Introduction to Statics

Lessandro Estelito O. GARCIANO

Associate Professor

Department of Civil Engineering

De La Salle University

What is Mechanics?

o Mechanics is the science which describes and predicts the conditions of rest or motion of bodies under the action of forces

o It is divided into three parts: mechanics of rigid bodies, mechanics of deformable bodies, and mechanics of fluids.

o Statics – the branch of mechanics that treat objects which are stationary (usually) or at constant velocity.

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What is Mechanics?

Engineering Mechanics

Mechanics of Fluids

Mechanics of Solids

Rigid Bodies

Deformable Bodies

Ideal fluids

viscous fluids

compressible fluids

Strength of materials

Theory of elasticity

Theory of plasticity

statics

dynamics

kinematics

kinetics

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What is Mechanics?

o Kinematics – the study of the relationship among displacement, velocity and acceleration without regard to the cause of the motion

o Kinetics – the branch of mechanics that examines the relationships between forces and acceleration

o Dynamics – combination of kinematics and kinetics

o The basic concepts used in mechanics are space, time, mass and force.

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Fundamental Concepts involved in Newton’s Laws

o Space- associated with the notion of the position of a point P given in terms of three coordinates measured from a reference point or origin.

o Time- definition of an event requires specification of the time and position at which it occurred.

o Mass- used to characterize and compare bodies, e.g., response to earth’s gravitational attraction and resistance to changes in translational motion. 5 /35

Fundamental Concepts involved in Newton’s Laws

o Force - represents the action of one body on another. A force is characterized by its point of application, magnitude, and direction, i.e., a force is a vector quantity

In Newtonian Mechanics, space, time, and mass are absolute concepts independent of each other. Force, however, is not independent of the other three. The force acting on a body is related to the mass of the body and the variation of its velocity with time.

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Newton’s Law of Motion

o First Law

If the resultant force acting on a particle is zero, the particle will remain at rest (if originally at rest) or will move with constant speed in a straight line (if originally in motion)

o Second Law

If the resultant force acting on a particle is not zero, the particle will have an acceleration proportional to the magnitude of the resultant and in the direction of the resultant force

o Third Law

The forces of action and reaction between bodies in contact have the same magnitude same line of action, and opposite sense.

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Units

o Kinetic Units: Length, time, mass and force.

o Three of the kinetic units, referred to as basic units, may be defined arbitrarily. The fourth unit, referred to as a derived unit, must have a definition compatible with Newton’s 2nd Law.

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Units

o International System of Units (SI): The basic units are length, time, and mass which are arbitrarily defined as the meter (m), second (s), and kilogram (kg). Force is the derived unit,

o U.S. Customary Units: The basic units are length, time, and force which are arbitrarily defined as the foot (ft), second (s), and pound (lb). Mass is the derived unit,

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Units

Units SI U.S.

Mass kilogram (kg) slug (-)*

Length meter (m) foot (ft)

Time second (s) second (sec)

Force newton (N)* pound (lb)

Note: The SI system is an absolute system, because a gravitational field is unnecessary for definition of the three basic units. On the other hand, the US customary system is a gravitational system.

*not base or fundamental units, but defined by Newton’s second law.

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Units

(SI)

F = ma

1 N = (1 kg) (1 m / s2)

1 N = 1 kg m / s2

So:

* 1 N is the force which will accelerate 1 kg at 1 m/s2

* 1 slug is the mass which a 1 –lb force will accelerate at at 1 ft/sec2

(US)

F = ma

1 lb = (1 slug) (1 ft / s2)

1 slug = 1 lb - s2 / ft

m F

a

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Law of Gravitation

The magnitude F of the mutual force of attraction is

where G is the universal gravitational constant.

m1, m2 = the masses of the two particles

r = the center-to-center distance between the particles

F F

r

m1 m2

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Law of Gravitation

Special case: mass m on surface of Earth

From , we have

or W = mg

where W is the magnitude of the force of gravitational attraction (the weight of m) and g = Gme/R2 is the acceleration of gravity.

me

m

R

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Law of Gravitation

There are absolute values for g relative to a nonrotating Earth. The common value for analyses relative to the rotating Earth are:

Note that W = mg may be obtained from Newton’s Second Law applied to a particle of mass m which is allowed to fall freely near the surface of the Earth.

m

a

F

m

W g

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Review of Vectors

o Vector: parameters possessing magnitude and direction which add according to the parallelogram law. Examples: displacements, velocities, accelerations.

o Vector classifications:

- Fixed or bound vectors have well defined points of application that cannot be changed without affecting an analysis.

- Free vectors may be freely moved in space without changing their effect on an analysis.

- Sliding vectors may be applied anywhere along their line of action without affecting an analysis. Force is a vector quantity.

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Vectors

o Scalar: parameters possessing magnitude but not direction. Examples: mass, volume, temperature

o Equal vectors have the same magnitude and direction.

o Negative vector of a given vector has the same magnitude and the opposite direction.

