Statics of Rigid Bodies
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1
STATICS OF RIGID BODIES Review Notes
Lecturer: Maria Victoria B. Munar
NEWTON’S LAWS OF MOTION First Law
A particle originally at rest, or moving in a straight line with constant velocity, will remain in this state provided the particle is not subjected to an unbalanced force. F1 F
F Second Law
Resolution It is the process of replacing a force with its components. RECTANGULAR COMPONENTS OF A FORCE
Components that are perpendicular with one another. These T hese components can be determined d etermined from trigonometry.
v
A particle acted upon by an unbalanced force F experiences an acceleration a that has the same direction as the force and a magnitude that is directly proportional to the force. force. If F is applied to a particle of mass m, this law may be expressed mathematically as F = ma. a F Third Law
Second Sem, SY 2019-2020
F y
F
F x = F cos cos sin F y = F sin
F
x
F x
d
F y
F
x
F x
F y
y
x
d
F x
d x 2
F
y 2
F
F x = F cos cos x F y = F cos cos y z y x
F'
VECTOR ADDITION
2
Rectangular Components of a Force in Three Dimensions z
F
W = mg where: g is the acceleration due to gravity (equals 9.81 m/s2 at standard location)
2
F x F y
d
y
The mutual forces of action and reaction between two particles are equal, equal, opposite, and collinear.
Weight
F z = F cos cos z y
F
2
F x
F y
2
2
F z
x
Unit Vector u A
Cartesian Unit Vectors z
A
k
j
y
A
Two vectors are added according to the parallelogram law. It states that “ Two forces on a body can be replaced by
i
x
Cartesian vector representation a force called with the resultant by drawing .” ofsingle the parallelogram sides equivalent to thethe twodiagonal forces .”
F = F u ; where: u is a unit vector
If the two vectors A and B are collinear , the parallelogram law rreduces educes to an algebraic or scalar addition.
F = F x i + F y jj + F z k
Subtraction is a special case of addition, where the – B). So the resultant may be expressed as R' = A – B = A + ( – rules of vector addition also apply to vector subtraction. FORCE
A force is the action exerted by one body upon another. Characteristics of a Force Force is a vector quantity, therefore a force is completely described by its characteristics: 1. Magnitude 2. Direction
3. Point of Application Components
Two separate forces equal to a single force.
Force Vector Along a Line z
B( x B , y B , z B)
F
A( x A , y A , z A)
y x
F F
x B x A i y B y A j z B z A k x B x A y B y A z B z A 2
2
2
DOT PRODUCT OR SCALAR PRODUCT A A · B = AB cos
where: 0 180 B
2
4. A cable exerts a force F = = 580 N at ring A. a. Find the component of the force along the x axis. b. Determine the component of F parallel parallel to the x-y plane.
Dot product of two vectors A · B = A x B B x + A y B B y + A z B B z Rectangular component of one vector along any direction This component is equal to the dot product of the vector with a unit vector in the desired direction. COMPONENT METHOD OF ADDING FORCES. RESULTANT OF CONCURRENT FORCES
The scalar components R x, Ry, and R z of the resultant R of several forces acting on a particle are obtained by adding algebraically the corresponding scalar components of the given forces. R x = F x ;
R y = F y ;
R z = F z
The magnitude and direction angles of the resultant R can be determined from the relations R
2
R x
R y
x = Arc cos z = Arc cos
2
R x
R z
2
y = Arc cos
R
R z
R y R
R
PROBLEMS: 1. For the given force shown, determine a. the x and y components b. the x and y’ components c. the x’ and y’ components d. the x’and y components ´
5. For the force F = 56 N acting on the bent pipe shown, a. determine the magnitudes of the component acting along line OA. b. What is the magnitude of the component that is perpendicular to line line AO?
y
F = = 1500 N
6. In the system shown, a force F acts from B to D. Find the magnitude of F if its component along line AC is is equal to 1200 lb.
x´
30
25
2. The x-component of the force P is equal to 450 N. a. What is the magnitude of the force? b. Find the magnitude of the corresponding y-component.
3. In the figure shown, the cable AB prevents bar OA from rotating clockwise about the pivot O. Determine the n and t components of this force acting at point A of the bar if the cable tension is 1200 N. t n
7. Three forces act on the bracket as shown. a. Determine the magnitude of F3 so that the resultant force is directed along the positive x' axis axis and has a magnitude of 1000 N. b. What is its direction ?
y
y'
F 1 = 450 N
A 45
2.0 m
O
B 1.5 m
60
F 2 = 200 N x
30
x'
F 3
3
8. a. Determine the magnitude of the force F so that the resultant force R of the three forces is as small as possible. b. What is the minimum magnitude magnitude of R?
MOMENT OF A FORCE ABOUT A POINT Moment of of a force – it it is the tendency of a force to rotate the body on which it acts about about a given point or axis Moment arm or lever arm – it it is the perpendicular distance from the point or axis to the line of action of the force Moment center – it is the point where the body rotates or tends to rotate O
M
9. Three forces, F1 = 136 N, F2 = 250 N, and F3 = 325 N, are applied with cables to the anchor block b lock shown. a. Determine the magnitude of the resultant of the three forces. b. Determine the direction angles defining the line of action of the resultant force.
M = Fd
where: M = = moment of the force F = = magnitude of the force = lever arm d =
d F
O
Varignon’s Theorem or Principle of Moments Moments
The moment of a force about a point is equal to the sum of the moments of its components about the same point. Principle of Transmissibility
The conditions of equilibrium or motion of a rigid body will remain unchanged if a force acting at a given point of the rigid body is replaced by a force of the same magnitude and direction, but acting at a different point, provided that the two forces have the same line of action. CROSS OR VECTOR PRODUCT OF TWO VECTORS
10. The resultant of the three forces acting at A is 675 lb directed downward. Find the tension in AB, AC , and AD.
It is defined as the product of their magnitudes by the sine of their included angle. The result is a new vector acting perpendicular to the plane of the vectors in the direction of the right-hand rule. n ˆ
A
B = AB sin n
A
B=
ˆ
B
A
A x B x i
A y A z B y B z j k
Moment of a Force About a Point F
MO
r
O d
11. If each cable can withstand a maximum tension of 1000 N, determine the largest mass of the cylinder for equilibrium.
MO = r
F
where: r is a position vector drawn from O to any point lying on the line of action action of F Moment of a Force About a Line or Axis z
F M x
O
r
y
M O Oxx = (r
F)· nOx
M O Oyy = (r
F)· n Oy
M Oz = (r
F)·
ˆ
ˆ
ˆ n
Oz
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Scalar Triple Product
The triple scalar product involves the dot product of a vector and the cross product of two vectors. It is written as A · B C or B C · A A x A y A z B x B y B z C x C y C z
A · B C =
Vector Representation of Moment
M x
2
M y
2
2. Line Load The magnitude of the resultant force is equal to the area under the load diagram and the line of o f action of the resultant force passes through the centroid of the area under the load diagram. PROBLEMS
MO = M x i + M y jj + M z k ;
M O
surface and the line of action of the resultant force passes through the centroid of the volume bounded by the load area and the load surface.
