Forces

The effect of force on an object can be characterized by its point of application, magnitude, and direction. As force has both magnitude and direction, it is a vector quantity. The unit of force is Newton.

Force Vectors

A force vector can be represented by an arrow with the length of arrow represents the magnitude of the force, the angle between the arrow and the coordinate axis defines the direction of force indicated by the line of action, and the arrow head indicates the sense of direction.

Forces in a Plane

When more than one force act on a point, they can be replaced by a single resultant force with teh same effect on the point of action.

The resultant force can be determined by means of vector addition.

• Parallelogram Law: The applied forces can be represented by the adjacent sides of the parallelogram and the resultant force can be obtained by drawing the diagonal of the parallelogram.

  

• Triangle Rule: The applied force can be represented by two sides of a triangle in sequence and the resultant force can be obtained by drawing the third closing side of the triangle.in the opposite sense. Similarly, the triangle rule can further extend to the polygon rule, by using polygon construction method to represent the resultant force in both magnitude and direction.

  

• Laws of Trigonometric Functions : This is an analytical method based on the geometry instead of using vector construction method with true scale of magnitude and direction.

  

  The magnitude of the resultant froce can be determined by pythagorean theorem or the law of cosines.

  

  The direction of the resultant froce can be determined by the law of sines.

  

  or be determined by the triangle rules.

  

Force decomposition

From the parallelogram Law, force can be resolved into two components. When the two components are perpendicular to each other in the form of a rectangle, they are called rectangular components.

or

Assumed two unit vectors, i and j, with unit magnitude along the x and y axis. Then

, , and , imply

The two components of the force vector can be obtained by

and

The magnitude of the force vector can also be obtained by

 

And the direction of the force vector can also be obtained by

Force Equilibrium

According to Newton's first law of motion, when a particle is in equilibrium, the resultant of all the forces acting on it is zero. That is

And graphically in the form of force polygon,

, imply

When there are only three applied forces, the problem can be reduced to a force triangle and be solved by trigonometry.

Analytically, forces can be resolved into rectangular components to form the equations of equilibrium. Imply

Therefore,

  and 

Forces in Space

In practical applications, forces are usually involved in three dimensional space. Forces in a plane is only a special three dimensional case with the third direction equals to zero.

Force Decomposition

A force can be represented by a force vector, F, in three dimensional space. The force can be regarded as a plane force by assuming a plane passing through the force vector and normal to coordinate plane zx.  And therefore force, F can be decomposed into a force, Fy along y axis and a force, Fzx in plane zx. Similarly, force, Fzx in plane zx can further be decomposed  into a force, Fx along x axis and a force, Fz along z axis.

Therefore force vector, F, in three dimensional space can be decomposed into Fx, Fy, and Fz. Applying the laws of trigonometric, the magnitude of the three rectangular force vector components are:

And the magnitude of force vector, F, can be obtained by applying the Pythagorean theorem, imply:

The resultant force vector, F, can be obtained by vector addition of the three rectangular force vector components by applying Parallelogram Law, imply:

Assuming three unit vectors, i, j, and k, with unit magnitude along the x, y, and z axis. then

Imply

The three rectangular force vector components  of the resultant force vector, F, can be arranged in a box, imply:

Direction Cosines

The direction of the force vector, F can be defined by the angles formed by the force vector and the three coordinate axes, θx,  θy, and θz. Imply

The magnitude of the three rectangular force vector components can be defined as:

The cosine of θx, θy, and θz are known as the direction cosines of the force vector, F. And the force vector becomes two dimensional force when one of the direction cosines equal to zero, that is angle equals 90 degree.

The magnitude of force vector, F can therefore be expressed in terms of  direction cosines as following:

Imply the sum of the squares of direction consines equals to 1, that is:

And the values of direction consines or the angles are not independent.

The force vector, F can also be expressed in terms of unit coordinate vectors and direction cosines as following:

The format of the force vector expression can be treated as the product of the force magnitude and the unit direction vector. Let λ be the unit direction vector. Then

Similarly, the unit direction vector can also decomposed to three rectangular vectors:

The magnitude of the unit direction vector equals to one because:

The relationship of force vectors and  unit vectors is:

Laws of Trigonometric Relations

The direction of a force in space along the line of action can also be respresented by a relative position vector, d.

Similar to the force vector, F, the relative position vector, d can be expressed as:

Applying the laws of trigonometric relations, imply:

And: