1. X, Y axis (Horizontal & Vertical)
Resolution of Forces
Lecture 2
Force: it is the action of one body on another, it tend to move a body in the direction of its action.
Force is a vector quantity, so the complete specification of the action of a force must include:
1 – Magnitude – Direction – Line of force action and point of application.
500 N 4 3
Body 
Resolution of forces
2 – D
3 – D (Space)
X, Y axis (Horizontal & Vertical)
Oblique Resolution
Fx, Fy (Slope or angle)
Sine Law, Cosine Law

2. Resolution of Forces Lecture 2
The process of replacing a force system by its components is called Resolution.
The component of a force is any one of two or more forces having the given force as a Resultant.
To resolve a force into x-y components means to express the force as the sum of two forces, one in the x-direction (the x-component) and one in the y-direction (the y-component), When resolving a force into x-y components, we must have information on the direction of the force and the magnitude of the force. x y
5 N
3 N
4 N 3 4 3 4
3 N
4 N
5 N

3. Resolution of Forces Lecture 2
When the angle of the force relative to the x- or y-axes is known, we can use trigonometry to find the components. Let ( θ ) be the angle that the force makes with the positive x-axis. Using trigonometry, we find the components ( Fx and Fy ) as follows:
It is usually easiest to find the magnitudes of the components from the acute angle of the triangle defined by the force and the axes. Note that the components may have positive or negative signs. You need to recognize the signs of the components so they agree with their senses.

4. Resolution of Forces Lecture 2
In general we use the triangle to find the magnitude of the components and the direction and sense of the force to find the signs of the components. Knowing two sides of a right triangle e.g., ( a and b ) you can either find the angle that the vector forms with the horizontal, or use similar triangles: proportions of (|Fx| : |Fy|: |F|) are the same as (a : b : a2+b2) , as follow: A b Fx Fy F q a B C
In general, use the triangle formed by the force and its components to find the magnitude of the components, and use the direction and sense of the force to find the signs of the components.
To determine force vector components:
when the angle between the force and the ( x or y axis ) is known use trigonometry.
when the force is parallel to the hypotenuse of a known triangle use similar triangles, or trigonometry.

5. Composition of Forces Lecture 2
The process of replacing a force system by its resultant is called composition.
The Resultant of a pair of concurrent (occurring at the same time and gathering at the same point) forces can be determined by means of Parallelogram Law, which states that:
Two forces on a body can be replaced by a single force called the resultant by drawing the diagonal of the parallelogram with sides equivalent to the two forces.
Stevinus ( ) was the first to demonstrate that forces could be combined by representing them by arrows to some suitable scale, and then forming a parallelogram in which the diagonal represents the sum of the two forces. In fact, all vectors must combine in this manner.
For example if F1 and F2 are two forces, the resultant ( R ) can be found by constructing the parallelogram.

6. Composition of Forces Body Lecture 2 The Triangular law:
If the two forces acting simultaneously on a body are represented by the sides of a triangle taken in order (the end of the 1st force represent the beginning of the 2nd force), then their resultant is represented by the closing side of the triangle in the opposite order.
To understand this, If we represent force F2 by (oa) and F1 by (ac), the resultant will be represented by (oc). Thus, we can add two forces either by parallelogram law or by triangle law.

7. Composition of Forces Lecture 2 The Oblique Forces:
If non-rectangular components of a force are needed, several methods are available for determining them.
The components of the force ( F ) shown as (OA) and (OB) can be determined graphically by drawing the parallelogram to any convenient scale. The magnitudes of the components can be determined algebraically from the law of sines & cosines [With reference to triangle (ABC) with sides of ( a, b, c ), the sine rule states] which states: F B O A C x y
Sine Law:
Cosine Law:

8. Composition of Forces Lecture 2
Using the sines and cosines laws in statics, the sum of two forces ( F1 and F2 ) which represents the Resultant by ( R ) could be found by using the parallelogram law with angles ( a, b & c ) shown in the figure:
Principle of Transmissibility:
The conditions of equilibrium or motion of a body remain unchanged if a force on the body is replaced by a force of the same magnitude and direction along the line of action of the original force.

9. Resolution of Forces in 3 – D space
Lecture 2
It is convenient to resolve a force in space into three mutually perpendicular components parallel to three coordinates axes.
The resultant ( R ) could be first resolved into two components along (AC) and (CD) by means the parallelogram law, and the component along (CD) could then be resolved into components along (AE) and (AF). From the figure it is apparent that:
The angles (qx , qy and qz ) are the angles between the resultant force and the positive coordinate axes. The cosines of these angles are called direction cosines. The summation of cosine’s squares of these angles is equal ( 1 ):
If the angle is higher than (90o), the cosine is negative, indicating that the component is opposite the positive direction.

10. Mechanical Component F = k .X Lecture 2
String or cable: A mechanical device that can only transmit a tensile force along itself. It represents by T.
Linear spring: A mechanical device which exerts a force along its line of action and proportional to its extension:
F = k .X
K is constant of proportionality which is a measure of stiffness or strength.

11. Mechanical Component Lecture 2 Cables:
Cables are assumed to have negligible weight and they cannot stretch.
They can only support tension or pulling (you can’t push on a rope!).
Frictionless Pulley: Pulleys are assumed to be frictionless.
Pulleys:
A device consisting of one or several (possibly different diameters) pulleys guiding a rope.
It combined with ropes, chains, gears or V-belts for transmission and amplification of forces.
A continuous cable passing over a frictionless pulley must have tension force of a constant magnitude.
The tension force is always directed in the direction of the cable.
For a frictionless pulley in static equilibrium, the tension in the cable is the same on both sides of the pulley.

12. Force Types Active Forces - tend to set the particle in motion.
Lecture 2
Active Forces - tend to set the particle in motion.
Reactive Forces - result from constraints or supports and tend to prevent motion.
Active force
Reactive force

13. Example 1
Lecture 2
1000 N Y X’
40o
30o X
Resolve the (1000 N) force acting on the pipe, into components in the (a) X and Y directions, and (b) X’ and Y directions.
Solution:
Part (a): note that the length of the components is scaled along the X and Y axes by first constructing lines from the tip of ( F = 1000 N ) parallel to the axes in accordance with the parallelogram law.
40o
1000 N X Y Fx Fy
Part (b): note carefully how the parallelogram is constructed. Applying the law of sines using data mentioned in figure, it will yield:
70o
1000 N X’ Y Fy
Fx’
60o
50o

14. Example 2
Lecture 2
The screw eye is subjected to two forces, F1 and F2. determine the magnitude and direction of the resultant force.
Solution:
Parallelogram Law: The parallelogram law of addition is shown in figure below. The two unknowns are the magnitude of ( FR ) and the angle ( q ).
Trigonometry: from the figure, the vector triangle is constructed. FR is determined by using the law of cosines:
The angle ( q ) is determined by applying the law of sines, using the computed value of ( FR ):
Thus, the direction ( f ) of ( FR ), measured from the horizontal is: