FORCE SYSTEMS

PROBLEMSon

1. The guy cables AB and AC are attached to the top of the transmission tower. The tension in cable AB is 8 kN. Determine the required tension T in cable AC such that

PROBLEMS

the net effect of the two cable tensions is a downward force at point A. Determine the magnitude R of this downward force. (2/13)

TAB =8 kN, TAC =? Net effect of two cable tensions is a downward force at point A. R =?

PROBLEMSR

ABT

ACTABT

ACT

A

R

kNjiT

ji

r

rnTT

mC

mBA

AB

AB

ABABABAB

25.65

5040

504088

)30,50(

)50,40()0,0(

03.64

22/

/

jiTTji

Tr

rTnTT ACACAC

AC

ACACACACAC

514.0857.0,3050

3050

31.58

22/

/

x

y

TAB =8 kN, TAC =? Net effect of two cable tensions is a downward force at point A. R =?

PROBLEMSR

ABT

ACTABT

ACT

A

R

kNRRTj

kNTTi

jiTjijR

jRTTR

jiTT

jiT

AC

ACAC

AC

ACAB

ACAC

AB

25.9,514.025.6

83.5,0857.050

514.0857.025.65

514.0857.0

25.65

2. In the design of the robot to insert the small cylindrical part into a close-fitting circular hole, the robot arm must exert a 90 N force P on the part parallel to the axis of the hole as shown. Determine the components of the force which the part exerts on the robot along axes (a) parallel and perpendicular to the arm AB, and (b) parallel and perpendicular to the arm BC. (2/28)

PROBLEMS

Determine the components of P along axes (a) parallel and perpendicular to the arm AB, and (b) parallel and perpendicular to the arm BC.

PROBLEMS

3. The unstretched

length of the spring is r.

When pin P is in an

arbitrary position q,

determine the x- and y-

components of the

force which the spring

exerts on the pin. Fspring=kd

Evaluate your answer for r = 400 mm, k = 1.4 kN/m and q = 40°.

Also when pin P is in the position q = 20°, determine the n- and

t-components of the force which the spring of modulus k = 1.4

kN/m exerts on the pin. Take r = 400 mm. (2/29)

PROBLEMS

l0=r. For q, determine x- and y-components of the force spring exerts on the pin. Find force for r = 400 mm, k = 1.4 kN/m and q = 40°. For q = 20°, determine the n- and t-components of the F.

PROBLEMS

sin

cos

springyspring

springxspring

FF

FF

springFyspringF

xspringF

Sine theorem

AP

r

APr

q

qsin2

sin

sinsin2

AP

r

AP

r

AP

r

qqqq

qq

sin2sin90

sin2sin90

90sin2

sinsinsin

11

11

q r

2r

A

P

O

springF

l0=r. For q, determine x- and y-components of the force spring exerts on the pin. Find force for r = 400 mm, k = 1.4 kN/m and q = 40°. For q = 20°, determine the n- and t-components of the F.

PROBLEMS

AP

rFF

AP

rFF

springspring

springspring

y

x

qq

qq

sin2sin90sin

sin2sin90cos

1

1

Cosine theorem

NF

NF

NllkkF

mAP

valuesgiventheFor

rAP

rrrrrrAP

y

x

spring

spring

spring

4.1945.62sin219

2.1015.62cos219

219)4.0557.0(1400)(

5.62557.0

40sin)4.0(2sin9040

557.040cos454.0

cos45

cos45cos)2)((2)2(

0

1

22222

d

q

qq

springFyspringF

xspringF

2r

q r

2r

A

P

O

springF

PROBLEMS

4. The circular cam has an offset e = 50 mm and a radius r = 100 mm. For the position where q = 30°, the smooth undersurface of the plunger exerts a downward force of 1600 N on the cam normal to the contacting surfaces. In designing the cam bearing it is necessary to calculate the rectangular component F' of this force along the line joining the contact point and the center of the shaft which turns the cam. Find F'.

PROBLEMS e = 50 mm, r = 100 mm. For q = 30°, smooth plunger exerts a downward force of 1600 N on the cam normal to the contacting surfaces. Calculate the rectangular component F' of this force along the line joining the contact point and the center of the shaft which turns the cam.

until it produces a tension Tin CD of 1.2 kN. Express as a vector in terms of its unit vectors , and . Determine the direction angles qx, qy and qz which makes with the positive x-, y-, and z-axes. Does it make any difference in the result which coordinate system is used? (2/107)

5. The rigid pole and cross arm assembly is supported by the three cables shown. A turnbuckle at D is tightened

i

T

j

k

T

PROBLEMS

PROBLEMS T =1.2 kN. = ? Determine the direction angles qx, qy and qz

which makes with the positive x-, y-, and z-axes. Does it make any difference in the result which coordinate system is used?

