Force is a Vector A Force consists of: Magnitude Direction –Line of Action –Sense Point of application –For a particle, all forces act at the same point of application (contrast with a scalar, which has only a magnitude)

Forces (Vectors) Add by the Parallelogram Law Can solve: –Graphically –Using trigonometry –Resolve into x and y components R is the resultant force acting on particle A

Simple Example

Solve using Trigonometry Law of sines Law of cosines

Resolving Vectors into x and y Components  φ Unit vectors:

Resultant Force Acting on a Particle Magnitude: Direction:

Example

Forces in 3D

Magnitude & Direction in 3D Magnitude: Direction:

Determining Forces from Unit Vectors dxdx dydy d z

Equilibrium Using Components: Graphically:

Equilibrium Problems In 2D, have 2 equations, so can solve for 2 unknowns –Find magnitudes of two forces with known directions –Find magnitude and direction of one force, knowing magnitude and direction of other force(s) In 3D have 3 equations, so can solve for 3 unknowns

Rigid Bodies: Equivalent Systems of Forces Chapter 3

Rigid Body Motion Translation –Caused by a Force Rotation –Caused by a Moment A Force acting at a distance from a Point –Several applied Moments: Determine magnitude and direction of single resultant moment, which determines direction of impending rotation.

Forces Principal of Transmissibility: –Can slide a force along its line of action

Moment of a Force about a Point Moment Vector: –Magnitude: Position Vector: Perpendicular distance = d

Determining Moments in 2D Determine + or - by the right hand rule For the picture here:

Determining Moments in 2D Right Hand Rule:

Moment Vector Position Vector: Determining Moments in 3D

Calculating a Vector Cross Product

Scalar Product Find the angle between two vectors, knowing the components Find the projection of a vector onto the line of action of another vector

Moment of a Force, F, About a Line, OL Called a “mixed triple product” M OL measures the tendency for the force, F, applied at A to cause the rigid body to rotate about line OL

Moment of a Couple A Couple consists of two forces: –Same magnitude –Parallel lines of action (but not co-linear) –Opposite sense Moment caused by the couple: d Direction determined by right hand rule

Moments Caused by Couples Moment vector caused by a couple is sometimes called a “couple vector” –Has components: Moment caused by a couple is a “free vector”; you can place it at any point

Moments Caused by Couples Can add moments caused by two or more couples –In 2D: Determine + or - by the right hand rule

Equivalent Couples Two couples are equivalent if they cause the same moment:

Force-Couple System A force, F, acting at point A can be replaced by the force, F, and a moment, M O, acting at point O.

Create a Force-Couple System at a Chosen Point FFFF F MAMA == d M A = d F ABABAB

Replace a Force-Couple System with Just Forces F MAMA = F F2F2 d2d2 F2F2 d 2 F 2 = M A A C A C

Reducing a System of Forces to a Resultant Force-Couple System (at a Chosen Point) F1F1 A F2F2 F3F3 r1r1 r2r2 r3r3 = MAMA R

Reduce a System of Forces to a Single Resultant Force F1F1 A F2F2 F3F3 r1r1 r2r2 r3r3 = MAMA R = MAMA R R R = R B B Using method from prior slide

R R R R RxRx RxRx RyRy RyRy dxdx MAMA d x R = –M A B B A A