Statics

Chapter 1Introduction to Statics

What is Statics?

Statics is the study of systems in which the motion of the object is not changing (no acceleration)• Static – Not Changing• Equilibrium – No net force, zero acceleration

0

What is systems is Statics used to describe or analyze?

Structures – bridges, buildings, frames, etc.Machines – robots, vehicles, etc.Material Properties – Stress, Strain, Strength, breakage, vibrations, etc.

Terminology• Space – geometric region occupied by a body. Described using linear and angular

measurements.• 1D – 1 linear component• 2D – 2 linear components or 1 linear and 1 angular component• 3D – 3 linear components or 2 linear and 1 angular component or 1 linear and

2 angular components• Time – Measure of successive events (not applicable to Statics)• Mass – Inertia of an object – Resistance to changing motion.• Force – Interaction between two objects. Defined by magnitude, direction and

point of application.• Particle – Object of negligible (or irrelevant) dimensions. Mass is concentrated at

a single point (center of mass).• Rigid Body – An object that does not change shape under the application of a

force.• Depends on material and how the force is applied• Deformation is beyond the scope of this course

Scalars and Vectors

Scalar – quantity that only has magnitude. ( or V or V)• Mass, time, density volume

Vector – quantity that has magnitude and direction. ( or V)• Velocity, acceleration, force, area

q

l V

Vectors can be added using parallelogram method (Fig. 1/2 (b)) or head to tail method (Fig. 1/2 (c))

𝑉=𝑉 1+𝑉 2 |𝑉|≠|𝑉 1|+|𝑉 2|

Vector Components

Vectors can be broken up into components. Components are used to describe part of the vector along each of the coordinate directions for your chosen coordinate system.

- The number of components is determined by the number of dimensions.

A 2D vector in the x-y plane has 2 components- 1 in the x direction and 1 in the y direction.

A 3D vector in spherical coordinates has 3 components

- 1 in the r direction, 1 in the q direction and 1 in the f direction.- All vector components are defined to be orthogonal (perpendicular) to each other.- Vector components are also vector quantities.

qx

y

A

xA

yA

yA

222

yx AAA

A

Ay

sin

A

Ax

cos

x

y

A

A

tan

yx AAA

Vectors can be moved as long as their length and orientation do not change!

When you square a vector, the result is a square of the magnitude. Directional information is removed.

sinAAy

cosAAx

These are only valid for the specified angle.

xA

You can determine magnitude and direction for 3D vectors in a similar fashion to what was done for 2D.

2222

zyx VVVV

V

Vxx cos

V

Vyy cos

V

Vzz cos

qx, qy and qz are the angles measured between the vector and the specified coordinate direction.

The magnitude can be found using the Pythagorean Theorem.

Vector notation using unit vectors

A unit vector is a dimensionless vector with a magnitude of 1 that is used to describe direction.Cartesian coordinate unit vectors:

x – direction =

y – direction = z – direction =

i

j

k

- The hat ‘^’ is used to distinguish unit vectors from other vectors.

The following notation is also used:

x – direction =

y – direction = z – direction =

x

y

z

kAjAiAA zyxˆˆˆ

Ax = magnitude of A along x – directionAy = magnitude of A along y – directionAz = magnitude of A along z – direction

Examples of vectors using unit vector notation:

Unit vectors can also be defined for any non-standard coordinate direction.

��=𝑉

|𝑉|

- Unit vector along direction of in terms of the coordinates of .

+

Dot Product (Scalar Product)

This product of two vectors results in a scalar quantity. You multiply one vector by the component of the second vector that is parallel to the first vector.

cosBABA

zzyyxx BABABABA

kBjBiBkAjAiABA zyxzyxˆˆˆˆˆˆ

1ˆˆ

1ˆˆ

1ˆˆ

kk

jj

ii

0ˆˆ

0ˆˆ

0ˆˆ

kj

ki

ji

If A = B: We use the same rules when multiplying a vector by itself.

2AAAAAAAAA zzyyxx The square of a vector only

gives magnitude.

2222zyx AAAAAA

Pythagorean Theorem!

Example: You have two vectors and .a) What are the coordinates of the resultant vector?b) What is the magnitude of the resultant?c) What is the direction of the resultant?

a) 𝑅=𝑉 1+𝑉 2¿2 ��+4 ��+6 ��+1 ��+2 ��+4 ��¿ (2+1 ) ��+(4+2) ��+(6+4 )��

𝑅=3 ��+6 ��+10 ��

b) 𝑅=√𝑉 𝑥2+𝑉 𝑦

2 +𝑉 𝑧2¿√9+36+100¿√145¿12.04 ≈12

c) Vx = Vcosqx

Vy = Vcosqy

Vz = Vcosqz

Newton’s Laws1. A Particle remains at rest or continues to move with uniform velocity (straight line,

constant speed) if there is no net force acting on it.• Inertia

2. The acceleration of a particle is proportional to the vector sum of forces acting on it, and is in the direction of this vector sum.• One object

3. The forces of action and reaction between interacting bodies are equal in magnitude, opposite in direction and colinear (lie on the same line). • Two different objects, force one object exerts on the other!

Units• Contains information regarding the meaning of a number. Nearly all numerical values will

have a unit.• Two systems of units

• English or U.S. Customary units (lb, ft, sec) (FPS)• SI units (m, kg, s) (MKS)

• You must be able to convert between unit in a single system as well as between systems!