Mae 241 - Lec3

MAE 241 - Statics Summer 2009

Dr. Konstantinos A. Sierros Office Hours: M and W 10:30 – 11:30 (263 ESB new add) [email protected] Teaching Blog: http://wvumechanicsonline.blogspot.com

*1–16. Two particles have a mass of 8 kg and 12 kg, respectively. If they are 800 mm apart, determine the force of gravity acting between them. Compare this result with the weight of each particle.

•2–1. If θ=30o and T = 6kN , determine the magnitude of the resultant force acting on the eyebolt and its direction measured clockwise from the positive x axis.

2–18. The truck is to be towed using two ropes. Determine the magnitudes of forces FA and FB acting on each rope in order to develop a resultant force of 950 N directed along the positive x axis.Set θ = 50°.

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

2.7 Position vectors

xyz coordinates • Positive z axis is directed upward • x, y axes lie in the horizontal plane Locate A(4m,2m,-6m)

2.7 Position vectors A position vector r is defined as a fixed vector which locates a point in space relative to another point (i.e from point O to P) • Starting at origin O, one ‘travels’ x in the +i direction, then y in the +j direction and z in the +k direction, we arrive at point P r = xi+yj+zk

2.7 Position vectors • In a more general case, the position vector r may be directed from point A to point B in space r=(xB-xA)i+(yB-yA)j+(zB-zA)k • The i,j,k components of r are formed by subtracting the coordinates of the tail A from the coordinates of the head B

2.8 Force vector directed along a line •Often, in 3D problems the direction of a force is specified by two points through which its line of action passes • F is directed along AB • F (as a Cartesian vector) has the same direction and sense as the position vector r • The common direction is specified by the unit vector u = r/r F=Fu=F(r/r)

2.9 Dot product • The dot product, which is a method for ‘multiplying’ two vectors is used in order to solve 3D statics problems. 2D problems can be solved using geometry and trigonometry A•B = AB cosθ Dot product

Laws of operation Cumulative law: A•B = B •A Multiplication by a scalar: α(A•B)=(α A) • B = A • (α B) Distributive law: A• (B+D) = (A•B) + (A•D)

2.9 Dot product • To determine the dot product of two Cartesian vectors, multiply their corresponding x,y, z components and sum these products algebraically A•B = AxBx + AyBy + AzBz

Applications of dot product in mechanics 2. The angle formed between two vectors or intersecting lines 3. The components of a vector parallel and perpendicular to a line