Mechanical Engineering Chapter 5 Homework Ay and NB can be determined by writing the moment

397

5–10.

Determine the components of the support reactions at the

fixed support Aon the cantilevered beam.

1.5 m

30

4 kN

6 kN

A

SOLUTION

Equations of Equilibrium:From the free-body diagram of the cantilever beam, Fig. a,

Ans:

398

Ans:

SOLUTION

Equations of Equilibrium. NA and By can be determined directly by writing the

moment equations of equilibrium about points B and A, respectively, by referring to

the beam’s FBD shown in Fig. a.

Using the result of NA to write the force equation of equilibrium along the x axis,

5–11.

Determine the reactions at the supports.

400 N/m

3 m

3

4

5

3 m

A

B

399

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*5–12.

6 m

AB

4 kN

2 m

30

Determine the horizontal and vertical components of

reaction at the pin Aand the reaction of the rocker Bon

the beam.

SOLUTION

Equations of Equilibrium:From the free-body diagram of the beam, Fig.a,NBcan

be obtained by writing the moment equation of equilibrium about point A.

Using this result and writing the force equations of equilibrium along the xand

Ans:

400

Ans:

SOLUTION

Equations of Equilibrium. NA and By can be determined directly by writing the

moment equations of equilibrium about points B and A, respectively, by referring to

the FBD of the beam shown in Fig. a.

a

+ΣMB=0;

600(6)(3) +

+ΣMA=0;

1

(300)(3)(5) –NA(6) = 0

Also, Bx can be determined directly by writing the force equation of equilibrium

along the x axis.

5–13.

Determine the reactions at the supports.

3

m3

m

AB

900 N/m

600 N/m

401

Ans:

SOLUTION

Equations of Equilibrium. NA can be determined directly by writing the moment

equation of equilibrium about point B by referring to the FBD of the beam shown

in Fig. a.

Using this result to write the force equations of equilibrium along the x and y axes,

5–14.

Determine the reactions at the supports.

B

A

3 m

800 N/m

3 m

1 m

402

Ans:

SOLUTION

Equations of Equilibrium. Ay and NB can be determined by writing the moment

equations of equilibrium about points B and A, respectively, by referring to the FBD

of the truss shown in Fig. a.

5–15.

Determine the reactions at the supports.

AB

2 m2 m2 m

2 m

6 kN

5 kN

8 kN

403

*5–16.

Determine the tension in the cable and the horizontal and

vertical components of reaction of the pin A.The pulley at

Dis frictionless and the cylinder weighs 80 lb.

SOLUTION

Equations of Equilibrium: The tension force developed in the cable is the same

throughout the whole cable.The force in the cable can be obtained directly by

summing moments about point A.

BA

D

C

5ft5ft

2

1

3ft

Ans:

404

5–17.

The man attempts to support the load of boards having a

weight Wand a center of gravity at G. If he is standing on a

smooth floor,determine the smallest angle at which he can

hold them up in the position shown. Neglect his weight.

u

SOLUTION

G

4ft

u

Ans:

405

5–18.

Determine the components of reaction at the supports Aand

Bon the rod.

AB

P

L

––

2

L

––

2

SOLUTION

Equations of Equilibrium:Since the roller at offers no resistance to vertical

movement, the vertical component of reaction at support is equal to zero.From

the free-body diagram, ,,and can be obtained by writing the force

equations of equilibrium along the and axes and the moment equation of

equilibrium about point ,respectively.

B

yx

MA

By

Ax

A

Ans:

406

5–19.

The man has a weight Wand stands at the center of the

plank. If the planes at Aand Bare smooth, determine the

tension in the cord in terms of Wand u.

SOLUTION

Solving Eqs. (1) and (2) yields:

B

Ans:

407

*5–20.

Auniform glass rod having a length Lis placed in the smooth

hemispherical

bowl having a radius r.Determine the angle of

inclination

for equilibrium.u

SOLUTION

By observation

.

Equilibrium:

T

ake the positive root

f=u

B

r

A

u

408

SOLUTION

5–21.

The uniform rod AB has a mass of 40 kg. Determine the

force in the cable when the rod is in the position shown.

There is a smooth collar at A.A

60

3 m

C

B

409

Ans:

SOLUTION

Equations of Equilibrium. NA can be determined directly by writing the moment

equation of equilibrium about point B by referring to the FBD of the beam shown

5–22.

If the intensity of the distributed load acting on the beam

is w = 3 kN

>

m, determine the reactions at the roller A and

pin B.

A

B

w

3 m

30

4 m

Ans:

SOLUTION

Equations of Equilibrium. NA can be determined directly by writing the moment

equation of equilibrium about point B by referring to the FBD of the beam shown

in Fig. a.

Using this result to write the force equation of equilibrium along x and y axes,

Thus,

5–23.

If the roller at A and the pin at B can support a load up

to 4 kN and 8 kN, respectively, determine the maximum

intensity of the distributed load w, measured in kN

>

m, so

that failure of the supports does not occur.

A

B

w

3 m

30

4 m

411

*5–24.

The relay regulates voltage and current. Determine the force

in the spring CD,which has a stiffness of k120 Nm, so

that it will allow the armature to make contact at Ain figure

(a) with a vertical force of 0.4 N. Also,determine the force

in the spring when the coil is energized and attracts the

armature to E,figure (b), thereby breaking contact at A.

50 mm 50 mm 30 mm

10°

D

k

C

CB

BEA

A

SOLUTION

From Fig. (a):

From Fig (b), energizing the coil requires the spring to be stretched an additional

Ans:

412

5–25.

Determine the reactions on the bent rod which is supported

byasmooth surface at Band by a collar at A, which is fixed

to the rod and is free to slide over the fixed inclined rod.

3ft3ft

3

4

5

100 lb

200lbft

2ft

B

12

5

13

A

SOLUTION

Ans:

413

5–26.

The mobile crane is symmetrically supported by two

outriggers at Aand two at Bin order to relieve the

suspension of the truck upon which it rests and to provide

greater stability. If the crane and truck have a mass of

18 Mg and center of mass at ,and the boom has a mass

of 1.8 Mg and a center of mass at ,determine the vertical

reactions at each of the four outriggers as a function of the

boom angle when the boom is supporting a load having a

mass of 1.2 Mg.Plot the results measured from to

the critical angle where tipping starts to occur.

u=0°

u

G2

G1

G

2

6.25 m

6m

SOLUTION

Since there are two outriggers on each side of the crane,

Ans:

414

Ans:

SOLUTION

5–27.

Determine the reactions acting on the smooth uniform bar,

which has a mass of 20 kg.

4 m

30ºA

B

60º

*5–28.

A linear torsional spring deforms such that an applied couple

moment M is related to the spring’s rotation

u

in radians by

the equation M = (20

u

)

N#m

. If such a spring is attached to

the end of a pin-connected uniform 10-kg rod, determine

the angle

u

for equilibrium. The spring is undeformed

when

u

= 0°.

SOLUTION

A

0.5 m

u

M  (20 u) N  m

Ans:

416

Ans:

SOLUTION

5–29.

Determine the force P needed to pull the 50-kg roller over

the smooth step. Take

u

= 30°.

A

B

P

300 mm

50 mm

u