Statics Mechanics isabranchofphysicsinwhichwestuythestate of rest or motion of boies subject to the action of forces. It can be ivie into two logical parts: statics, where we investigate the equilibrium of boies uner the influence of forces, an ynamics, where we investigate the motion of boies uner the influence of forces. Newton s laws of motion N1. particle remains at rest or continues to move in a straight line (with constant velocity) if the resultant force acting upon it is zero. N2. The acceleration of a particle is proportional to the resultant force acting upon it an is in the irection of this force. N3. Two boies exercise mutual forces upon each other, equal in magnitue but oppose in irection. There are four funamental quantities which occur in mechanics: time (measure in secons, abbreviation s), istance (measure in metres, abbreviation m), mass (measure in kilograms, abbreviation kg), an force (measure in Newtons, abbreviation N).
orces orce is a vector quantity (experimentally proven). orces can be represente by irecte line segments. The length of the line segment gives the magnitue of the force. The irection of the line segment gives the irection of the force. The mechanical effect of a force on a boy is what the force oes to the boy. Examples inclue: translation, when the boy moves without any rotation or eformation; rotation, when the boy changes its orientation about an axis; eformation, when theshapeofthe boyischange by theforcesacting upon it.
rigi boy is a boy of which the eformation is negligible. Mechanical effect oes not epen only on the magnitue an irection of the force, but also where the force is applie to the boy. The point where a force is applie to a boy is calle the point of application of the force. In the figures above, the points,, an are all points of application. The straight line through the point of application an parallel to the force is calle the line of action of the force. line of action force is only fully specifie by the provision of its irection, magnitue, an position of its point of application. Two forces are equivalent (that is, they have the same mechanical effect) if an only if they are equal vectors with the same line of action.
If two forces 1 an 2 are applie at a point of a boy, then they can be replace by an equivalent force which is applie at the same point an is the vector sum of the original two forces. = 1 + 2 1 1 P 2 P 2 In the figure above, is calle the resultant of the system of forces. If two forces 1 an 2 areacting uponaboyan areparallel to each other, then they can be replace by an equivalent force which is the vector sum of the original two forces. 1 x 1 x 2 2 Note that the istances x 1 between 1 an, an the istance x 2 between 2 an, satisfy x 1 = 1. x 2 2
Two parallel forces, equal in magnitue but opposite in irection an with ifferent lines of action,cannot be reuce to a single force. Such a system of forces is calle a couple, which we inicate by {, }. couple can never be reuce to a single force! couple has a purely rotational effect on a boy. The moment of a force is the rotational tenency of a boy ue to the force acting upon it. r θ The moment of a force is a vector quantity. The magnitue of the moment of a force at the point is M =.
It follows from the figure that M = = rsinθ = r. Since the vector prouct appears in the previous equation, we conclue that the moment of a force is a vector which is perpenicular to the plane containing the vectors r an, an so M = r. The irection of this vector is given by the right han rule. Equivalent forces have the same moment about a given point. The resultant rotational effect of a number of forces about a given point is given by the sum of their moments about this point: M = M,1 +M,2 + +M,n = r 1 1 +r 2 2 + +r n n = r i i. The moment of a couple {, } about all points is the same: M = s = = ˆn. s The moment of a couple is a free vector, meaning that all couples with the same moment are mechanically equivalent, an contains complete information about the couple s mechanical effect. number of couples are jointly equivalent to a single couple, the moment of which is the vector sum of that of the respective couples. M = s 1 +s 2 = M 1 +M 2
2 1 s 1 2 The resultant of a system of forces is the most simple system that is mechanically equivalent to it. The resultant of a system consists, in general, of: a single force at an arbitrary point P, which is the vector sum of all the forces in the system, an a couple, where the moment of the couple is the total moment M of the given system about P. In symbols: = i, M = s i i. If M = 0, then the system can be replace by either an equivalent force or a couple.
Equilibrium of boies particle isaboythat canberegareassmall enoughinagiven situation to be escribe as a mass at a geometric point. particle is in equilibrium if an only if the vector sum of the forces acting on the particle is the null vector, that is, i = 0 or in component form. x,i = x,1 + x,2 + + x,n = 0, y,i = y,1 + y,2 + + y,n = 0, z,i = z,1 + z,2 + + z,n = 0 This conition of equilibrium can be applie to particles couple to each other by means of ros, ropes or springs, by applying it to each particle separately. rigi boy is in equilibrium if an only if the resultant of the system of forces acting upon the rigi boy is the null vector. rigi boy is therefore in equilibrium if: the sum of the forces on it is zero, an the total moment of these forces about any point is zero. = M P = i = 0 s i i = 0 or a boy in a plane we obtain the equations x = x,i = 0, y = y,i = 0, M P = M z,i = 0.