Units

 SI U.S. Mass M kilogram kg slug Length L meter m feet ft Force F Newton N pound lb Time T second s second sec

Relation between units is based on the equation F=ma:

1 N = (1 kg) (1 m/s2)

1 lb= (1 slug) (1 ft/sec2 )

 Example of calculating mass in U.S. system: The mass, m, of a particle which weighs W=10 lb and is in a gravitational field of with an acceleration of gravity g=32.2 ft/sec2 is

Unit conversion:

 l lb = 4.4482 N 1 slug = 14.5938 kg 1 ft = 0.3048 m 1 ft = 12 in 1 mile = 5,280 ft 1 kip = 1,000 lb 1 ton = 2,000 lb

Rounding numbers:

As a general rule for engineering problems, the data are seldom known with an accuracy greater than 0.2%.  Therefore, it is usually appropriate to record parameters beginning with “1” with four digits and with three digits in all other cases, i.e.,  40.2 m and 15.58 m.

Homogeneous Rule:

A + B = C

All terms in an equation must have the same dimensions.

Scalars and Vectors

• Scalar: A quantity like mass or temperature which only has a magnitude
• Vector: A quantity like heat flux or force which has both a magnitude and a direction (denoted by a bold faced character, an underlined character, or a character with a arrow on it)

Notation:

 Book By hand Scalar a a Vector a  (bold faced) a  or
• Vector Addition: Vector Addition follows the parallelogram law described be the figure

• Resolution of a Vector: A vector can be resolved along different directions using the parallelogram rule. The figure shows how one resolves vector c into components a and b which are along the given directions

• The math you need:
• For a right triangle:

a2+ b2 = c2

tan(q) = b/a

sin(q) = b/c

cos(q) = a/c

• For a general triangle:

a+b+g=180o

Sine law:

Cosine law:

o        A line intersecting parallel lines:

Unit vector: A vector of unit length

Components of a vector in orthogonal bases: Unit vectors i and j are along the x and y directions

Addition of vectors using the components:

Vectors in 3-D

Unit vector: A vector of unit length.

Base vectors for a rectangular coordinate system: A set of three mutually orthogonal unit vectors

Right handed system: A coordinate system represented by base vectors which follow the right-hand rule.

Rectangular component of a Vector: The projections of vector A along the x, y, and z directions are Ax, Ay, and Az, respectively.

Magnitude of a Vector:

Direction Cosines: Cos(a), Cos(b), Cos(g)

Unit vector along a vector: The unit vector uA along the vector A is obtained from

Addition of vectors: The resultant vector FR obtained from the addition of vectors F1, F2, …, Fn is given by

Coordinates of points in space: The triplet (x,y,z) describes the coordinates of a point.

The vector connecting two points: The vector connecting point A to point B is given by

A unit vector along the line A-B: A unit vector along the line A-B is obtained from

A vector along A-B: A vector F along the line A-B and of magnitude F can be obtained from

The dot product: The dot product of vectors A and B is given by

Projection of a vector by using the dot product: The projection of vector A along the unit vector u is given by

Examples: