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 16.1, Problem 16.36P

(a)

To determine

The angular acceleration of gear A (αA).

(a)

Expert Solution
Check Mark

Answer to Problem 16.36P

The angular acceleration of gear A is (αA) 6.06rad/s2_.

Explanation of Solution

The mass of the gear A and B (mAandmB) is 9 kg.

The radius of gyration (kA,B) is 200 mm.

The mas of the gear C (mC) is 3 kg.

The radius of gyration (kC) is 75 mm.

The magnitude of couple (M) is 5 N-m.

Calculation:

Find the tangential acceleration (at) of gear teeth:

at=αr

Here, r radius of gear.

Calculate the tangential acceleration of gear A:

at=(250mm×1m1000mm)αA=0.25αA (1)

Calculate the tangential acceleration of gear B.

at=(250mm×1m1000mm)αB=0.25αB (2)

Calculate the tangential acceleration of gear C.

at=(100mm×1m1000mm)αC=0.1αC (3)

Equate Equation (1) and (2).

0.25αA=0.25αBαA=αB

Equate Equation (1) and (3).

0.25αA=0.1αCαC=2.5αA

Show the free body diagram and kinetic diagram of Gear B as in Figure (1).

VECTOR MECH...,STAT.+DYNA.(LL)-W/ACCESS, Chapter 16.1, Problem 16.36P , additional homework tip  1

Take moment about point B.

Since, the system of external forces is equivalent to system of effective forces. Therefore, express the system of external moment:

MB=(MB)eff (4)

Take counterclockwise moment as positive.

Take moment about point A MB for external force using Figure (2).

MB=FBCrB

Take moment about point A (MB)eff for effective force using Figure (2).

(MB)eff=IBαB

Substitute FBCrB for MB, and IBαA for (MB)eff in Equation (4).

FBCrB=IBαAFBC=IBαArB

Show the free body diagram and kinetic diagram of Gear C as in Figure (2).

VECTOR MECH...,STAT.+DYNA.(LL)-W/ACCESS, Chapter 16.1, Problem 16.36P , additional homework tip  2

Calculate the moment of inertia (IC) of gear C:

IC=mCkc2

Substitute 3 kg for mC and 0.075 m for kC.

IC=(3kg)(0.075m)2=0.016875kgm2

Since, the system of external forces is equivalent to system of effective forces. Therefore, express the system of external moment:

MC=(MC)eff (5)

Take counterclockwise moment as positive.

Take moment about point C MC for external force using Figure (3).

MC=FBCrC+FACrC

Take moment about point C (MC)eff for effective force using Figure (3).

(MC)eff=ICαC

Substitute FBCrC+FACrC for MC, and ICαC for (MC)eff in Equation (5).

FBCrC+FACrC=ICαC

Substitute IBαArB for FBC, 2.5αA for αC, 0.1 m for rC, and 0.25 m for rB.

IBαArBrC+FACrC=IC(2.5αA)FACrC=(2.5ICαA+IBαA0.25m0.1m)FAC=1rC(2.5IC+0.4IB)αA . (6)

Show the free body diagram and kinetic diagram of gear A as in Figure (3).

VECTOR MECH...,STAT.+DYNA.(LL)-W/ACCESS, Chapter 16.1, Problem 16.36P , additional homework tip  3

Since the system of external forces is equivalent to system of effective forces, express the system of external moment:

MA=(MA)eff (7)

Take clockwise moment as positive.

Take moment about point C MA for external force using Figure 4.

MA=MAFACrA

Take moment about point C (MA)eff for effective force using Figure 4.

(MA)eff=IAαA

Substitute 0.25F for MAFACrA and IAαA for (MA)eff in Equation (7).

MAFACrA=IAαA

Substitute 1rC(2.5IC+0.4IB)αA for FAC, 0.1 m for rC, and 0.25 m for rA.

MA(1rC(2.5IC+0.4IB)αA)rA=IAαAMA=IAαA+0.250.1(2.5IC+0.4IB)αAMA=IAαA+(6.25IC+IB)αAMA=(IA+IB+6.25IC)αA (8)

Calculate the moment of inertia of gear C.

IC=mCkC2

Substitute 3 kg for mC, and 0.075 m for kC.

IC=(3kg)(0.075m)2=0.016875kgm2

Calculate the moment of inertia (IA) of gear A using the relation:

IA=mAkA2

Substitute 9 kg for mA, and 0.2 m for kA.

IA=(9kg)(0.2m)2=0.36kgm2

The moment of inertia of gear A (IA) is equal to moment of inertia of gear B (IB).

Substitute 0.016875kgm2 for IC, 5Nm for MA, and 0.36kgm2 for IA, and 0.36kgm2 for IB in Equation (8).

MA=(IA+IB+6.25IC)αA5Nm=(0.36kgm2+0.36kgm2+6.25(0.016875kgm2))αAαA=5Nm(0.36kgm2+0.36kgm2+6.25(0.016875kgm2))αA=6.06rad/s2

Thus, the angular acceleration of gear are (αA) 6.06rad/s2_.

(b)

To determine

The tangential force (FA) which gear C exerts on gear A.

(b)

Expert Solution
Check Mark

Answer to Problem 16.36P

The tangential force (FA) which gear C exerts on gear A is 11.28N_.