1. A 90-N force is applied to the control rod AB as shown. Knowing that the length of the rod is 225 mm, determine the moment of the force about point B.
2 M z
Moment of a Couple – consists of tw Couple – two o forces that are equal in magni magnitude, tude, opposite in direction, and have parallel (noncollinear) lines of action. It produces a purely rotational effect, and the moment of a couple is the same about any point in the plane of the couple; i.e., it is independent of the moment center.
2. In order to raise the lamp post from the position shown, the force F on the cable must create a counterclockwise counterclockwise moment of 1500 lb-ft about point A. Determine the magnitude of F that must be applied to the cable.
M = Fd ;;
where: F = = magnitude of the force d = = perpendicular distance between the forces RESULTANTS OF FORCE SYSTEMS Coplanar Force Systems ĵ R = R R x îî + R y ĵ
where: R x = F x and
R y = F y
and the magnitude R and direction angle θ , measured from the x-component, are
R
2
R x R y
2
and = tan1
R y
R x The location of the line of action of the resultant with respect to an arbitrary reference point, say O, can be determined by applying the principle of moments. Hence, R
M O
3. The towline exerts a force P = 4 kN at the end of the 20-m long crane boom. If = 30°, a. determine the placement x of the hook at A so that this force creates a maximum moment about point O. b. What is this moment?
M O
Non-coplanar Force Systems R = R x îî + R y ĵ ĵ + R z k ˆ
where:
R x = F x , R y =F y , R z =F z 2
R cos θ
x
R x
R x
R y
2
cos θ
y
R
R z
R y
2
cos θ
z
R
R z R
M = M x îî + M y ĵ ĵ + M z k ˆ
where:
M x = M x , M y = M y ,
x
M M Distributed Normal Loads
2
2
M y
M z = M z 2
M z
1. Surface Load The magnitude of the resultant force is equal to the volume of the region between the load area and the load
4. Determine the moment of force F about about point O.
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5. Two parallel 75-N forces are applied to a lever as shown. Determine the moment of the couple formed by the two forces.
10. Three couples are applied to a bent bar as shown in the
6. Determine the magnitude of force F in cable AB in order to produce a moment of 500 lb-ft about the hinged axis CD, which is needed to hold the panel in the position shown.
7. A couple of magnitude M = = 540 mN-m and three forces shown are applied to an angle bracket. a. Find the resultant of this system of forces. b. Locate the point where the line of action of the resultant intersects the line AB. c. Locate the point where the line of action of the resultant intersects the line BC .
8. Two parallel forces of opposite sense F1 = (125i + 200 j + 250k) N and F2 = (125i 200i 250k) N act at points A and B of a body as shown in the figure. a. Determine the moment of the couple. b. Find the perpendicular distance between the two forces.
9. For the loading system shown, a. compute the magnitude of the resultant force. i ntersects b. Determi ne where the resultant’s line of action intersects the member measured from A.
figure. a. Determine the magnitude of the resultant couple b. Compute the direction angles associated with the unit vector used to describe the normal to the plane of the resultant couple.
11. Determine the magnitudes of F1 and F2 and the direction of F1 so that the loading creates a zero resultant force and couple on the wheel.
12. a. Replace the loading system system acting on the post by a single resultant force. b. Where is its point of application on the post measured measured from point O?
6
13. a. Determine Determine the magnitude of the resultant resultant of the two forces and one couple acting on the I-beam. b. Locate the line of action action of the resultant fforce orce with respect to the left end of the beam.
17. The turnbuckle is tightened until the tension in cable AB is 1.2 kN. Calculate the magnitude of the moment about point O of the force acting on point A.
14. a. Determine the resultant of the four forces and one couple acting on the plate shown. b. Locate the point where the resultant’s line of action intersect line AC measured measured from A. c. Find the point where resultant’s line of action intersect edge AB of the plate from A. EQUILIBRIUM OF PARTICLES Conditions for Equilibrium F x = 0,
F y = 0 ,
F z = 0
Free-body Diagram (FBD) – a drawing that shows the particle with all the forces, known and unknown, that act on it.
In drawing a free-body diagram of a body, certain assumptions are made regarding the nature of the forces (reactions) exerted by other bodies on the body of interest. Three common assumptions are the following: 1. Springs 15. Find the x and y coordinates of the point where the resultant of the three forces crosses the plate.
ℓ
ℓ o
F = ks where:
s
k = = stiffness or spring constant s = spring deflection s = l – l lo where l is the stretched length and lo is the original length
F 2. Cables and Pulleys
Cables (or the like) can support only a tension or pulling force.
16. Replace the two forces and single couple by an equivalent force-couple system at point A.
3. Smooth or Frictionless Surfaces The action (or reaction) of the body on the other is directed normal to the surface of 90
R
contact .
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EQUILIBRIUM OF RIGID BODIES IN TWO DIMENSIONS Equilibrium equations F x = 0;
F y = 0;
M = = 0
Idealization of Two-dimensional Supports and Connections Type of Connection
Reaction
Number of Unknowns
1. Cable
One unknown. The reaction is a tension force which acts away from
θ
θ
2. Weightless link θ
the member in the direction of the cable.
R
θ R
One unknown. The reaction is a force which acts along the axis of the link.
θ
or
R
One unknown. The reaction is a force which acts perpendicular to the surface at the point of contact.
3. Roller 90º
R
4. Roller or pin in confined smooth slot
One unknown. The reaction is a
or
90º R
90º
force which acts perpendicular to the slot. R
5. Rocker 90º
One unknown. The reaction is a force which acts perpendicular to the surface at the point of contact.
R
6. Smooth contacting surface
One unknown. The reaction is a force which acts perpendicular to the surface at the point of contact.
θ θ
R
7. Smooth pin or hinge
Two unknowns. The reactions are two components of force, or the magnitude and direction of the resultant force.
R H RV
8. Member pin connected to collar on smooth rod
R
R
or
θ
θ
One unknown. The reaction is a force which acts perpendicular to the rod.
θ
9. Member fixed connected to collar on smooth rod
Two unknowns. The reactions are the couple moment and the force which acts perpendicular to the rod.
10. Fixed support R H M RV
Three unknowns. The reactions are the couple moment and the two force components, or the couple moment and the magnitude and direction of the resultant force.
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Equilibrium of a two-force body
Equilibrium of a three-force body
A rigid body subjected to two forces acting at only two points is commonly called a two-force body. If a two-force body is in equilibrium, the two forces must have the same magnitude, the same line of action , and opposite sense. F 1
If a rigid body in equilibrium is subjected to forces acting at only three points, the lines of action of the three forces must be either concurrent or parallel. F 1
C A
O
F 3
A B B
F 2
F 2
EQUILIBRIUM OF RIGID BODIES IN THREE DIMENSIONS F x = 0
F y = 0
F z = 0
M x = 0
M y = 0
M z = 0
Supports and Connections Type of Connection
1. Cable
Reaction
Number of Unknowns
One unknown. The reaction is a force which acts away from the member in the known direction of the cable.