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Addition of Vectors

o Trapezoid rule for vector addition

o Triangle rule for vector addition

o Law of cosines

o Law of sines

QPR

BPQQPR cos2222

A

C

R

B

Q

A sinsinsin

B

B

C

C

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Addition of Vectors

o Vector addition is commutative

o Vector subtraction

PQQP

B

B

C

C

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Addition of Vectors

o Addition of three or more vectors through repeated application of the triangle rule

o The polygon rule for the addition of three or more vectors.

Vector addition is associative,

o Multiplication of a vector by a scalar

SQPSQPSQP

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Unit Vectors

V

kjiV

kjiV

kVjViV

kVjViV

VVVV

of cosinesdirection

1

coscoscos

coscoscos

222

l,m,n

nml

nml

zyx

zyx

zyx

zyx

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Rectangular Components

Fx and Fy are the x- and y- scalar components of F.

Note that scalar components include sign information.

sin

cos

FF

FF

y

x

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Dot or Scalar Product

Dot or Scalar Product

May be regarded as

Magnitude of P times component Qcos of Q in the direction of P.

Or

Magnitude of Q times component Pcos of P in the direction of Q.

cosPQQP

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Dot product definition

0jkkjkiijji

kkjjii

1

zzyyxx

zyxzyx

QPQPQP

QQQPPP

kjikjiQP

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Cross or Vector Product

Cross or Vector Product

Where en is a vector perpendicular to the plane formed by P and Q in the right-hand sense.

From the definition,

nePQ sinQP

0

kkjjii

jkiijkkij

jikikjkji

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Dot product definition

kji

kjikjiQP

xyyxzxxzyzzy

zyxzyx

QPQPQPQPQPQP

QQQPPP

This cross product may be evaluated from the determinant

zyx

zyx

QQQ

PPP

kji

QP

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Resultant of Two Forces

o force: action of one body on another; characterized by its point of application, magnitude, line of action, and sense.

o Experimental evidence shows that the combined effect of two forces may be represented by a single resultant force.

o The resultant is equivalent to the diagonal of a parallelogram which contains the two forces in adjacent legs.

o Force is a vector quantity.

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Resultant of several concurrent forces

o Concurrent forces: set of forces which all pass through the same point.

A set of concurrent forces applied to a particle may be replaced by a single resultant force which is the vector sum of the applied forces.

o Vector force components: two or more force vectors which, together, have the same effect as a single force vector. 27 /35

Method of Problem Solution

o Problem Statement: Includes given data, specification of what is to be determined, and a figure showing all quantities involved.

o Free-Body Diagrams:

Create separate diagrams for each of the bodies involved with a clear indication of all forces acting on each body.

o Fundamental Principles:

The six fundamental principles are applied to express the conditions of rest or motion of each body. The rules of algebra are applied to solve the equations for the unknown quantities.

o Solution Check:

Test for errors in reasoning by verifying that the units of the computed results are correct,

test for errors in computation by substituting given data and computed results into previously unused equations based on the six principles,

always apply experience and physical intuition to assess whether results seem “reasonable”

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Numerical Accuracy

o The accuracy of a solution depends on

accuracy of the given data, and

accuracy of the computations performed. The solution cannot be more accurate than the less accurate of these two.

o The use of hand calculators and computers generally makes the accuracy of the computations much greater than the accuracy of the data. Hence, the solution accuracy is usually limited by the data accuracy.

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Accuracy and Approximation

o Answers in statics will be displayed to three significant figures (four if the number begins with a one).

o Be sure to store intermediate results in the calculator storage registry so as to avoid roundoff errors.

Answer displayed on calculator Answer displayed on paper

4.13459 4.13

4.13594 4.14

1.33333… 1.333 or (4/3)

0.96592 0.966

934.25 934

279,340 279,000

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Accuracy and Approximation

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Sample Problem

The two forces P and Q act on a bolt A. Determine their resultant. R = 98 N, = 35o

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Sample Problem

A barge is pulled by two tugboats. If the resultant of the forces exerted by the tugboats is 5000 N directed along the axis of the barge, determine (a) the tension in each of the ropes for a = 45o, (b) the value of a for which the tension in rope 2 is a minimum. (a) T1 = 3700 N, T2 = 2600

N (b) T1 = 4330 N, T2 = 2500 N, a = 60o 33 /35

Sample Problem

Two forces are applied to an eye bolt fastened to a beam. Determine graphically the magnitude and direction of their resultant using (a) the parallelogram law, (b) the triangle rule. R = 8.40 kN, a = 19.0o

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Check your understanding

To steady a sign as it is being lowered, two cables are attached to the sign at A. Using trigonometry and knowing that a = 25o, determine (a) the required magnitude of the force P if the resultant R of the two forces applied at A is to be vertical, (b) the corresponding magnitude of R. (a) 489 N (b) 738 N

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