2D – Force Systems• Choose appropriate coordinate system for situation. Standard x-horizontal and y-vertical

system may not be the best choice! • Relate forces to the chosen coordinate system using proper component notation.• Determine magnitude and direction of a vector from components.

MomentsForces that are applied to an object can cause translation or rotation. Forces that result in rotation are called moments (torques). Moments are rotational forces – forces in a direction perpendicular to the plane containing the rotational motion.

��=𝑟 × �� - Moment [M] = Nm, FtLb - Lever arm or location of applied force relative to rotation axis. - Force associated with the moment|𝑀|=|𝑟||𝐹|𝑆𝑖𝑛𝜃

• The order of multiplication is important! r x F ≠ F x r.• Using the scalar form of the cross-product only provides magnitude information.• The angle, q, is the angle measured from r to F in the positive angular direction.• You must define the axis you are rotating about in order to properly define r.• Maximum moment occurs when r and F are perpendicular. M = rF

Vector Product (Cross-product)

BAC

sinBAC

sinBABA

There are two different methods for determining the vector product between any two vectors: The Determinant method and the Cyclic method

Determinant Method

zyx

zyx

BBB

AAA

kji

BA

ˆˆˆ

yzzy BABAi ˆ xyyx BABAk ˆ xzzx BABAj ˆ

Cyclic Method 0ˆˆˆˆˆˆ kkjjii

i

j k

kBjBiBkAjAiABA zyxzyxˆˆˆˆˆˆ

iBAjBAiBAkBAjBAkBA yzxzzyxyzxyxˆˆˆˆˆˆ

xyyxzxxzyzzy BABAkBABAjBABAi ˆˆˆ

xzzx BABAj ˆRewrite the j term as to get an identical expression

Varignon’s Theorem – The moment of a force about any point is equal to the sum of the moments of the components of the force about the same point.

where Mo – Moment about point o.R – Resultant Force.P – Non-Cartesian component of R.Q – Non-Cartesian component of R.

��𝑜=𝑟 ×( ��+��)

��𝑜=𝑟 ×𝑃+𝑟 × ��• This is true for any number of components!