ME 221 Lecture 3 1

ME 221 Statics

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Sections 2.2 – 2.3

ME 221 Lecture 3 2

Announcements• Help Sessions – ME Help Room – 1522EB – Cubicle #2

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Fridays – 8:00am to 11:00am

Will begin on Tuesday, September 2

Hours also posted on Angel

ME 221 Lecture 3 3

Announcements•HW#1 Due on Wednesday 09/03

Chapter 1 - 1.1, 1.3, 1.4, 1.6, 1.7

Chapter 2 – 2.1, 2.2, 2.11, 2.15, 2.21

• Quiz #1 on Friday 09/05

ME 221 Lecture 3 4

Last Lecture

• Vectors, vectors, vectors

• Law of Cosines• Law of Sines

• Drawing vector diagrams

• Chapter 1: Basics

• Example 1. Addition of Vectors

ME 221 Lecture 3 5

Law of CosinesThis will be used often in balancing forces

c

b

a

180 2 2 2 2 cosa b c bc 2 2 2 2 cosb a c ac 2 2 2 2 cosc a b ab

ME 221 Lecture 3 6

Law of Sines

Again, start with the same triangle

c

ba

sin sin sin

a b c

ME 221 Lecture 3 7

300 lb200 lb

45o25o

Example

Note: resultant of two forces is the vectorial sum of the two vectors

ME 221 Lecture 3 8

25o

155o

45o

= 90o+25o-200 lb

300 lb

R

25o

110o

300 lb

R

200 lb

ME 221 Lecture 3 9

Scalar Multiplication of VectorsMultiplication of a vector by a scalar simply

scales the magnitude with the direction unchanged

Line of action

A

Line of actionstays the same

0.5 x A

ME 221 Lecture 3 10

Forces• Review definition• Shear and normal forces• Resultant of coplanar forces• Problems

ME 221 Lecture 3 11

Characteristics of a Force• Its magnitude

– denoted by |F|

• Its direction

• Its point of application– important when we discuss moments later

ME 221 Lecture 3 12

Further Categorizing Forces• Internal or external

– external forces applied outside body

P

P

• A section of the body exposes internal body

P

Cut plane through body

Internaltension

ME 221 Lecture 3 13

Shear and Oblique• Shear internal force has line of action

contained in cutting plane

P

PIntenalshearforces

ME 221 Lecture 3 14

Oblique Internal Forces• Oblique cutting planes have both normal

and shear components

P

S

N

Where N + S = P

ME 221 Lecture 3 15

Transmissibility• A force can be replaced by a force of equal

magnitude provided it has the same line of action and does not disturb equilibrium

B

A

ME 221 Lecture 3 16

Weight is a Force• Weight is the force due to gravity

– W = mg • where m is mass and g is gravity constant• g = 32.2 ft/s2 = 9.81 m/s2

• English and metric– Weight lb or N– Mass slugs or kg

ME 221 Lecture 3 17

Resultant of Coplanar ForcesA body’s motion depends on the resultant of all the

forces acting on it

In 2-D, we can use the Laws of Sines and Cosines to determine the resultant force vector

In 3-D, this is not practical and vector components must be utilized

• more on this in the future

ME 221 Lecture 3 18

x

y

Ax

Ay

x

A

y y

Perpendicular Vectors

Ax is the component of vector A in the x-direction

Ay is the component of vector A in the y-direction

Ax

x

Ay

x

y

A

Ay

Ax

ME 221 Lecture 3 19

Vector ComponentsVector components are a powerful way to

represent vectors in terms of coordinates.

x

y

x

y

A

whereAx = |A| cos x

Ay = |A| cos y

= |A| sin x

x

y

Ax

Ay

A=Ax

Ay

ME 221 Lecture 3 20

Ax = |A| cos x

Ay = |A| cos y

= |A| sin x

cos x = Ax / |A|

cos y = Ay / |A|

= x

= y

x and y are called direction cosines

x2 + y

2 = 1

Note: To apply this rule the two axes must be orthogonal

Vector Components (continued)

ME 221 Lecture 3 21

Summary• External forces give rise to

– tension and compression internal forces– normal and shear internal forces

• Forces can translate along their line of action without disturbing equilibrium

• The resultant force on a particle is the vector sum of the individual applied forces

ME 221 Lecture 3 22

Sample Problem