Explanation of Solution

Given information:

The mass of the gear A and B (mAandmB) is 9 kg.

The radius of gyration (kA,B) is 200 mm.

The mas of the gear C (mC) is 3 kg.

The radius of gyration (kC) is 75 mm.

The magnitude of couple (M) is 5N-m.

Calculation:

Calculate the tangential force (FA) which gear C exerts on gear A:

Substitute 0.0169kgm2 for IC, 0.36kgm2 for IB, 6.06rad/s2 for αA, and 0.1 m for rC in Equation (6).

FA=1rC(2.5IC+0.4IB)αA=10.1m(2.5(0.0169kgm2)+0.4(0.36kgm2))6.06rad/s2=(60.572)(0.1862)=11.28N

Therefore, the tangential force (FA) which gear C exerts on gear A is 11.28N_.

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Chapter 16 Solutions

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

Ch. 16.1 - Prob. 16.4PCh. 16.1 - A uniform rod BC of mass 4 kg is connected to a...Ch. 16.1 - A 2000-kg truck is being used to lift a 400-kg...Ch. 16.1 - The support bracket shown is used to transport a...Ch. 16.1 - Prob. 16.8PCh. 16.1 - A 20-kg cabinet is mounted on casters that allow...Ch. 16.1 - Solve Prob. 16.9, assuming that the casters are...Ch. 16.1 - 16.11 A completely filled barrel and its contents...Ch. 16.1 - Prob. 16.12PCh. 16.1 - The retractable shelf shown is supported by two...Ch. 16.1 - Bars AB and BE, each with a mass of 4 kg, are...Ch. 16.1 - At the instant shown, the tensions in the vertical...Ch. 16.1 - Three bars, each of mass 3 kg, are welded together...Ch. 16.1 - Members ACE and DCB are each 600 mm long and are...Ch. 16.1 - 16.18 A prototype rotating bicycle rack is...Ch. 16.1 - Prob. 16.19PCh. 16.1 - The coefficients of friction between the 30-lb...Ch. 16.1 - Prob. 16.21PCh. 16.1 - Prob. 16.22PCh. 16.1 - For a rigid body in translation, show that the...Ch. 16.1 - 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Disk B is at rest when it is brought into contact...Ch. 16.1 - Prob. 16.45PCh. 16.1 - Prob. 16.46PCh. 16.1 - For a rigid body in plane motion, show that the...Ch. 16.1 - Prob. 16.48PCh. 16.1 - Prob. 16.49PCh. 16.1 - Prob. 16.50PCh. 16.1 - Prob. 16.51PCh. 16.1 - A 250-lb satellite has a radius of gyration of 24...Ch. 16.1 - A rectangular plate of mass 5 kg is suspended from...Ch. 16.1 - Prob. 16.54PCh. 16.1 - A drum with a 200-mm radius is attached to a disk...Ch. 16.1 - A drum with a 200-mm radius is attached to a disk...Ch. 16.1 - The 12-lb uniform disk shown has a radius of r =...Ch. 16.1 - Prob. 16.58PCh. 16.1 - Prob. 16.59PCh. 16.1 - Prob. 16.60PCh. 16.1 - Prob. 16.61PCh. 16.1 - Two uniform cylinders, each of weight W = 14 lb...Ch. 16.1 - Prob. 16.63PCh. 16.1 - Prob. 16.64PCh. 16.1 - A uniform slender bar AB with a mass m is...Ch. 16.1 - Prob. 16.66PCh. 16.1 - 16.66 through 16.68A thin plate of the shape...Ch. 16.1 - 16.66 through 16.68A thin plate of the shape...Ch. 16.1 - A sphere of radius r and mass m is projected along...Ch. 16.1 - Solve Prob. 16.69, assuming that the sphere is...Ch. 16.1 - A bowler projects an 8-in.-diameter ball weighing...Ch. 16.1 - Prob. 16.72PCh. 16.1 - A uniform sphere of radius r and mass m is placed...Ch. 16.1 - A sphere of radius r and mass m has a linear...Ch. 16.2 - A cord is attached to a spool when a force P is...Ch. 16.2 - Prob. 16.5CQCh. 16.2 - Prob. 16.6CQCh. 16.2 - Prob. 16.7CQCh. 16.2 - Prob. 16.5FBPCh. 16.2 - Two identical 4-lb slender rods AB and BC are...Ch. 16.2 - Prob. 16.7FBPCh. 16.2 - Prob. 16.8FBPCh. 16.2 - Show that the couple I of Fig. 16.15 can be...Ch. 16.2 - Prob. 16.76PCh. 16.2 - 16.77 In Prob. 16.76, determine (a) the distance r...Ch. 16.2 - A uniform slender rod of length L = 36 in. and...Ch. 16.2 - In Prob. 16.78, determine (a) the distance h for...Ch. 16.2 - An athlete performs a leg extension on a machine...Ch. 16.2 - Prob. 16.81PCh. 16.2 - Prob. 16.82PCh. 16.2 - Prob. 16.83PCh. 16.2 - A uniform rod of length L and mass m is supported...Ch. 16.2 - 16.84 and 16.85 A uniform rod of length L and mass...Ch. 16.2 - 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