978-0073380292 Chapter 2 Part 4

Statics 2e 93

Problem 2.46

A Caterpillar Ultra High Demolition machine is shown. The distances

between points

A

and

B

is

12:5

m, points

B

and

C

is

2:8

m,

C

and

D

is

7

m, and

D

and

E

is

2:5

m. Determine the position vectors

ErAB

,

ErBC

,

ErCD

, and

ErDE

, where

ErAB

is the position vector from point

A

to point

B, and so on. Add these vectors to determine the position vector ErAE .

Problem 2.47

The actuator at point

C

of the robotic arm is positioned by specifying angles

˛

and

ˇ

, where

˛

is measured positive counterclockwise from the positive

x

axis and

ˇ

is measured positive clockwise from the

AB

direction to the

BC

direction. Determine the position vector from point

O

to point

C

in terms of

angles ˛and ˇ.

Problem 2.48

Two ropes are used to lift a pipe in a congested region. Determine the ratio

F2=F1so that the resultant of E

F1and E

F2is vertical.

Problem 2.49

A welded steel tab is subjected to forces

F

and

P

. Determine the largest value

P

may have if

FD1000 lb

and the magnitude of the resultant force cannot

exceed 1500 lb.

Problem 2.50

The mast of a ship supports forces from three cables as shown. If

FD400 lb

,

determine the value of

˛

that minimizes the magnitude of the resultant of the

three forces. Also, determine the magnitude of that resultant.

Problem 2.51

The mast of a ship supports forces from three cables as shown. If

˛D0ı

,

determine the value of

F

that will make the magnitude of the resultant of the

three forces smallest.

Problem 2.52

A short cantilever beam is subjected to three forces. If

FD8kN

, determine

the value of

˛

that minimizes the magnitude of the resultant of the three forces.

Also, determine the magnitude of that resultant.

Problem 2.53

A short cantilever beam is subjected to three forces. If

˛D45ı

, determine the

value of

F

that will make the magnitude of the resultant of the three forces

smallest.

Problem 2.54

An eyebolt is loaded by forces

F1

and

F2

. If the eyebolt has a maximum

working load of 1200 lb, determine if the working load multipliers given in

Fig. 1 of Example 2.6 are met for the following loading scenarios:

(a) Only F1is applied.

(b) Only F2is applied.

(c) Both F1and F2are applied simultaneously.

Problem 2.55

An important and useful property of vectors is they may be easily transformed

from one Cartesian coordinate system to another. That is, if the

x

and

y

compo-

nents of a vector are known, the

t

and

n

components can be found (or vice versa)

by applying the formulas

EvDvxO{CvyO|DvtO

tCvnOn; (1)

where vtDvxcos Cvysin ; (2)

vnD vxsin Cvycos ; (3)

or vxDvtcos vnsin ; (4)

vyDvtsin Cvncos : (5)

In these equations,

O

t

and

On

are unit vectors in the

t

and

n

directions, respectively;



is measured positive counterclockwise from the positive

x

direction to the positive

t

direction; and the

y

and

n

directions must be oriented

90ı

counterclockwise

from the positive xand tdirections, respectively.

(a)

Derive the above transformation that gives

vt

and

vn

in terms of

vx

and

vy

.

Hint: First consider a vector

Evx

that acts in the

x

direction, and resolve this

into components in

t

and

n

directions. Then consider a vector

Evy

that acts

in the

y

direction, and resolve this into components in

t

and

n

directions.

Vectorially adding these results yields the transformation.

(b)

For the eyebolt and post of Example 2.7, the

x

and

y

components of the

resultant force are given by Eq. (4) of Example 2.6. Use these

x

and

y

components with the preceeding transformation equations to obtain the

t

and

n

components of the resultant force, and verify these are the same as

those in Eq. (4) of Example 2.7.

Problem 2.56

A box weighing

200

N rests on an inclined surface. A worker applies a hori-

zontal force

F

to help position the box. Determine the

x

and

y

components of

the resultant of forces

W

and

F

. Also determine the

t

and

n

components of

the resultant force vector. Comment on why the

t

and

n

components might be

useful to know.