T

T

kNkjiT

kjiT

kjin

kji

r

rn

mDmC

nTT

CD

CD

CDCD

CD

962.0641.0321.0

802.0535.0267.02.1

802.0535.0267.0

5.435.1

5.435.1

)0,3,0()5.4,0,5.1(

612.5

222/

/

CDr /

CDn

ji

k

PROBLEMS T =1.2 kN. = ? Determine the direction angles qx, qy and qz

which makes with the positive x-, y-, and z-axes. Does it make any difference in the result which coordinate system is used?

T

T

32.143,802.0cos

66.57,535.0cos

51.74

267.02.1

321.0cos,321.0cos

coscoscos

zz

z

y

y

y

x

xxxx

n

zyxzyx

T

TT

T

T

TTT

kjiTkTjTiTT

CD

qq

qq

q

qq

qqq

It does not matter which coordinate system is used. x-y-z or x'-y'-z' systems can be used. Both systems have the same unit vectors.

xq yq

zq

kji

802.0535.0267.0

6. Three forces are acting on the eye bolt. If the

resultant force has a magnitude and direction as

shown,

determine the

magnitude and

direction angles

of .

RF

3F

PROBLEMS

PROBLEMS Given determine the magnitude and direction angles of .

RF

3F

PROBLEMSA

35°60°

25°40°

5 km

y

x

z

2 kmB

O

7. At a certain

instant, the locations

of a plane at A and a

train at B are

monitored by a radar

antenna at O.

Determine the

distance d between A

and B for this instant.

PROBLEMSA

35°60°

25°40°

5 km

y

x

z

2 kmB

O

Determine the distance d between A and B

kmrd

kjir

kjir

kmB

B

kmA

A

BA

BA

BA

72.618.582.222.3

18.582.222.3

)85.0(33.439.143.117.105.2

)85.0,39.1,17.1(

)25sin2,40cos25cos2,40sin25cos2(

)33.4,43.1,05.2(

)60sin5,35sin60cos5,35cos60cos5(

222

/

/

/

BAr /

8. Two cables exert

forces on the pipe.

Determine the

magnitude of the

projected

component of

along the line of

action of . Also

determine the angle

q between the two

cables.

1F

2F q

PROBLEMS

q

PROBLEMS Determine the magnitude of the projected component of along the line of action of andthe angle q between the two cables.

1F

2F

9. The access door

is held in the 30°

open position by

the chain AB. If

the tension in the

chain is 100 N,

determine the

projection of the

tension force onto

the diagonal axis

CD of the door.

(2/116)

PROBLEMS

PROBLEMS TAB =100 N, determine the projection of TAB onto the diagonal axis CD of the door

NkjiT

kjinTT

NT

kjikjinTT

kjin

kji

r

rn

NkjiT

kjikji

n

nnTT

mB

mA

A

CD

DCCDCD

CD

DCABCD

DC

CD

CDDC

AB

AB

ABABABAB

8.1392.238.36

3.052.08.046

46)3.0)(30()52.0)(52()8.0)(80(

3.052.08.0305280

3.052.08.0

50.1

45.078.02.1

305280

3.052.08.050.1

45.078.02.1

100

)9.0,0,0(

)45.0,78.0,2.1(

)30sin9.0,30cos9.0,2.1(

/

/

BAT /

CDr /

x y

z

10. An overhead crane is used to reposition the boxcar within a railroad car-repair shop. If the boxcar begins to move along the rails when the x-component of the cable tension reaches 3 kN, calculate the necessary tension T in the cable. Determine the angle qxy between the cable and the vertical x-y plane. (2/115)

PROBLEMS

PROBLEMSboxcar begins to move when the x-component of the cable tension reaches 3 kN, calculate tension T in the cable. Determine the angle qxy between the cable and the vertical x-y plane.

11. The spring of constant k = 2.6 kN/m is attached to the disk at point A and to the end fitting at point B as shown. The spring is unstretched when qA and qB are both zero. If the disk is rotated 15° clockwise and the end fitting is rotated 30°

counterclockwise, determine a vector expression for the force which the spring exerts at point A.

PROBLEMS

PROBLEMS k = 2.6 kN/m, spring is unstretched when qA and qB

are both zero. Determine a vector expression for the force the spring exerts at point A.