Integral: Tables

1. Laying out the first vector
2. Moving the second parallel to itself so that its tail touches the head of the first vector
3. The resultant vector R (result of adding F₁ and F₂) is obtained by joining the tail of the first to the head of the second

The magnitude of R is obtained (see above relation)
vector add

- results in a new vector C
- C is normal to both A and B
- direction of C is given by the right hand rule. As the fingers of the right hand curl from A to B the thumb is in the direction of C
- order of the vectors being multiplied is important

- P is any point on the line of action of F
- OP is the displacement vector r_OP (vector from O to point P)

I. Method of Joints can be used for all truss problems
II. Method of Sections is used where the forces are required only in selected members.

(a) Identify the support reactions for the truss supports.

rigid biody

(b) Solve for the member forces by cosidering the FBD of each joint until all member forces are obtained. Alway choose joints where there are as many unknowns as the number of equations.
joints

(a) Identify the support reactions for the truss supports.

rigid biody

(b) Solve for the required member forces by sectioning the truss through the members. Draw the FBD of the sectioned truss and solve for the member forces. Always section the truss so that tthe number of unknowns match the number of equations.

rigid biody

between two contacting surfaces and lies in the plane of contact
opposes motion of the rigid body
depends on the type of surfaces in contact (established experimentally)

Topic	Equations	Description
1. Cosine Law	Note the angles and the side lengths in the formula: Cosine Law: Sine Law:
2. Vector addition	Two vectors F₁and F₂ are added by 1. Laying out the first vector 2. Moving the second parallel to itself so that its tail touches the head of the first vector 3. The resultant vector R (result of adding F₁ and F₂) is obtained by joining the tail of the first to the head of the second The magnitude of R is obtained (see above relation)
2. Resolution of vector (2D) x,y components	The vector/force resolution in xy coordinates requires the coordinates (x,y axes), the force F, and information on the angle between the vector and one of the axes. (a) If a is given, then (b) If L_x and L_y are given , then
3. Resolution of a Vector - 3D
4. Resolution of a vector - 3D	The angles shown are called the direction cosines
5. Unit Vector (rectangular system)	The formula is illustrated on the figure. The vector is between the two points D and E. It goes from D to E. The unit vector uses the coordinate locations of the two points.
5. Equilibrium - Particle	For several forces on the particle, the sum of the forces must be zero. Note that if the sum of the forces are zero then the vectors must close on itself when added
6. Dot Product (Scalar Product) of two vectors	The dot product between two vectors A and B (2D or 3D) is the product of the magnitudes of the vectors and the cosine of the angle between them. The result of this vector product is a scalar.
7. Cross Product (Vector Product) of two vectors	The cross product between two vectors A and B (2D or 3D) - results in a new vector C - C is normal to both A and B - direction of C is given by the right hand rule. As the fingers of the right hand curl from A to B the thumb is in the direction of C - order of the vectors being multiplied is important
8. Moment of a force about a point	The Moment of the force F about the point O is M - P is any point on the line of action of F - OP is the displacement vector r_OP (vector from O to point P)
9. Moment of a force about an axis	The moment of a force F about an axis (RS) through the point O is the dot product of the moment of F about O and the unit vector along MN (e_RS )
10. Couple	The couple is a moment created by two forces F and -F that are equal and opposite and seperated by a distance. The moment due to the couple can be calculated in several ways
11. Moving a Force away from its line of action	A force F acting through A is moved parallel to itself to act at B. At B the force F must be accompanied by a moment M that can be calculated through
12. Equivalence	The effect of a force or several forces acting on a rigid body can be replaced by an equvalent system of a single force and a single moment acting at any chosen point on the rigid body. Here a system of two forces, force F_A at A and force F_Bat B, is replaced by a single force F_C at C and a single couple M_C at C. F_C = F_A + F_B M_C = r_CA x F_A + r_CB x F_B
13. Equilibrium of Rigid Body	The concept of equilibrium for the rigid body is the same as in the case of the particle described earlier. Equilibrium imples that the rigid body does not "move" or change its "orientation". This is ensured if two equations are satisfied: These equations are wriiten with respect to a FBD of the problem. Each equation above yields two/three equations - one along each of the axes. This gives a total of four/six equations which allows solving for 3/6 unknowns - depending if the problem is 2D/3D If convenient, another moment at another point, can be used instead of the force equation.
14. Truss - Method of Joints	A truss is a structure that is designed to handle large loads, using as little material as possible.. You can see them in roof designs, bridges, buildings etc. Trusses are usually made of straight beams that are typically connected together by joints or pins. In addition, the forces in the members are along the axis of the members. These forces try either to extend the member (tension) or shorten the member (compression). In designing trusses it is necessary to know the forces handled by all of the members of the truss. Two methods are used to solve truss problems. I. Method of Joints can be used for all truss problems II. Method of Sections is used where the forces are required only in selected members. I. Method of Joints: The sequence used to solve the problem involves two distinct steps. (a) Identify the support reactions for the truss supports. (b) Solve for the member forces by cosidering the FBD of each joint until all member forces are obtained. Alway choose joints where there are as many unknowns as the number of equations.
15. Truss - Method of Sections	II. Method of Sections: The sequence used to solve problems involves two distinct steps (a) Identify the support reactions for the truss supports. (b) Solve for the required member forces by sectioning the truss through the members. Draw the FBD of the sectioned truss and solve for the member forces. Always section the truss so that tthe number of unknowns match the number of equations. The same approach is used for machines, frames, and mechanisms. In thoses cases it is not necessary that the forces on the members should be oriented along the axis of the members
16. Dry friction	Friction is a force between two contacting surfaces and lies in the plane of contact opposes motion of the rigid body depends on the type of surfaces in contact (established experimentally) Calculating the force due to friction requires the assumption of impending motion, that is, motion is about to happen. For this case Once the body starts moving, the friction is then calculated using the coefficient of kinetic friction, which is usually less than coefficient of static friction
17 Centroid of Area