Problem 2.57

A motor-driven gear is used to produce forces

P1

and

P2

in members

AB

and

AC of a machine. Member AC is parallel to the yaxis.

(a)

Determine the

x

and

y

components of the resultant force vector at

A

due

to forces P1and P2.

(b)

Determine the

t

and

n

components of the resultant force vector. Comment

on why the tand ncomponents might be useful to know.

Solution

Problem 2.58

Traction is a medical procedure used to treat muscle and skeletal disorders by

strategically applying one or more forces to a person’s body for a specific length

of time.

(a)

Determine the

x

and

y

components of the resultant force vector at

A

due

to the 40 N and 60 N forces.

(b) Determine the tand ncomponents of the resultant force vector.

Solution

Part (a)

Problem 2.59

A structure supports forces from a bar and cable as shown.

(a)

Determine the

x

and

y

components of the resultant force vector at

A

due

to the 6kip and 10 kip forces.

(b) Determine the tand ncomponents of the resultant force vector.

Solution

Problem 2.60

Two people apply forces P1and P2to the handle of a wrench as shown.

(a)

Determine the

x

and

y

components of the resultant force vector applied

to the handle of the wrench.

(b)

Determine the

t

and

n

components of the resultant force vector. Comment

on why the tand ncomponents might be useful to know.

Problem 2.61

Bar

AC

is straight and has 106 in. length,

B

is a pulley that supports forces

W

and

F,Wis vertical, and the tdirection is parallel to bar AC:

(a)

If

FD150 lb

and

˛D30ı

, determine the coordinates of point

C

so that the

t

component of the resultant of Fand Wis zero.

(b)

If

FD150 lb

and

C

is located at

.56; 90/ in.

, determine angle

˛

so that the

t

component of the resultant of Fand Wis zero.

(c)

If

C

is located at

.56; 90/ in.

and

˛D30ı

, determine

F

so that the

t

component

of the resultant of Fand Wis zero.

Hint: For each of these questions, first find the

x

and

y

components of the resultant

force and then use the transformation given in Prob. 2.55 to obtain the tcomponent.

Statics 2e 111

2

0

-2

-4

1◦2◦4◦

3◦

Rt(lb)

α

α≈2.584◦

Using vector transformation (Eq. (2) in Prob. 2.55 with

DD

58:11ı

), the component of the resultant force in the

t

direction, which

is zero according to the problem statement, is

RtD.150 lb/.cos ˛/.cos 58:11ı/

C.150 lb sin ˛100 lb/.sin 58:11ı/D0; (10)

Equation (10) has one unknown, namely

˛

, and the equation may

be solved using computer algebra software (e.g., Mathematica or

Maple), by trial and error, or by simply plotting Rtversus ˛, as shown at the right. In this plot, the location

where RtD0corresponds to

˛D2:58ı:(11)

Part (c)

With point

C

located at

.56; 90/ in.

, the orientation



of the bar is given by Eq. (6), namely

D58:11ı. The resultant of E

Fand E

Wis given by

E

RD.100 lb/O|CF .cos 30ıO{Csin 30ıO|/ (12)

DFcos 30ıO{C.F sin 30ı100 lb/O|(13)

DRxO{CRyO| : (14)

Using vector transformation (Eq. (2) in Prob. 2.55 with

DD58:11ı

), the component of the resultant

force in the tdirection, which is zero according to the problem statement, is

RtDF .cos 30ı/.cos 58:11ı/C.F sin 30ı100 lb/.sin 58:11ı/D0: (15)

Solving Eq. (15) for F, we find that

F.cos 30ı/.cos 58:11ı/C.sin 30ı/.sin 58:11ı/D.100 lb/.sin 58:11ı/)FD96:3 lb (16)

Problem 2.62

Screw

AC

is used to position point

D

of a machine. Points

A

and

C

have

coordinates

.185; 0/ mm

and

.125; 144/ mm

, respectively, and are fixed in space

by the bearings that support the screw. If point

B

is

52 mm

from point

A

,

determine the position vector ErAB and the coordinates of point B.

Solution