Mechanics 4th | 1st term

يارب الأول زي كل سنة

Scalar quantity

Is determined by a magnitude only such as: mas , volume , length , time.

Vector quantity

Is determined by a magnitude and a direction like velocity , force , displacement weight.

The Directed Line Segment (Vector)

• Is a line segment which has:

  • a starting point,

  • an ending point and a

  • direction from the start to the end.

Two directed line segments are said to be equivalent if:

1. They have the same norm (length).

2. They have the same direction.

Norm equals to

sqrt(X² + Y²)

Position Vector to a given point with respect to the origin:

Is the directed line segment which its starting point is the origin point and the given point is its ending point.

The Polar form of the position Vector:

The vector OA makes an angle 𝜃 with the positive direction of X-axis its polar form is (‖ AB ‖ , 𝜃).

To convert from the polar form to the Cartesian coordenate form:

If the polar form of the vectors A is: (‖ AB ‖, 𝜃)

Then the coordinates of the vectors A is: (‖AB‖ cos𝜃 , ‖ AB ‖ sin 𝜃)

Some Properties of vectors:

Expressing the vector in terms of the fundamental unit vectors:

A = xi + yj
where i and j are unit vectors

Parallel and Perpendicular of two vectors:

Parallel formula

X_1 Y_2 - X_2 Y_1

A = k B
then A and B are Parallel vectors

Perpendicular formula

X_1 X_2 + Y_1 Y_2

Expressing 𝑨𝑩 in terms of the position vectors of its ends:

If A (X_1, Y_2), B (X_2, Y_2)

Then AB = B - A

Effectiveness of the force depends on

• Magnitude

• Direction

• Point of action

Kg_f (Kg.wt) =

9.8 Newton

Newton =

10⁵ Dyne

Units of Magnitude of the force

• Kg.wt

• Newton

• Gm.wt

• Dyne

Smallest measuring unit of force is

Dyne

1 N = 10^5 Dyne

F1 and F2 are two forces meeting at a point and the measure of the angle between them is α and the magnitude of their resultant “R” makes an angle ɵ with F1 , then

R = Sqrt [ (F₁)² + (F₂)² +2F₁F₂ * Cos(𝛂) ]

If R is perpendicular on F1

F1 + F2 cos a = 0

if 𝛂 = 0

R is Maximum

if 𝛂 = 180

R is Minimum

F₁ =

F₁ = R Cos(𝜽_𝟏)

F₂ =

F₂ = R Cos(𝜽₂)

If the system of forces is in equilibrium, then

• X = 0

• Y = 0

R = 0

The body is in equilibrium under the action of two forces if the forces are:

• equal in magnitude

• opposite in direction

• their lines of actions are on the same straight line

If three forces meeting at a point and acting up on a particle equilibrium, then the magnitude of each force is proportional to the sine of the angle between the two other forces.

Lami’s Rule

The triangle of forces rule

If three forces acting at a point are in equilibrium and a triangle is drawn whose sides are parallel to the lines of action of the forces and taken in the same cyclic order, then the lengths of the sides of the triangle are proportional to the magnitudes of the corresponding forces