2. Smooth surface support One unknown. The reaction is a force which acts perpendicular to the surface at the point of contact. 3. Roller
One unknown. The reaction is a force which acts perpendicular to the surface at the point of contact.
4. Ball and Socket Three unknowns. The reactions are three rectangular force components.
5. Single journal bearing
6. Single journal bearing with square shaft
7. Single thrust bearing
Four unknowns. The reactions are two force and two couple-moment components which act perpendicular to the shaft. Note: The couple moments are generally not applied if the body is supported elsewhere. Five unknowns. The reactions are two force and three couple-moment components. Note: The couple moments are generally not applied if the body is supported elsewhere. Five unknowns. The reactions are three force and two couple-moment components. Note: The couple moments are generally not applied if the body is supported elsewhere.
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Type of Connection
Reaction
8. Single smooth pin
9. Single hinge
Number of Unknowns
Five unknowns. The reactions are three force and two couple-moment components. Note: The couple moments are generally not applied if the body is supported elsewhere. Five unknowns. The reactions are three force and two couple-moment components. Note: The couple moments are generally not applied if the body is supported elsewhere.
10. `Fixed support Six unknowns. The reactions are three force and three couple-moment components.
PROBLEMS
1. Four forces act on the particle shown. Determine the magnitude and direction angle of force F4 for equilibrium of the particle.
2. Find the smallest value of P for which the crate shown will be in equilibrium in the the position shown.
3. Determine the stretch in each spring for equilibrium of the 2kg block. The springs are shown in their equilibrium position.
4. Cables AB, BC , and CD support the 10-kg and 15-kg traffic lights at B and C , respectively. a. Determine the tension in cable AB. b. Determine the tension in cable CD. c. Determine the value of θ .
5. Two 10-in diameter pipes and a 6-in diameter pipe are supported in a pipe rack as shown in the figure. The 10-in diameter pipes each weigh 300 lb and the 6-in diameter pipe weighs 175 lb. Assume all surfaces to be smooth. a. Determine the force exerted by the support on the pipe at contact surface B. b. What is the reaction at C ? c. What is the reaction at A?
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9. Determine the force P required to pull the 250-lb 25 0-lb roller over the step shown in the figure.
6. The smooth disks D and E have have a weight of 200 lb and 100 lb, respectively. Determine the largest horizontal force P that can be applied to the center of disk E without without causing the disk D to move up the incline.
7. A 3000-lb cylinder is supported by a system of cables as shown in the figure. Determine the tensions in cables A, B, and C .
8. A container of weight W is suspended from ring A to which cables AC and AE are attached. A force P is applied to the end F of of a third cable which passes over a pulley at B and through ring A and which is attached to a support at D. Knowing that W = = 1000 N, determine the magni magnitude tude of P.
F
10. A cylinder is supported by a bar and a cable as shown in the figure. The weight of the cylinder is 150 lb and the weight of the bar is 20 lb. If all surfaces are smooth, a. determine the reaction at support C of of the bar b. determine the tension in the cable.
11. The framework is supported by the member AB which rests on the smooth floor. When loaded, the pressure distribution on AB is linear as shown. Determine: a. the length d of of member AB b. the intensity w for this case.
12. Spring CD remains in the horizontal position at all times due to the roller at D. The spring is unstretched when = 0 and the stiffness is = 1. 1.5 5 kN kN/m /m. a. Determine the smallest angle θ for for equilibrium. b. Determine the horizontal and vertical components of reaction at pin A.
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13. The floor crane and the driver have a total weight of 2500 lb with a center of gravity at G. Determine the largest weight of the drum that can be lifted without causing the crane to overturn when its boom is in the position shown.
17. The bent rod of negligible mass is supported by a ball-andsocket joint at A and a slider bearing at B; the rod leans against a smooth surface at D. Find all forces acting on the rod when the force P = 960 N is applied.
14. Determine the force P needed to support the 20-kg mass using the Spanish Burton rig.
PLANE TRUSSES Zero-force Members Zero-force members in a truss usually arise in one of two general ways:
15. A loading car is at rest on a track forming an angle of 25° with the vertical. The gross weight of the car and its load is 5500 lb, and it is applied at a point 30 in. from the track, halfway between the two axles. The car is held by a cable attached 24 in. from the track. a. Determine the tension in the cable b. Find the reaction at each pair of wheels.
1. When only two members form a non-collinear truss joint and no external load or support reaction is applied to the joint, then the members must be zero-force members. 2. When three members form a truss joint for which two of the members are collinear and the third forms an angle with the first two, then the non-collinear member is a zero-force member provided no external force or support reactions applied to that joint. The two collinear members carry equal loads.
Exercises 1.
B
A
2.
C B A
16. Determine the force developed in cords BD, CE , and CF and the reactions of the ball-and-socket joint A on the block.
3.
C
D
D
E
G
F
H
I
J
J K
E
K
I
L
O N
H F
G
L
M
M
N
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Method of Joints
The method of joints is based on the fact that if the entire truss is in equilibrium, then each of its joints is also in equilibrium. The free-body diagram of each joint is used to obtain the member forces acting at the joint. Since the members of a plane truss are straight two-force members lying in a single plane, each joint is subjected to a force system that is coplanar and concurrent; hence, only F x = 0 and F y = 0 need to be satisfied for equilibrium equilibrium.. Method of Sections
Analyzing the free-body diagram of a part of a truss that contains two or more joints is called the method of sections. It is used when the force of only a few members of a truss are to be found.
4. The diagonal members in the center panels of o f the truss shown are very slender and can act onlycounters in tension (counters). Determin Determine e the forces in the that are acting under the given loading.
PROBLEMS
Determine the force in member BG and member CF of of the truss loaded as shown. 1. Determine the force in member BG and member CF of of the truss loaded as shown.
2. Each member of the truss is a uniform 20-ft bar weighing 400 lb. a. Find the reaction at the roller support.
b. .. c. Calculate Determinethe theaverage averageforce forceininmember member AE ED
3. For the truss loaded as shown, a. determine the force in member BC . b. determine the force in BI . c. find the force in member HI .
5. A truss is subjected to two point loads at A as shown. a. Find the reaction at H . b. Find the force acting in member EF . c. Determine the force in member EH .
6. For the truss loaded as shown, a. determine the force in member DE . b. Find the force in member FI . c. Determine the force in member EI .
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FRAMES AND MACHINES Frame – – aa structure that always contains at least one member acted on by forces at three or more points. Frames are structures which are designed to support applied loads and are usually fixed in s position. Machines or mechanism – frame-like frame-like structures that are not fully constrained. They are structures which contain moving parts and are designed to transmit forces or couples from input values to output values.
The forces acting on each member of a connected system are found by isolating the member with a free-body diagram and applying the established equations of equilibrium. The principle of action and reaction must be carefully observed when we represent the forces of interaction on the separate free-body diagrams. PROBLEMS
1. Determine the reaction at the roller F for for the frame loaded as shown.
2. The aircraft landing gear consists of a spring and a nd hydraulically-loaded piston and cylinder D and the two hydraulically-loaded pivoted links OB and CB. If the gear is moving along the runway at a constant speed with the wheel supporting a stabilized constant load of 24 kN, calculate the total force which the pin at A supports.
4. The two-member frame shown carries a distributed load and a concentrated load F = 500 N. a. Determine the horizontal and vertical components of reaction at pin A. b. Determine the horizontal and vertical components of reaction at pin B.
5. Determine the reactions at the supports of the compound beam loaded as shown. shown.
6. The compound beam is pin-supported at C and and supported by rollers at A and B. There is a hinge (pin) at D. Neglect the thickness of the beam. a. Determine the reaction at A. b. Determine the reaction at B. c. Determine the components of reaction at the support C .
3. The shipboard crane is supporting a load of 4 tons in the position shown where = 30°. The hoisting drum B is operated by a high-torque electric motor. a. Calculate the added compression P in the hydraulic cylinder due to the effect of the 4-ton load. b. Determine the magnitude R of the additional force supported by the pin at O.
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7. Determine the horizontal and vertical components of reaction at pin C .
2. Horizontal distance between joints are equal
Horizontal component of reactions:
Solve for R A using ∑ M B = 0
Solve for R B from ∑F v = 0
The tensions in the cables are and
Hd = = M 2
T A
T B
2
R A H R B
2
H
2
PROBLEMS
8. The axis of the three-hinge arch ABC is a parabola with
1. Cable ABCD supports the 10-kg lamp E and and the 15-kg lamp F . Determine the maximum tension in the cable and the sag of point B.
vertex at B. Knowing that P = 112 kN and Q = 140 kN, determine a. the components of the reaction at A. b. the components of the force exerted at B on segment AB.
2. Two loads are suspended as shown from the cable ABCD. Knowing that h B = a. hC 1.8 , m, determine the distance b. the components of the reaction at D, c. the maximum tension in the cable.
CABLES Flexible Cables with Concentrated Loads
When a cable of negligible weight supports several concentrated loads, the cable takes the form of several straight-line segments. Any portion of cable between successive loads can therefore be considered as a two-force member, and the internal forces at any point in the cable reduce to a force of tension directed along the cable. 1. Slopes or angles of inclination are given
Apply the equations of equilibrium at the joints
6 kN
10 kN
3. If each cable segment can support a maximum tension of 75 lb, determine the largest load P that can be applied.
4. If d C = 8 ft, determine C = a. the reaction at A, b. the reaction at E .
2
15
300 lb 300 lb
200 lb
5. A string supported at A and B, at the same level over a span of 30 m is loaded as shown in the figure given below. If the depth of the point D is 8 m from the supports, find a. the tensions in CD, b. the tension in DE , c. the horizontal thrusts in the strings at A and B.
When the supports A and B of the cable have the same elevations, the distance L between the supports is called the span of the cable and the vertical distance h from the supports to the lowest point is called the sag of the cable. 1. Supports are of the same level 2
Horizontal component of reactions:
H
wL
8d 2
wL H 2 2
The tension in the cable is
Approximate length of the cable:
T A
2
S L
4
8d
32d
3 L
2
5 L
6. The cable supports the three loads shown. Determine the sags y B and y D and of points B and D. Take P1 = 400 lb, P2 = 250 lb.
2. Supports are of different levels
Horizontal component of reactions: Use ∑ M A = 0] Hd 1 wx1
x or H 2 1
x 2 or H 2
∑ M B = 0] Hd 2 wx2 Cables Subjected to a Distributed Load
Parabolic Cable
If a cable carries a load that is uniformly distributed along the horizontal, the curve formed by the cable is a parabola. The shape of the curve formed by the cable is defined by the wx 2 . From the figure shown, equation y 2T
The tensions are
T A
T A
2
wx H wx H
2
2
2d 1
1
2
wx1
wx2
2
2d 2
and
2
2
T
T o
2
2
w x
2
and
tan
wx
; where T and θ are
T o
respectively the tension and slope at any point in the cable.
3. When the distance from the lowest point of the cable to the chord joining the supports is known
Horizontal component of reactions: H
Solve for R B using ∑ M A = 0
wL
2
8d
16
Solve for R B from ∑F v = 0
The tensions in the cables are
2
2
2
2
T A
R A H
T B
R B H
and
Catenary Cable
If a cable carries a load that is uniformly distributed along the cable itself, the shape of the cable is a catenary curve. Cables hanging under their own weight are loaded in this way. The tension at any point in the cable can be determined from the equation If
c
T o
T
T o
2
2
w s
2
a. Determine the maximum tension in the cable. b. Determine the minimum tension in the cable.
3. A cable is suspended and loaded as shown in the figure. a. Compute the length of the cable. b. Compute the horizontal component of the tension the cable. c. Determine the magnitude and position of the maximum tension occurring in the cable.
, the leng length th of tthe he cable, s, under consideration
w
is equal to
s
c sinh
x
c
and the equation of the curve is defined by the relation x y c cosh
c
4. The power transmission cable has a weight per unit length of 15 lb/ft. If the the lowest point of the cable cable must be at least 90 ft above the ground, a. determine the maximum tension developed in the cable. b. What is the total length of the cable from A and B?
– s = c2. The relation of y and s is y – 2
2
From the above equations, T = = wy. When the supports at A and B of the cable have the same elevation, the distance L between the supports is called the span of the cable and the vertical distance h from the supports to the lowest point C is is called the sag of the cable which is equal to h = y A – – c.
5. Cable AB supports a load uniformly distributed along the horizontal as shown. Knowing that at B the cable forms an angle θ B = 35° with the horizontal, determine a. the maximum tension in the cable, b. the vertical distance a from A to the lowest point of the cable.
PROBLEMS
1. Determine the maximum uniform distributed loading wo N/m that the cable can support if it is capable of sustaining a maximum tension of 60 kN.
FRICTION Coulomb’s Theory of Dry Friction Friction
The maximum value of static friction (when motion is impending) is proportional to the normal force; i.e.,
2. A cable supports a load of 50 kg/m uniformly distributed with respect to the horizontal and is suspended from the two t wo fixed points located as shown.
N ; F max = s N where: s = coefficient of static friction
Once the block starts to slip relative to the surface, the friction force will decrease to
17
F m N ; where kk = = coefficient of kinetic friction max ax = kk N
The total force R exerted by the supporting surface on the block bl ock is the resultant of F and and N , therefore, 2
R N
2
F
and
tan
F
N
At the point of impending motion, R
2
N
F max
2
and F max
Since F m N , max ax = s N
s
tan
F max
N
and tan = = s, where = = angle
N
of static friction.
2. The 200-lb crate is being moved by a rope that passes over a smooth pulley. The coefficient of friction between the crate and the floor is 0.30. Assume that h = 4 ft and determine the force P necessary to produce impending motion.
