AM+DG I.
Mechanics, a physical science, describes and predicts the conditions of rest or motion of bodies under the action of forces. It is divided into three parts: mechanics of rigid bodies, mechanics of deformable bodies and mechanics of fluids.
II.
Mechanics does not have the empiricism found in some engineering sciences; it does not solely rely on experience and observation. It greatly resembles mathematics in the sense that it puts emphasis on deductive reasoning. However, it is not as abstract; mechanics is an applied science.
III.
Mechanics of fluids is further subdivided into the study of incompressible and compressible fluids. Hydraulics, which deals with problems involving water, is also covered by the former.
IV.
Mechanics of rigid bodies is further subdivided into statics, which focuses on objects at rest, and dynamics, which deals with objects in motion.
V.
In this study, bodies are assumed to be perfectly rigid. It should be noted, however, that this is not the actual case. Machines and structures undergo deformations due to the forces they are subjected to. Fortunately, these deformations are usually minute that they barely affect the conditions for motion or equilibrium of the body in consideration.
VI.
The study of Newtonian mechanics dates back to the time of Aristotle and Archimedes. However, the formulation of its fundamental principles was only given several centuries later, by Newton. The validity of these principles remained unchallenged until Einstein formulated his theory of relativity, which built the foundations for relativistic mechanics.
VII.
Though impossible to truly define, the basic concepts in mechanics are space, time, mass, and force. The first three are absolute concepts, as they are independent of each other (in Newtonian mechanics only). Force, on the other hand, relies, on the mass of the body it is acting upon and the rate at which its velocity changes with time.
VIII.
Space is associated with the position of a point P. This position can be defined by using three lengths measured from a reference point, the origin, in three given directions. These lengths are referred to as the coordinates of P.
IX.
To define an event, space is not sufficient. Time must be indicated as well.
X.
Mass is defined as the amount of matter in an object. It is used to compare bodies on the basis of certain fundamental mechanical experiments, especially those involving gravitation and motion (inertia).
XI.
A force represents the action of one body on another. It can be exerted by actual contact or at a distance (as with gravitation and magnetism). A force is characterized by its point of application, magnitude, and direction. As a result, a force may be represented as a vector. Its direction is characterized by its line of action and the sense of the force.
XII.
A particle is a very small amount of matter which may be assumed to occupy a single point in space. A rigid body is a collection of a large number of particles that have a fixed position with respect to each other. Fernando, Don Lawrence Carl A. 2010-24598
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XIII.
The Parallelogram Law for the Addition of Forces states that two forces acting on a single particle may be replaced by a single resultant force which can be obtained by drawing the diagonal of the parallelogram which has sides equal to the two original forces. B A
A+ B
XIV.
The Principle of Transmissibility states that the conditions for motion or equilibrium of a body will not change if the point of application of a force, acting on that body, is moved along its line of action, as long as the force maintains the same magnitude and direction.
XV.
Newton’s Three Fundamental Laws:
XVI.
First Law: If the resultant force acting on a body is zero, then it will remain at rest, if originally at rest, or will remain in motion along a straight line at a constant speed, if originally in motion. That is:
XVII.
Second Law: If the resultant force acting on a body is not zero, then it will have an acceleration that is proportional to the magnitude of the resultant and in the direction of the said force. That is:
XVIII.
Third Law: The forces of action and reaction between bodies in contact have the same magnitude, same line of action, and opposite sense.
XIX.
Polygon Rule for the addition of vectors: R A
D C
B
Note that if the formed polygon is closed, the resultant is zero. XX.
For each force F, there exists A3 an infinite number of possible components. A2 A1
R B3 B1
B2
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XXI.
Two forces are equal if they have the same magnitude and direction.
XXII.
Two forces are equivalent if they have the same magnitude, direction and line of action.
XXIII.
