Introduction to Basic Mechanics, Resolution of Vectors

Introduction to Basic Mechanics, Resolution of Vectors

Chapter 1

Mechanics?

• Mechanics is a branch of the physical sciences that is concerned with the state of rest or motion of bodies that are subjected to the action of forces.

• It deals with the effect of forces upon material bodies.

Division of Mechanics

• Mechanics of fluids, a phase of is called hydraulics• Mechanics of materials, more often called strength

of materials, as subject which deals with the internal forces or stresses in bodies

• Analytic mechanics or mechanics of engineering, a study of external forces on bodies, ordinarily rigid bodied or bodies considered to be rigid, and of the effects of these forces on the motions of bodies.

Analytic mechanics

Analytic mechanics includes the study of: • Statics, deals with the equilibrium of bodies,

that is, those that are either at rest or move with a constant velocity

• Dynamicsis, which deals with the accelerated motion of bodies.

Our Concern: Statics

• We can consider statics as a special case of dynamics, in which the acceleration is zero.

• However, statics deserves separate treatment in engineering education since many objects are designed with the intention that they remain in equilibrium.

Base of Analytical Mechanics

Application of Newton’s Laws

• Law I define the condition of equilibrium and from it develop the first part of the work- Statics.

• Law III applies to both Statics and Dynamics. • The study of Dynamics is developed from Law

II.

Analytic Mechanics: Dealing with Forces

• In mechanics, a force arises out of the interaction of two bodies and causes or tends to cause the motion of the bodies. A body which is at rest or is moving with a constant velocity is said to be equilibrium.

• Force is a vector quantity. The characteristics of a force vector are that has (1) magnitude (2) sense or direction and (3) line of action.

Vector Addition & Subtraction

• Vector quantities, such as force, acceleration, velocity and momentum, cannot be added or subtracted as are scalar quantities, which posses magnitude only.

• Then How??? (See Page 3-5 of Analytic Mechanics 3rd Edition, Virgil Moring Faires)

Laws of cosine

•  R2 = F12 + F2

2 – 2F1F2 cos(180 - α)

• or, R2 = F12 + F2

2 + 2F1F2 cos α ……(1) [Since, cos(180 - α) = -cosα]

• Where, α is the angle between the vectors F1 and F2. Also from Fig. (a)

• tan θ = F2 sin α / (F1 + F2 cos α) …… (2)

Rectangular components

• For α = 90º, we get the special case of components which are perpendicular to each other. Since cos 90º = 0, we have from equation (1) and also from the right triangle AKB of Fig. (c)

• R2 = F12 + F2

2 or, R = (F12 + F2

2)1/2 ………….. (3)

• Components of a resultant that are at angles to each other are called rectangular components:

• Fx = F cosθ and Fy = F sinθ ..………………….(4)

• And, tanθ = Fy / Fx

• The process of finding components of a force is called resolution.

Simple Math Probs.

• Find the resultant of a horizontal force of Fx = 400 lb, acting toward the right, and a vertical force of Fy = - 300 lb, the negative sign indicating that the force acts in the negative direction, downward.

• A block with a 1 ft x 4 ft. rectangular section is 20 ft long and weighs W= 9000 lb. If a 2000 lb force acts 8 ft from the base, what is the magnitude of the resultant R of these two forces and where does the line of action of R intersect the base of the block?

• A force of 5000 lb. acts upward toward the right at an angle of θ = 30 º with the horizontal. What are its horizontal and vertical components?

Classification of Force System

• Based on the planes, Force System may be classified as:

• Coplanar force system: The force vectors are all in the same plane.

• Non-coplanar force system: The forces are not all in the same plane.

Classification of Force System

• Based on Line of Action, Force system may also be classified as:• Collinear force systems: All the forces act along the same line of

action. A collinear system is necessarily coplanar.• Concurrent force system: All lines of action intersect at one

point. A concurrent force system may be either coplanar or non-coplanar provided that there are more than two forces.

• Non-concurrent force system: The lines of action of the force vectors do not intersect at a point. A non-concurrent system may be either coplanar or non-coplanar.

• Parallel force system: The lines of action of all force vectors are parallel. A parallel force system may be either coplanar or non-coplanar.

Welcome to Basic Mechanics

• In Fig, let F = 3600 lb and θ = 45º. Assume both pulleys to have no friction so that the tension in the cable CD is 3600 lb. Solve the problem algebraically for the force on the shaft at B and A.

Closure• The composition of forces as presented in this chapter may be

carried out either by use of the Parallelogram Law or Triangle Law.

• Parallelogram Law: If two coplanar force vectors are laid out to scale from their point of intersection, both pointing away from the point of intersection, and if a parallelogram is completed with these force vectors as two sides, then the diagonal of the parallelogram that passes through the point of intersection represents the resultant in magnitude and direction.

• Triangle Law: If two coplanar force vectors are laid out to scale with the tail of one at the point of other, the third side of a triangle of which these two vectors are two sides represents the resultant in magnitude with a sense from the tail of the first vector to the point of the second vector.

!!Assignments!!

From the Book:• No. 16 (Similar to the previous problem)• No. 22

!!Assignments!!

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!!Assignments!!

Hints: You have to use Sine Law to solve the problems. Search Google If you don’t know the Sine Law.

THANK YOU