When a block rests on an inclined surface and is acted on only by gravity, the resultant of the normal and friction forces must be collinear. The angle between the resultant and the normal force can never be greater than the angle of static friction; therefore, the steepest inclination for for which the block will be in equilibrium is equal to the angle of static friction. This angle is called the angle of repose. Wedges – it it is a block that has two flat faces that make a small angle with each other which are often used in pairs to raise heavy loads.
3. A lightweight rope is wrapped around a drum as shown in the figure. The coefficient of friction between the drum and the ground is 0.30.
Flexible Belts The relationship between the tensions on the ropes for problems involving a flat belt passing over a fixed cylinder can be determined from the formula
a. Determine the maximum angle such that the drum does not slip. b. Determine the tension in the cable for this angle if the drum weights 100 N. N.
T 2
T 1e
where: T 2 > T 1 μ = coefficient of friction β = = angle of contact in radians Direction for impending motion (or motion) of belt relative to the drum
Theequation: following points should be kept in mind when using the above 1. T 2 is the belt tension that is directed opposite the belt friction. Thus, T 2 must always refer to the larger of the two tensions. 2. For impending motion, use = = s. If there is relative motion between the belt and the the cylinder, use = = kk .. .. 3. The angle of contact must be expressed in radians. 4. Since the equation is independent i ndependent of r , its use is not restricted to circular contact surfaces; it may also be used for a surface of arbitrary shape. PROBLEMS 1. The block in figure weighs 500 lb and the coefficient of friction between the block and the floor is 0.2. a. Determine if the system would be in equilibrium for P = 400 lb. b. Calculate the minimum P to prevent motion. c. Determine the maximum P for which the system is in equilibrium.
4. Two blocks A and B, each having a mass of 6 kg, are connected by the linkage shown. If the coefficient of static friction at the contacting surfaces is B = 0.8 and A = 0.2, determine the largest vertical force P that may be applied to pin C without without causing the blocks to slip. Neglect the weight of the links.
5. The 200-N board is placed across the channel and a 400-N boy attempts to walk across. If the coefficient of static friction at A and B is s = 0.4, determine if he can make the crossing; and if not, how far will he get from A before the board slips?
18
6. How many turns of rope around the capstan are needed for the 300-N force to resist the 120-kN pull of a docked ship? The coefficient of static friction between the capstan and the rope is 0.20.
8. A wedge is being forced under an 80-kg drum as shown in the figure. The coefficient of friction between the wedge and the drum is 0.10 while the coefficient of friction is 0.30 at all other surfaces. Assuming a wedge angle θ of of 25º and that the weight of the wedge may be neglected, determine the minimum force P necessary to insert the wedge.
7. A pair of wedges is used to move a crate of weight W = = 2400 N. The coefficien coefficientt of friction is the same at all surfaces and the weight of the wedges are negligible. If the coefficient of static friction is 0.30 and the wedge angle is is 20, determine the force P necessary to insert the wedge. 9. The force P applied to the brake handle enables the band brake to reduce the angular speed of a rotating drum. If the tensile strength of the band is 17 kN, a. find the maximum safe value of P b. find the corresponding braking torque acting on the drum. Assume that the drum is rotating clockwise.
10. If a force of P = 200 N is applied to the handle of the bell crank, determine the maximum torque M that can be resisted so that the flywheel is not on o n the verge of rotating clockwise. The coefficient of static friction between the brake band and the rim of the the wheel is = 0.3.
CENTROIDS AND CENTERS OF GRAVITY
it is used to denote the point in a system of particles or physical body where the mass can be conceived to be Center of mass – it concentrated – the Center of gravity – the point in the body through which the weight acts Centroid of Areas and Lines
x A ∑ x ∆ A or
x L
∑ x ∆ L
x A
x dA or x L x dL
y A = ∑ y ∆ A or y A y dA
y L = ∑ y ∆ L or y L
y dL
19
Centroids of the Common Geometric Shapes Shape
Triangular area
x
y
h
h
y
3 b
b
bh
2
2
2
Semi-circularr Area Semi-circula
4 r
0
r
r
Quarter-circular Area
Area
3π
4 r
2
πr
4 r
2
πr
3π
3π
2
4
a y = kx2
h
Parabolic spandrel
3a
3h
10
4
O
ah
3
r
Circular Sector
2 r sin
O
3
Shape
x
Semicircularr arc Semicircula O
r
O
π
π
πr
2 r π
2
Length
2 r
2 r
Quarter-circular Quarter-cir cular arc
r
y
0
r
0
r
2
r
Arc of a circle O
r sin
0
2 r
x
Shape
x
y
Volume
r
Hemisphere
3a 8
2 3
3
r
20
Shape
x
y
Volume
h a
Semiellipsoid of Semiellipsoid revolution
3h
2
a
2
h
a
2
h
r
2
h
3
8
h a
Paraboloid of revolution
h
3
1
2
h h
r
Cone
4
1
3
h
a
Pyramid
h
1
4
3
abh
b
PROBLEMS
1. Locate the centroid of the wire bent in the shape shown.
3. Locate the centroid of the wire shown.
2. Locate the centroid of the given cross-sectional area. 4. Locate the centroid of the section shown.
200 mm
1.in
6.in
40 mm
1.in
100
6.in 20 mm
C •
1.in
d
x
21
5. Locate the centroid of the plane area shown.
MOMENT OF INERTIA
y
6 in
Moment of Inertia of an Area
8 in
In the application of mechanics, if a load is distributed continuously over an area on which they act, the computation of the loading distribution about an axis perpendicular to the area will involve a quantity called the moment of inertia or the second moment of the area .
8 in 4 in 12 in
2
2
I xx y dA and I yy x dA
x
6. Where is the centroid of the shaded area shown?
The moment of inertia about the pole O or about the z axis is
y
I O r 2 dA
60 mm 30 mm
These moments of inertia about the x and y axes are called the rectangular moment of inertia while the moment of inertia about the origin or about the z axis is called the
30 mm O
2
45 mm
7. Determine the distance h to which a hole must be bored into the cylinder so that the center of mass of the assembly is located at x = 64 mm. The material has a density of 8 Mg/m3.
2
2
, the moment of inertia of an area is always positive and is expressed in units of length to the fourth power. I O I x I y
polar moment of inertia. Since r
x
x
y
The Parallel Axis Theorem for an Area
The moment of inertia of an area about any axis that is parallel to the centroidal axis can be determined by the parallel-axis theorem theorem. I x I x A d
2
; where d is the distance between the axes
Radius of Gyration I x
2
k x A or
k x
I x A
The distance k x is referred to as the radius of gyration of the area with respect to the x axis.
