MAE 241 - Statics

Summer 2011

Dr. Konstantinos A. SierrosOffice Hours: M and W 10:30 – 11:30 (263 ESB new add)

kostas.sierros@mail.wvu.eduTeaching Blog: http://wvumechanicsonline.blogspot.com

Chapter 2:Force vectors

Objectives

• To show how to add forces and resolve them into components using the Parallelogram Law• Cartesian vectors• Introduce dot product

2.1 Scalars and vectors

A scalar is any positive or negative physical quantity that can be completely specified by its magnitude (eg. length, mass, time) A vector is any physical quantity that requires both a magnitudeand a direction for its complete description (eg. force, moment)

2.2 Vector operations

Multiplication and division of a vector by a scalar• If a vector is multiplied by a positive scalar, its magnitude is increased by that amount• If a vector is multiplied by a negative scalar it will also change the directional sense of the vector

2.2 Vector operationsVector addition• All vector quantities obey the parallelogram law of addition• If we have two ‘component’ vectors A and B, they can be added to form a ‘resultant’ vector R=A+B-First join the tails of the ‘component’ vectors-From the head of B draw a line parallel to A. Draw another line from head of A that is parallel to B. These lines intersect at point P-The diagonal of the parallelogram forms R

Vector addition• We can also add B to A using the triangle rule which is a special case of the parallelogram law• B is added to A in a ‘head to tail’ fashion• R extends from the tail of A to the head of B• Vector addition is cumulative (i.e R=A+B=B+A)

2.2 Vector operations

2.2 Vector operations

Vector addition• A special case is when A and B are collinear (they act in the same line)• The parallelogram law reduces to an algebraic or scalar additionR=A+B

2.2 Vector operations

Vector subtraction• The resultant of the difference between two vectors A and B may be expressed as:

R´=A-B=A+(-B)• The rules of vector addition apply to vector subtraction

2.3 Vector addition of forcesFinding a resultant force• Two common problems in statics involveeither finding the resultant force or resolving aknown force to its components• Forces F1 and F2 can be added together to formthe resultant force FR (FR=F1+F2)• In order to find the magnitude and direction ofFR we need to use the law of cosines or the law of sines

2.3 Vector addition of forces

Finding the components of a force• In order to resolve a force into twocomponents construct a parallelogramby drawing lines starting from the tip of Fparallel to u and v axes• Fu and Fv are then established by joining the tail of F to the intersection point

2.3 Vector addition of forces

Addition of several forces• If we have more than two forces to add, successive applications of the parallelogram rule need to be carried out• First find F1+F2

• Then add to F3 in order to get FR

FR=(F1+F2)+F3

• However, adding more than two forces iscomplicated enough in terms of geometry and trigonometry. An easier method will be presented later (section 2.4)

Procedure of analysis

Problems that involve addition of two forces can be solved using:• Parallelogram law…and…• Trigonometry-Redraw half portion of the parallelogram to illustratethe triangular head to tail addition of components-From the triangle the resultant force can be calculatedfrom the law of cosines and its direction from the law of sines

2.4 Addition of a system of coplanar forces

Scalar notationF=Fx+Fy

Fx=Fcosθ and Fy=Fsinθ

• The direction of F can be also defined using a small ‘slope’ triangle• Since the ‘small’ triangle is similar with the largershaded triangle we have;

When a force is resolved into two components alongx and y axes, the components are called rectangular

components• For analysis purposes these components can be represented in either scalar or Cartesian vector notation

2.4 Addition of a system of coplanar forces

Cartesian vector notation• x and y components can be represented in terms of Cartesian unit vectors i and j

F = Fxi+Fyj

Coplanar force resultants• We can use both the two methods described to determine the resultant of several coplanar forces

• F1=F1xi+F1yj

F2=-F2xi+F2yj

F3= F3xi-F3yj

FR= F1+ F2+ F3=

=F1xi+F1yj-F2xi+F2yj+F3xi-F3yj=

=(F1x- F2x+ F3x)i+(F1y+F2y -F3y)j=

=(FRx)i+(FRy)j

…or…

(+ ) FRx=F1x-F2x+F3x

(+ ) FRy=F1y+F2y-F3y

2.4 Addition of a system of coplanar forces

2.4 Addition of a system of coplanar forces

Coplanar force resultants• We can represent the components of the resultant force of any number of coplanar forces symbolically using the algebraic sum of the x and y forces

FRx=ΣFx

FRy=ΣFy

2.5 Cartesian vectors

Right-handed coordinate system• If thumb points in the direction of positive z axis, the rectangular coordinate system is called right-handed

2.5 Cartesian vectors

Rectangular components of a vector• A vector A may have one, two or three rectangular components along the x,y,z coordinate axes

A=Ax+Ay+Az

2.5 Cartesian vectors

Cartesian unit vectors• i, j, k are used to designate the directions of the x,y,z axes respectively

2.5 Cartesian vectors

Cartesian vector representation

A=Axi+Ayj+Azk

• This form is advantageous because it separates the magnitude and the direction of each component vector and makes it more simple when we have to deal with 3D problems

2.5 Cartesian vectors

Magnitude of a Cartesian vector• The blue shaded right angle triangle gives us:

• The gray shaded right angle gives us:

Combining the above two…

2.5 Cartesian vectors

Direction of a Cartesian vector

• We define the direction of A by definingThe coordination direction angles α,β and γ

2.6 Addition of Cartesian vectors

Solving 3D problems…

FR=ΣF= ΣFxi+ ΣFyj+ ΣFzk