For two-component problems, if one component and the resultant are given, the triangle rule can be used to solve for the second component. Trigonometry can be used to solve for the relevant angles and the magnitude of the missing component, specifically with the Law of Cosines.
XXIV.
For two-component problems, if the lines of action of the two components and the resultant are given, the parallelogram law can be used to determine the magnitude of the two components graphically. Their magnitudes can be computed with the Law of Sines.
In this equation, side A is opposite of angle , side B is opposite of angle , while side C is opposite of angle . XXV.
For minimum-maximum problems, always show all cases. For example: A barge is pulled by two tugboats. If the resultant of the forces exerted by the tugboats is a 5000-N force directed along the +x-axis, determine the value of the angle
such that
the tension in the second rope is a minimum.
R B1
B4
B3
B2
ALine of Fernando, Don Lawrence Carl A. 2010-24598 action
AM+DG
Note that minimum tension occurs in the second rope when the two ropes are perpendicular to each other. XXVI.
The rectangular components of a force in a plane:
Fy
F Fx
Fx and Fy are scalar components while Fxi and Fyj are vector components. For multiple (non-rectangular) components:
XXVII.
The rectangular components of a force in space:
Fy F
Fx
Fz
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XXVIII.
If defined by its magnitude and two points (A and B) on its line of action only, the vector representation of these forces is found by multiplying the magnitude F, a scalar, to the unit vector uAB that has the same sense as vector AB. That is:
FA uA
A B
B
B
XXIX.
Addition of concurrent forces in space:
Fernando, Don Lawrence Carl A. 2010-24598
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XXX.
For three-dimensional minimum tension problems, symmetries may be used to reduce the problem to a two-dimensional one. In the case that this is possible, having perpendicular tension vectors results to minimum tension for one of the ropes.
XXXI.
For a particle to be in static equilibrium, resultant, and its components effectively, must be zero. That is:
XXXII.
For 3-dimensional equilibrium tension problems, the following systems of equation is used to solve for the unknown values:
TA
TAD
C
TA
W
B
XXXIII.
External forces represent the action of other bodies on a rigid body in consideration. They are entirely responsible for the external behavior of the body. Internal forces are the ones which hold together the particles and (macro) parts that form the body.
XXXIV.
The scalar product of vectors A and B is defined as the sum of the component-wise products of the two vectors. That is:
Pertinent equations are:
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XXXV.
If two vectors are perpendicular, then their dot product is zero. That is:
XXXVI.
The scalar projection of a vector A onto B is defined as:
XXXVII.
The vector projection of a vector A onto B is defined as the scalar projection of A onto B multiplied by the unit vector in the direction of B. That is:
XXXVIII.
The vector product of two vectors A and B is defined as the vector that is perpendicular to the plane containing the two said vectors, has a magnitude that is equal to the product of the magnitudes of the two vectors and the sine of the angle between them, and has a direction that can be obtained with the right-hand rule. Pertinent equations are:
XXXIX.
The vector product does not follow the cancellation law. That is:
Fernando, Don Lawrence Carl A. 2010-24598
AM+DG However, note that:
Consider the case where A and (B – C) are neither zero. Obviously, the two must be parallel. This implies that the two are scalar multiples; that is:
XL.
For a non-zero vector A, if A
(B – C) = 0 and A (B – C) = 0, then vector (B – C) must be
parallel and perpendicular to a vector A. Such a situation is only possible if B – C is zero; this implies that B = C. XLI.
i,j, and k form a right-handed triad. Their cross products can easily be remembered through the use of the counterclockwise circle. j i
k XLII.
The graphical representation of the magnitude of A x B is the area of a parallelogram that has A and B for its sides. Therefore: AB B B’ A
XLIII.
The scalar triple product of three vectors A, B, and C is defined as the dot product of a vector A with the cross product of two other vectors, B and C. The geometric representation of the scalar triple product is the volume of a parallelepiped which has vectors A, B, and C as its sides.
