.222PCh. 15.7 - Prob. 15.223PCh. 15.7 - Prob. 15.224PCh. 15.7 - The bent rod shown rotates at the constant rate of...Ch. 15.7 - The bent pipe shown rotates at the constant rate 1...Ch. 15.7 - The circular plate shown rotates about its...Ch. 15.7 - Prob. 15.228PCh. 15.7 - Prob. 15.229PCh. 15.7 - Prob. 15.230PCh. 15.7 - Prob. 15.231PCh. 15.7 - Using the method of Sec. 15.7A, solve Prob....Ch. 15.7 - Prob. 15.233PCh. 15.7 - Prob. 15.234PCh. 15.7 - Prob. 15.235PCh. 15.7 - The arm AB of length 16 ft is used to provide an...Ch. 15.7 - The remote manipulator system (RMS) shown is used...Ch. 15.7 - A disk with a radius of 120 mm rotates at the...Ch. 15.7 - Prob. 15.239PCh. 15.7 - Prob. 15.240PCh. 15.7 - Prob. 15.241PCh. 15.7 - Prob. 15.242PCh. 15.7 - Prob. 15.243PCh. 15.7 - Prob. 15.244PCh. 15.7 - Prob. 15.245PCh. 15.7 - Prob. 15.246PCh. 15.7 - Prob. 15.247PCh. 15 - A wheel moves in the xy plane in such a way that...Ch. 15 - Two blocks and a pulley are connected by...Ch. 15 - A baseball pitching machine is designed to deliver...Ch. 15 - Prob. 15.251RPCh. 15 - Prob. 15.252RPCh. 15 - Knowing that at the instant shown rod AB has zero...Ch. 15 - Rod AB is attached to a collar at A and is fitted...Ch. 15 - Prob. 15.255RPCh. 15 - A disk of 0.15-m radius rotates at the constant...Ch. 15 - Prob. 15.257RPCh. 15 - Prob. 15.258RPCh. 15 - In the position shown, the thin rod moves at a...
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