introduction to basic mechanics, resolution of vectors chapter 1
TRANSCRIPT
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Introduction to Basic Mechanics, Resolution of Vectors
Introduction to Basic Mechanics, Resolution of Vectors
Chapter 1
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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.
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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.
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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.
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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.
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Base of Analytical Mechanics
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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.
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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.
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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)
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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)
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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.
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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?
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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.
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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.
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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.
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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.
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!!Assignments!!
From the Book:• No. 16 (Similar to the previous problem)• No. 22
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!!Assignments!!
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!!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.
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THANK YOU