Fernando, Don Lawrence Carl A. 2010-24598
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A C B Note that the three triple product representations for the volume also follow the counterclockwise circle. j i
k XLIV.
If the scalar triple product of A, B, and C is zero, then the three vectors must be coplanar since the parallelepiped that they defined would be flat and have no volume.
XLV.
The vector triple product of three vectors A, B, and C is defined as the cross product of one of the three vectors with the cross product of the other two. Pertinent equations are as follows:
XLVI.
A vector quadruple product can be defined as the cross product of the cross products of vectors A and B, and C and D. Pertinent equations are as follows:
Also, note that:
Fernando, Don Lawrence Carl A. 2010-24598
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XLVII.
One of the factors that affect the effect of a force on a given body is its point of application. The magnitude of the moment of a force F about a point measures the tendency of that force to rotate the rigid body about a fixed axis directed along Mo. Mo is the cross product of the position vector r from the (reference) point O to the force’s point of application A and the force vector. That is:
M o
r
F
XLVIII.
The moment is not affected by the actual position of the point of application of the force along its line of action. Conversely, the moment of a force F does not characterize the position of the force’s point of application. However, the moment of a force F of given magnitude and direction completely defines the line of action of F; the distance of the line of action from the reference point must be equal to the quotient of M o and F and its location is determined by the sense of Mo.
XLIX.
XXII can be restated as: “Two forces are equivalent if they are equal and have equal moments about a point O.”
L.
A positive moment denotes one that points out of the plane and rotates the rigid body in a counterclockwise manner while a negative moment is one that points into the plane and rotates the body in a clockwise manner.
LI.
Varignon’s Theorem states that the moment about a given point O of the resultant of several concurrent forces is equal to the sum of the individual moments of the various forces about the same point O. That is:
Therefore, in resolving the position vector r and a force F into its components, the following is obtained:
Fernando, Don Lawrence Carl A. 2010-24598
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LII.
For a more generalized formula for the moment of a force with an arbitrary point of application B about an arbitrary point A, the position vector r is simply replaced by the vector from the reference point to the point of application. That is:
LIII.
Two-dimensional moment problems can be treated as three-dimensional ones by setting the unused dimension components to zero.
LIV.
The moment of a force F about a given axis OA measures the tendency of the force to rotate a rigid body about the fixed axis. This can be found by finding the projection of Mo onto uOA. That is:
Axis OA M
M
o
A
uO
r
F
A
LV.
Just as the components Fx, Fy, and Fz of a force F acting on a rigid body measure, respectively, the tendency of F to move the body in the x, y, and z directions, the moments Mx, My, and Mz of F about the coordinates axes measure the tendency of F to rotate the body about the x, y, and z axes, respectively.
LVI.
For a more generalized formula for the moment of a force with an arbitrary point of application C about an arbitrary axis defined by AB, the position vector r is simply replaced by the vector from the reference point on the axis to the point of application. That is:
Fernando, Don Lawrence Carl A. 2010-24598
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Note that the choice of the reference point on the axis does not matter because a different reference point won’t change the value of the moment. This is true since:
Axis AB Mo MA B
uA
F B C
B
A B LVII.
A C
Two forces, F and –F, having the same magnitude, parallel lines of action and opposite senses are said to form a couple. The resultant force of the two is zero but their resultant moment is not. That is:
M F F
Fernando, Don Lawrence Carl A. 2010-24598
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LVIII.
Two couples, one consisting of F1 and –F1 and the other of the forces F2 and –F2, will have equal moments if their F1d1 = F2d2 and if the two couples lie in parallel planes and have the same sense.
LIX.
Two couples, whether they act on the same plane or parallel ones, having the same moment M are equivalent.
LX.
The moments of couples are added the same way as vectors are. This is supported by Varignon’s Theorem. That is:
R
F2
F1
F1 F
R
2
Fernando, Don Lawrence Carl A. 2010-24598
