Representation of Force
There are two ways of representation of force. The method used depends on the type of problem being solved and the easiest approach to finding a solution.
1. Scalar Notation
2. Vector Notation
Vector Notation Of Forces
1. Two Dimensional Force Systems (Coplanar Forces) It is also possible to represent the x and y components of a force in terms of Cartesian unit vectors i and j.
where the scalars Fx and Fy are the x and y scalar components of the vector F.
2. Three Dimensional Force Systems(Non-Coplanar Forces)
Unit Vector Vectors having unit magnitude and represents only the direction of vectors is called a unit vector. It is usually denoted by n. A vector V may be expressed mathematically by multiplying its magnitude V by a vector n whose magnitude is one and whose direction coincides with that of V. The unit vectors along the Rectangular Coordinate axis x, y and z are |
PROBLEMS
1. A force vector F = 700i + 1500j is applied to a bolt. Determine the magnitude of force and the angle it forms with the horizontal.
2. A force of 500 N forms angles 60^{0}, 45^{0} and 120^{0} respectively with x, y and z axes. Write the vector form of force.
Position Vector A position vector r is defined as a fixed vector which locates a point in space relative to another point. a) Position Vector of P relative to origin
b) Position vector of B with respect to A |
A. RESULTANT OF NON COPLANAR FORCES -By Vector Notation
1. Resultant of Non Coplanar Concurrent Forces
In vector notation, the scalar components of the resultant vector can be obtained by adding algebraically the sum of the corresponding scalar components of the force vectors.
where
B. MOMENT AND COUPLE â€“ In Vector Notation
Moment
Moment is a vector quantity whose direction is perpendicular to the plane of the body. The right-hand rule is used to establish the sense of direction of moment.
Vector Formulation
Varignonâ€™s Theorem
Vector Formulation
Couple
Vector Formulation
Couple vectors are free vectors, i.e., the point of application is not significant.
Note : Cross Product of vectors |
2. Resultant of Non Coplanar Non-Concurrent Forces PROBLEMS 1. A table exerts the four forces shown on the floor surface. Reduce the force system to a forceâ€“ couple system at point O. Determine the resultant of the following force and its location 2. Replace the two forces acting on the post by a resultant force and couple moment at point O. Express the results in Cartesian vector form. |
C. EQUILIBRIUM OF NON COPLANAR FORCES â€“ By Vector Notation
1. Equilibrium of Non-Coplanar Concurrent Forces
In vector notation, the equation of equilibrium can be summarized as