Moment of inertia of the common geometric shapes
y
3
h
I yO
yO
3
bh
Rectangle
I x O
I y
I x
O
xO
x b
3
3
b h
3
3
bh
12
b h
12
I O
22
h
Triangle
bh
xO
I yO
I x O
I y
I x
3
bh
12
I O
3
36
x b y
r
4
r
4
Circle O
4
or
xO
d
r
4
r
2
or 4
64
or
d
4
4
d
4
32
64
y or yO
Semicircle
r
O
xO x 4r 3
4
r
8
r
4
8
0.11 r
4
r
4
8
y yO
r
Quartercircle
r
O
xO 4r
4
16
r
16
4
4
r 0.055
0.055 r
4
x
3
PROBLEMS
1. Determine the polar moment of inertia and the polar radius of gyration of the shaded area shown with respect to point P. y
a a
P a
a
x a
a
2. Determine the polar moment of inertia of the area shown with respect to a) point O; b) the centroid of the area.
23
3. Determine the moment of inertia of the composite area about
4. Determine the moments of inertia of the area shown about the centroidal x and y axes.
the x axis.
BOARD PROBLEMS 1. The magnitude of a force is 80 kN. The coordinates of its tail is (0 m, 4 m, 3 m) and the coordinates of its tip are (4.5 m, 0 m, 3 m,). What is the moment of this force about point O with coordinates (0 m, 0 m, 0 m)? A. 308.6 kN B. 338.8 kN-m C. 425.9 kN-m D. 278.4 kN-m-m
c)
Which of the following gives the y’ -component -component with respect to x’ and and y’ axis. axis.
4. For the machine part shown in the figure, the load 150 kN is acting at A.
2. A force of 60kN is acting horizontally, then another force of 50kN is acting upward to the right. If the resultant of these two forces is 95.4 kN. a) At what angle does the resultant make with the horizontal? A. 27º B. 32º C. 24º D. 30º b)
At what angle does the the 50-kN force makes with the horizontal? B. 72º A. 60º C. 54º D. 48º
c)
Compute the horizontal component of the resultant? A. 85 kN B. 78 kN C. 88 kN D. 65 kN
3. From the given figure shown, a force P = 100 kN is acting at 30° from the horizontal axis.
a)
When force at A is transferred to B, what is the equivalent couple in kN.m b) What is the force at C . c) What is the force at B. 5. Find the resultant of four parallel forces acting on a horizontal bar AB with length of 11 m; 400 kN download at the left most end, 800 kN downward at 9 m from the left end and 300 kN upward at 11 m from the left end. A. 1200 kN force force acting upward B. 1200 kN-m couple acting clockwise C. 1200 kN force force acting downward D. 1200 kN-m couple couple acting counterclockwise counterclockwise
6. For the forces in the figure shown
68.3 N
a) b)
Which of the following gives the y-component of P with respect to x and y axis. Which of the following following gives the y-component of P with respect to x’ and and y axis.
a) Which of the following most nearly gives the magnitude of the horizontal component of the resultant of the force system? A. 25 kN B. 50 kN C. – 5 50 0 kN D. – 25 25 kN b) Which of the following most nearly gives the magnitude of the resultant force?
24
c)
A. 25 kN B. 50 kN C. 35 kN D. 42 kN Which of the following gives the ang angle le that the equilibrant makes with the horizontal axis. A. – 135º 135º B. – 4 45º 5º C. 45º D. 135º
7. A concurrent force system in space is composed of 3 forces described as follows: P1 has a magnitude of 100 kN and acts through the origin and points x = 3, y = 4, z = 2. P2 has a magnitude of 60 kN acts through the origin and points x = 4, y = 1, z = – 2. 2. and P3 has a magnitude of 80 kN and acts through the origin and points x = 2, y = – 3, 3, z = 3.
1) Which of the following gives the x-component of the resultant force? A. 371.06 kN B. 392.40 kN 2)
C. 410.32 kN D. 401.30 kN Which of the following gives the y-component of the resultant force? A. 590.67 kN B. 620.23 kN C. 583.20 kN D. 556.59 kN
3) Which of the following gives the z-component of the resultant force? A. 620.23 kN C. 583.20 kN
1) Which of the following most nearly gives the xcomponent of the resultant force. B. 136 kN A. 142 kN C. 128 kN D. 159 kN 2) Which of the following most nearly gives the ycomponent of the resultant force. B. 50 kN A. 36 kN C. 42 kN D. 28 kN 3) Which of th thee following following most nearly gives the zcomponent of the resultant force. A. 25 kN B. 36 kN C. 48 kN D. 62 kN
B. 742.12 kN D. 401.30 kN
10. A concurrent force system in space is composed of 3 forces a nd acts described as follows. P1 has a magnitude of 93 kN and through the origin and point x = 5, y = – 4 and z = – 6 6.. P2 has a magnitude of 126 kN and acts through the origin and point x = – 1 1,, y = 5 and z = 3. P3 has a magnitude of 38 kN and acts through the origin and point x = 4, y =1 and z = 3. 1) Which of the following gives the x-component of the resultant of the sets of forces? 2) Which of the following gives the y-component of the resultant of the sets of forces? 3) Which of the following gives the the resultant resultant of the sets sets of forces.? 11. The coplanar force system shown consists of two forces and a couple. All coordinates are in meters.
8. From the given sets of parallel forces shown,
a) Which of the following gives the resultant force? A. 100 kN B. 90 kN C. 110 kN D. 135 kN b) Which of the following gives the location of the resultant force from the left support? A. 5 m B. 6 m C. 7 m D. 8 m c) Which of the foll following owing gives gives the reaction at the left support? A. 55.1 kN B. 57.4 kN C. 53.4 kN D. 52.5 kN 9. The resultant of the concurrent forces has a magnitude of 1000 kN and acts through the origin and points x = 2, y =3 and z = 4.
a)
b)
c)
Which of the following most nearly nearly giv gives es the resultant of the force system in Newtons. A. 16.8 B. 14.3 C. 12.2 D. 13.1 Which of the following following most nearly gives the angle angle that the resultant forces make with the horizontal axis in degrees (positive counterclockwise)? counterclockwise)? A. 72.4 B. 55.2 C. 68.9 D. 65.2 Which of the following following most nearly gives the yintercept of the resultant force in meters. A. 3.14 B. 6.38 C. 2.46
D. 5.34
25
12. A force system in space is shown below.
a)
b)
c)
Which of the following most nearly gives gives th thee mome moment nt of the forces about the x-axis, in Newton-Meter? A. 720 B. 2500 C. 2865 D. 1200 Which of the following following most nearly gives the magnitude of the resultant moment in Newton-Meter? A. 1200 B. 720 C. 2500 D. 2865 Which of th thee following following most nearly gives the direction cosines of the resultant moment? 0.419; cos θ z = 0.873 A. cos θ x = 0.251; cos θ y = – 0.419; θ θ – θ x = 0.251; cos y = 0.214; cos z = 0.771 B. cos C. cos θ x = 0.324; cos θ y = – 0.419; 0.419; cos θ z = 0.771 0.214; cos θ z = 0.873 D. cos θ x = 0.324; cos θ y = – 0.214;
13. A simply supported beam with a span of 6.0 m carries a vertical load that increases uniformly from zero at the left end to a maximum value of 9 kN/m at the right end. The larger reaction occurs at the right and has a value in kN of A. 27 B. 4.5 C. 18 D. 9
14. A concrete block is supported by two guy wires attached to an anchor ring as shown. sho wn. Determine the following: a) Resultant force on the anchor ring. b) Angle of resultant force with respect to the horizontal. c) Weight W of of the concrete block with a factor of safety of 1.25 to prevent uplift.
b) If α = 60°, what is the value of force C such such that the resultant of forces A, B, and C acts acts along the x-axis? c) For the forces A, B, and C to to be in equilibrium, what is the magnitude of the resulting force C ?
