statics in mechanics

8/10/2019 statics in Mechanics

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Problem 3.41 A single force P acts at C in a directionperpendicular to the handle BC of the crank shown.Knowing that the moments about "O' are Mx= 20 Nm, My= -8.75 Nm and Mz= -30 Nm, determine the magnitude of Pand the values of and . Components of moment about"0"

mN20=Mx

mN8.75-=My

mN30-=Mz

Find ,,Pr

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Problem 3.43 The 23 inch vertical rod CD is welded to themidpoint C of the 50 inch rod AB. Determine themoment about AB as a result of the 235 lb force P.

If P = 235 lbs (also acts at G)

and AB = 50"

Find MAB:

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3.12 Moment of a Couple

If two forces F-andFrr

have the same magnitude, but of

opposite sense and act at different points, but have parallellines of action, are said to form a _________.

Although these forces sum up to zero and hence are in a

(static) translational equilibrium, the 0MCG r

thereby

____________.

Let points A & B be connected to the origin "0" by position

vectors r&r BArr

, respectively.

)Fx(-r+Fxr=M BA0r

r

r

rr

)Fxr(-Fxr=M BA0r

r

r

rr

LawtiveDistribu_Fx)r-r(=M BA0r

rrr

Let )Fxr(=Msoandr=r-r 0BAr

rrrrr

is the moment of the

couple, perpendicular to the plane containing the twoforces.

The magnitude of Fd=sinrF=MM 00 rr

where d is the

_____________ distance between the lines of action of

F&F-rr

If two couples exist, one consisting of F1-&F1rr

and the

second , they will have equalF-&F 22rr

Mr

if

and if the two couples have the same sense and lie in

parallel planes.

dF=dF 2211

3.14 Addition of Couples

M )F+F(xr=Rxr=r

r r rr

21

r

Fxr+Fxr=M 21rrrr

r

3.15 Couples May Be Represented By Vectors

where Mx represents couples in the yz plane

My represents couples in the xz plane

Mz represents couples in the xy plane

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3.17 Resolution of a Given Force Into a Force at "0" and a Couple

Consider Fr

acting on a rigid body at point A connected to point "0" by position vector rr

. If we would rather the force act at

point "0", one could attach twoadditional forces to point "0"; one

equal to and the otherr

, this

will not modify the action of

Fr

F

F

-r

onthe rigid body.

As a result, Fr

is now applied at "0",while the other two forces form a

couple of momentr

Fxr=M0rr

So any acting on a rigid body may be moved to an arbitrary point "0", provided that a couple is

added, whose moment is equal to the moment of F

Fr

r

about "0".

The couple represented by a couple vector M0r

is ___________the plane containing Fandrr

r

.

Since is a free vector it can be applied anywhere; for convenience, the couple vector is

usually attached at point "0" together with

M0r

Fr

. The resulting combination is referred to as a force-couple system.

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Problem 3.55 Given the drill exerts a 12 Nm couple,determine the magnitude of the resultant force applied tothe nails at "A" & "B"

Problem 3.57 Find the diameter (d) of the pegs, knowing that the resultant couple applied to theboard is 1132.5 in lb (ccw)

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Problem 3.59 Replace the two couples shown by a singleequivalent couple, specifying it's magnitude and thedirection of it's axis

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Problem 3.63 Replace the force by an equivalent force-couple system at "c".

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Problem 3.65 30==N;300= r

P . Replace force Pr

by

a) equivalent force - couple system at "B", and b) equivalentsystem formed by two parallel forces applied at B & C

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Problem 3.67 Replace the resulting force and couple by a single forcelocated at C and determine the distance x.

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Problem 3.71 TBC= 570 lb.; replace the force that the cable BCexerts at point "B" by an equivalent force-couple system atpoint "A".

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Problem 3.73 Find the equivalent force-couple system at A

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3.17 Reduction of a System of Forces to One Force and One Couple

Consider a system of forces shown. may be moved from AF1r

1to a given point "0" if a couple of

moment is added to point "0"Fxr=M 111r

rr

All M are c.c.w.

Because they are concurrent, a resultant force and moment can be determined.

While the corresponding moments (or couple vectors) are perpendicular to the correspondingforce, the resultant force and the _________________ will not in general beperpendicular to each other.

Once a system has been reduced to a resultant force and a couple about some point "0". Thetransfer to another point (0') can be easily determined.

The Rr

force (resultant) will remain unchanged, but a new moment must be determined.

where Sr

is the _________ between the points 0 and 0' and Rr

the resultant force vector.

=

3.18 Equivalent Systems of Force Two systems of forces are equivalent, if they may be reduced to the same force-couple

system at a given point

M=

MandF=F

00

rr

rr

Note with respect to pt "0"; but if truly equivalent it must hold for any point, therefore

F=FF=FF=F zzyyxx

M=MM=MM=M zzyyxx

Two systems of forces are equal if; i) they impart the same transitional motion in the x,y& z directions, and ii) the same rotation about the x,y and z axis.

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Problem 3.77 Find the resultant loading and replace with an equivalent force couple system atpoint "A".

a)

f)

g)

Problem 3.79 Determine the location of the resultant force and its distance from

point "A" to its line of action for Problem 3.77a only

N500-=R&.)W.(CNm1000-=MR

A

r

r

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Problem 3.85 Find the resultant of this system @ A&B for the threeforces shown and M = 80 in-lb

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Problem: Find the location of the resultant line of action for Ry& Rxalong the line segment ABand BC

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Problem 3.87 Find the resultant force acting on the truss and the

point of intersection of this force Rr

with the line AB.

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Problem 3.89 Determine the magnitude of the forces exerted at B and C knowing that these

forces are equivalent to the force couple system at A consisting of kR+jR+iN)(25-=R zyr

and

iNm13.5-=MR

A

r

. Also determine the force vector at C and the resultant force vector.

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Problem 3.115: A force Pr

acts on the frame shown at point E.

Knowing that the absolute value of the moment of Pr

about a linejoining points F and B is 300 ft-lb, determine the magnitude of the

force Pr

.

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Problem 3.123: Determine the resultant forces and location of the line ofaction for P = 160N only

statics in mechanics

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