Chapter 2 Review and Summary PDF

# Review And Summary ## Forces - Forces are _vector quantities_ and are described by a _point of application_, a _magnitude_ and a _direction_. - Forces are added according to the _parallelogram law_ (Fig. 2.35). - The _magnitude_ and _direction_ of the resultant _R_ of two forces _P_ and _Q_...

# Review And Summary ## Forces - Forces are _vector quantities_ and are described by a _point of application_, a _magnitude_ and a _direction_. - Forces are added according to the _parallelogram law_ (Fig. 2.35). - The _magnitude_ and _direction_ of the resultant _R_ of two forces _P_ and _Q_ can be determined by: - Using the _law of cosines_ - Using the _law of sines_ ## Components of a Force - A _force_ acting on a particle can be _replaced_ by two or more forces. - For example, a force _F_ can be replaced by components _P_ and _Q_ by drawing a _parallelogram_. ## Rectangular Components of a Force - A force _F_ can be _resolved_ into two _rectangular components_ if they are perpendicular to each other and are directed along the coordinate axes (Fig. 2.37). - Introducing the _unit vectors_ _i_ and _j_ along the _x_ and _y_ axes respectively, we write: - *Fx = Fi* - *Fy = Fyj* - *F = Fi + Fj* - Where *Fx* and *Fy* are the _scalar components_ of _F_. - They can be positive or negative and are defined by the relations: - *Fx = F cos θ* - *Fy = F sin θ* - If the _rectangular components Fx_ and _Fy_ of a force _F_ are given, the angle *θ* defining the _direction_ of the force can be obtained by: - *tan θ = Fy / Fx* - The magnitude of a force can be obtained by using the _Pythagorean theorem_: - *F = √(Fx² + Fy²)* ## Resultant of Several Coplanar Forces - When three or more _coplanar forces_ act on a particle, the _rectangular components_ of their resultant _R_ can be obtained by _adding algebraically_ their corresponding components. - *Rx = ΣFx* - *Ry = ΣFy* - The magnitude and direction of _R_ can then be determined from relations similar to equations 2.9 and 2.10. ## Forces in Space - A _force_ in three-dimensional space can be resolved into _rectangular components_. - Denoting by *ux*, *uy*, and *uz* the angles that _F_ forms with the _x_, _y_ and _z_ axes (Fig. 2.38), we have: - *Fx = F cos ux* - *Fy = F cos uy* - *Fz = F cos uz* - Cosines of *ux*, *uy*, *uz* are known as the _direction cosines_ of the force _F_. - Introducing the _unit vectors_ _i_, _j_ and _k_ along the coordinate axes, we write: - *F = Fi + Fj + Fk* - *F = F (cos uxi + cos uyj + cos uzk)* - The magnitude of _F_ is the product of its _magnitude_ and a _unit vector_: - *A = cos uxi + cos uyj + cos uzk* - Sine of *θ* must equal one: - *cos² ux + cos² uy + cos² uz = 1* - If the rectangular components of _F_ are given, the _magnitude_ of the force is found by: - *F = √(Fx² + Fy² + Fz²)* - Direction cosines of _F_ are obtained by: - *cos ux = Fx/F* - *cos uy = Fy/F* - *cos uz = Fz/F* ## Resultant of Forces in Space - When two or more forces act on a particle in three-dimensional space, the _rectangular components_ of their resultant _R_ can be obtained by adding algebraically the corresponding components of the given forces. - *Rx = ΣFx* - *Ry = ΣFy* - *Rz = ΣFz* - The magnitude and direction of _R_ can then be determined from relations similar to equations 2.18 and 2.25. ## Equilibrium of a Particle - A particle is in _equilibrium_ when the _resultant_ of all the forces acting on it is zero. - When the particle is in equilibrium, it remains at rest or moves with constant speed in a straight line. ## Free-Body Diagram - To solve a problem involving a particle in _equilibrium_, a _free-body diagram_ of the particle showing all the forces acting on it should be drawn. - If only three _coplanar forces_ act on the particle, a _force triangle_ may be drawn to express that the particle is in equilibrium. - Using graphical methods of trigonometry, this triangle can be solved for no more than two unknowns. - If more than three _coplanar forces_ are involved, the equations of equilibrium should be used. - ΣFx = 0 - ΣFy = 0 - These equations can be solved for no more than two unknowns. ## Equilibrium in Space - When a particle is in _equilibrium in three-dimensional space_, the three equations of equilibrium should be used. - ΣFx = 0 - ΣFy = 0 - ΣFz = 0 - These equations can be solved for no more than three unknowns.