16. The force R = 600 kN is the resultant of the forces P, Q, and 50 kN. a) Determine the value of P. b) Determine the value of Q. c) Determine the distance x that specifies the line of action of Q.
4m
17. Three identical cylinders are stacked within a rigid bin as shown in the figure weight of each cylinder c ylinder is 500 N. There is no friction at any contact surface.
a) A Which of the following most nearly gives the force at , in Newtons? A. 425 B. 754 D. 289 C. 567 b) Which of the following most nearly gives the force at C , in Newtons? A. 425 B. 354 C. 144 D. 236 c) Which of the following most nearly nearly giv gives es the force at at B, in Newtons? A. 865 B. 650 C. 536 D. 750 18. From the figure shown, the spring is subjected to an initial tension of 200 N and has a spring constant of 10.2 kN/m. A force P is applied at B and D as shown.
15. The hook is subjected to three forces A, B, and C as as shown. A = 35 kN, B = 45 kN. a) If the resultant of the forces is 80 kN and is acting along the positive x-axis, find the angle α.
26
a) Which of the following is the force acting on the spring? A. 1.567 kN B. 1.223 kN C. 1.687 kN D. 1.454 kN b) Which of the following following gives the force acting on AB? A. 0.8953 kN B. 0.9047 kN C. 0.9578 kN D. 0.7893 kN c) Which of th thee following following gives the value value of P? B. 1.3283 kN A. 0.9047 kN C. 0.9578 kN D. 0.7893 kN 19. From the given frame shown compute the following: a) Which of the following gives the force F if the reaction at B is 100kN? B. 140 kN A. 120 kN C. 100 kN D. 110 kN b) Which of the following gives the reaction at A? A. 156 kN B. 136 kN C. 148 kN D. 162 kN c) Which of the follow following ing gives the angle which th thee resultant reaction at A makes with the horizontal axis measured counterclockwi counterclockwise? se? A. 230º B. 50º C. 160º D. 130º
21. From the given figure shown. a) Which of following gives the reaction reaction at C b) Which of following following gives the shear at a distance of 7 m. from A? c) Which of fol following lowing gives gives the the moment at a distance of 3 m. from A?
22. The beam shown weighs 30N/m.
a) b) c)
Which of the following gives the tension force? Which of the following gives the reaction at the pinned support? Which of the following following gives the angle that the reaction makes with the horizontal measured in counterclockwisee direction? counterclockwis
23. The force system shown consists of the couple C and four forces. The resultant of this system is a 500 kN-m counterclockwisee couple. counterclockwis
20. A rod is connected to a pin at A and a chord at B as shown. It holds a cylindrical drum which weighs 176 N. The drum has a diameter of 2 m. a) Which of the following gives the force between the drum and the rod? b) Which of the following following gives the force in the chor chord d BC ? c) Which of the following gives the reaction at the pin at A? a)
b)
c)
Which of the following most nearly nearly giv gives es the value of P in kN? A. 230 B. 150 D. 200 C. 250 Which of the following following most nearly gives the value value of Q in kN? A. 240 B. 260 C. 300 D. 200 Which of the following most nearly nearly giv gives es the value of C in in kN-m? A. 1070 B. 1140 C. 1440 D. 1210
27
24. A load W is is to be lifted using the crane which is hinged at B as shown in the figure. The value of x1 is 10 m, x2 is 8 m, and h is 18 m. Neglecting the weight of the crane, a) determine the force in cable AC . b) determine the resultant reaction at B. c) determine the largest load that can be lifted if the maximum force of cable AC is is 50 kN.
25. A load of 6 kN is supported as shown. The pulley weighs 2 kN.
a)
b)
c)
Which of the following most nearly nearly g gives ives the reaction at C in in kN? A. 8 B. 10 C. 9 D. 11 Which of the following following most nearly gives the reac reaction tion at B in kN? A. 30 B. 32 C. 41 D. 36 Which of th thee following following most nearly gives the angle angle that the reaction at B make with the horizontal? A. 21.5° B. 29.6° C. 24.8° D. 22.4°
27. The total load W is is to be lifted using the mast hinged at B. The mast is of uniform cross section and weighs 8 kN. a) What is the tensile force in the cable if W = = 36 kN? b) What is the vertical reaction at B if W = = 36 kN? c) If the allowable tensile force in the cable c able AC is is 45 kN, what is the maximum load W that that can be lifted?
28. The figure shows a circular steel plate p late supported on 3 posts. A, B, and C are are equally spaced along its circumference. A load W = = 1350 N is at a distance x = 0.5 m from the post at A along the x axis. Diameter of steel plate is 1.8 m. a) Find the reaction at post A. Neglect weight of the steel plate. b) Find the reaction at post B. Neglect weight of the steel plate. c) Compute the reaction at C considering considering the weight of the plate if it has a thickness of 16 mm and has a unit weight of 77 kN/m3.
29. In the given figure, a = 1 m, P1 = 1.8 kN, P2 = 0.9 kN, = 30°, β = = 45°. P3 = 0.45 kN, θ = a) Determine the resultant of the three forces. b) Determine the vertical reaction at B. c) Determine the horizontal reaction at B.
26. A load W = = 30 kN is lifted by a boom BCD making an angle α = 60° from the vertical axis. Neglect the weight of the boom. a) Determine the angle β between between the cables AC and and AD. b) Determine the horizontal reaction at B. c) Determine the tension in the cable AC .
30. A force P acting at an angle α = 45° from the x axis along the xy plane prevents the pole weighing 375 N from falling. The pole leans against a frictionless wall at A. Given x = 3.15 m, z = 3.15 m, y = 4 m. a. What is the force P in Newtons? b. Determine the reaction at the wall at A in Newtons. c. Calculate the vertical reaction at B in Newtons.
28
34. A cantilever truss is pin-connected at joint D and is supported by a roller at G. Spacing of trusses trusses is 3 m. If the the wind load is 1.44 kPa, a. Determine the horizontal reaction at the hinged support. b. Determine the stress in member AB. c. Determine the stress in member BE .
31. A tripod supports the load W as as shown in the figure. a. Determine the maximum load W that that can be supported by the tripod if the capacity of each leg is limited to 10 kN. b. If the load W = = 50 kN, calculate the force in the leg AD. c. If the load W = = 50 kN, calculate the force in the leg AB.
32. Pole AB is 12 m long and its weight W = = 35 kN. It is being lifted using cables BC and and BD. When the pole is tilted at an angle of 60° from the x axis, the resultant force acts at point A. a. Find the tensile force in cable BC . b. Find the tensile force in cable BD. c. What is the value of the resultant acting at point A?
33. The magnitude of the force acting in member BC of the truss shown is B. 3 A. 2 C. 4 D. 5
35. A plane truss as shown in the figure is acted upon by 480 N downward load at joints C and E and and a 1200 N load at J directed as shown. a) Find the reaction at G. b) Find the force acting on member member AH . c) Find the force acting on member JD.
36. The transmission tower is subjected to lateral forces as shown. Given a = 2.55 m, b = 1.8 m, c = 1.8 m, d = = 1.5 m, F 1 = 3 kN, F 2 = 5 kN, F 3 = 7 kN. a. Find the resultant reaction at support A. b. What is the resultant reaction at support B? c. What is the force in member FJ ?
29
37. A three-hinged arch is shown in the figure.
a)
b)
c)
Which of the following most nearly gives gives th thee verti vertical cal reaction at A in KiloNewtons: A. 31 B. 36 C. 28 D. 23 Which of the following following most nearly gives the tot total al reaction at the hinge in KiloNewtons: A. 45.2 B. 85.6 C. 54.1 D. 32.3 Which of the following most nearly gives gives th thee total reaction at B in KiloNewtons: A. 32.3 B. 45.2 D. 54.1 C. 85.6
40. A suspension cable is supported at A and B 120 m horizontally apart with B higher than A by 48m. Concentrated loads of 100 kN, 200 kN and 100 kN were applied at a distance of 30 m, 60 m and 90 m respectively from A. The cable sags a distance of 30 m measured from the chord AB at the point where the 200 kN is applied. Compute the horizontal reaction at the supports. A. 300 kN B. 200 kN C. 400 kN D. 500 kN
38. The figure shows a portable seat braced by a cable FG. The permissible tension in the cable is 1800 N. Surface C , A, and E are are frictionless. frictionless. a. What load W can can the seat safely carry? b. If W = = 1500 N, what is the reaction at C ? c. If W = = 1500 N, what is the reaction at A? 41. For the cable loaded as shown in the figure.
a)
Which of the following gives the the value of the stress of BC ? b) Which of the following following gives the value of β 1? c) Which of the following gives the the total length of the cable? 39. A suspension cable is supported at A and B 120 m horizontally apart with B higher than A by 15 m. The cable sags a distance of 10 m. from the chord joining A and B at the midspan. Compute the horizontal reaction at the supports. A. 1800 kN B. 1400 kN C. 1200 kN D. 1000 kN
42. For the cable shown
30
a)
b)
c)
Which of the following most nearly nearly g gives ives the tension in segment CD, in kilonewton: A. 7.3 B. 6.8 C. 5.2 D. 6.9 Which of the following following most nearly gives the tens tension ion in segment AB, in kilonewton: A. 6.8 B. 5.2 D. 7.3 C. 6.9 Which of the following most nearly gives gives th thee value of h in meters: A. 2.74 C. 2.32
B. 2.57 D. 3.12
43. The horizontal distance from A at one end of the river to frame C at at the other end is 20 m. The cable carries a load W = = 50 kN. a. At what distance from A is the load W such that the tension in segment AD of the cable is equal to that in segment CD? b. When the load W is is at distance x1 = 5 m from A, the sag in the cable is 1 m. Calculate the tension in segment DC of the cable. c. If the sag in the cable is 1 m at a distance x1 = 5 m, what is the total length of the cable?
46. The coefficient of friction between the 60 kN block and the plane shown is 0.30. If the block is to remain in equilibrium, what is the maximum allowable magnitude for the force P? A. 12 B. 18 D. 21 C. 15
47. A 200 kg. crate impends to side down do wn a ramp inclined at an angle of 19.29° with the horizontal. What is the frictional resistance? Use g = 9.81 m/s 2. A. 618.15 N B. 638.15 N C. 648.15 N D. 628.15 N
44. The suspended girder shown is supported by a series of hangers, uniformly spaced along a parabolic cable. a. What is the tension in the cable at midspan? b. What is the vertical reaction at support A? c. What is the resulting sag if the maximum tension in the cable is 300 kN?
45. The idealized model for a suspension bridge is shown. The trusses are pin connected at D, on hinged support at C , and on roller support at E . The parabolic cable is supported on towers at AC and and at BE . a. Determine the tension in the cable at midlength. b. Determine the vertical reaction at the pin at D. c. Determine the total vertical force at the tower at AC . d. Determine the maximum force in the cable.
48. A 40-kg block is resting on an inclined plane making an angle of 20° from the horizontal. If the coefficient of friction is 0.60, determine the force parallel to the incline that must be applied to cause impending motion motion down the plane. Use g = 9.81. B. 82 A. 87 C. 72 D. 77
31
49. A 40 kg block is resting on an inclined plane making an angle of 20° from the horizontal. If the coefficient of friction is 0.60, determine the force parallel to the inclined plane that must be applied to cause impending motion up the t he plane. A. 36.23 kg B. 28.42 kg C. 19.62 kg D. 42.46 kg
50. The uniform 50-kg plank is resting on rough surfaces at A and B. The coefficients of static friction are shown in the figure. A 100-kg woman starts walking from A toward B. It is required to determine the distance x when the plank starts to slide.
52. The weight of the cylindrical tank is negligible in comparison to the weight of water it contains. The coefficient of static friction between the tank and the horizontal surface μs. a. Assuming a full tank, find the smallest force P required to tip the tank. b. Find the smallest coefficient of static friction that would allow tipping to take place. c. If the force P = 6.5 kN initiates tipping, determine the depth of water in the tank.
53. A 5 cm. × 5 cm. square is cut from a corner of a 20 cm × 30 cm cardboard. Find the centroid from the longest side. A. 10.33 B. 12.06 C. 11.32
a) Which of the follow following ing most nearly gives th thee total reaction at B in Newtons. A. 680 B. 690 C. 700 D. 670 b) Which of the following following most nearly give givess the total reaction at A in Newtons. A. 758 B. 742 C. 817 D. 863 c) Which of the follow following ing most nearly gives th thee value of x in meters. A. 1.73 B. 1.85 C. 1.62 D. 1.54 51. To prevent the ladder weighing 600 N from sliding down, the man exerts a horizontal force at C . The coefficient of friction at A = 0.2 while surface B is frictionless. a. Find the vertical reaction at A. b. Find the horizontal reaction at A. c. Find the horizontal force exerted by the man at C .
D. 13.02
54. For the shaded area a given in the figure.
a) Determine the area, in square meters. A. 15.5 B. 39.4 C. 26.4 D. 44.5 b) Determine the the x-coordinate of the centroid, in meters. A. – 0.18 0.18 B. – 0.16 0.16 C. +0.18 D. +0.16 c) Determine the y-coordinate of the centroid, in meters. A. 1.62 B. 1.45 C. 2.01 D. 1.75 55. The moment of inertia of an isosceles trapezoid with base b = 600 mm and top a = 400 mm and depth d = = 900 mm is nearest to: A. 50 × 109 mm4 B. 40 × 109 m4 9 4 D. 30 × 109 mm4 C. 60 × 10 m
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