Vectors, Forces, and Equilibrium

[Pages:5]Vectors, Forces, and Equilibrium

Cartesian unit vectors: = + + The magnitude of a vector: |||| = 2 + 2 + 2

Finding the angle between a vector and an axis (for 2D models): - Simply use the method of sine, cosine, and tangent to determine the angle between a vector and an axis.

=

cos =

tan =

=

-1

()

=

cos-1

()

=

-1

( )

Adding and Subtracting vectors to find the Resultant Force: 1. Break down each force into its Cartesian components. 2. Add all the x-components together and add all the y-components together. 3. Using the new x and y-components, find the magnitude and angle of the Resultant Force. = (1 + 2 . . . ) + (1 + 2 . . . ) + (1 + 2 . . . )

Coordinate direction angles (used for 3D models):

- alpha () - against the x-axis cos() =

||||

- beta () - against the y-axis cos() =

||||

- gamma ()- against the z-axis cos() =

||||

Unit-Force Vectors - used to calculate a Force along a material of specific length, such as a rod/rope. The unit force is equal to the force along the rope (usually in Newtons) multiplied by the individual components divide by the total magnitude of those components.

=

=

(,,) ()

+ + = 2+2+2

In equilibrium, all the forces acting on an object must cancel out: = = = 0

*The equilibrium eq. can also be used to find unknown forces such as the weight of an object. For all vector problems, first draw a free body diagram and separate all known forces into its components.

*An FBD needs to show all external forces acting on an object.

Types of external forces

Weight Always points directly downwards, no matter what the surface angle

Normal force Is always perpendicular to the surface of contact.

Springs Cables Pulleys

( = ) Treat as a single force

Are a tensive force; treat as a single force

if the pulley is smooth/frictionless, then both sides of the cable have the same

tension

Spring Force

Pulleys

1 = 2

Moments and Couple Forces

Moment ? the tendency of a force to cause rotation about a point. = ? or = ?

In equilibrium, all the forces and moments acting on an object must cancel out: *The equilibrium equations can also be used to find unknown forces, such as reaction forces. = = = 0 and = = = 0

*Right hand rule: the distance (r) and force (F) must be perpendicular in which CCW(+) and CW(-)

Finding the Moment about a point (): *Always place (r) before (F) (-)

= ? = [ ]

= ( - ) + ( - )(-) + ( - ) = ( - ) - ( - ) + ( - )

Rule of thumb for cross-multiplying specific vector components:

Example Problems

1. Determine the force in each cable needed to support the 20-kg flowerpot.

For online solution, visit pter%203.pdf (Ch. 3, Question #47)

2. Determine the moment about point B of each of the three forces acting on the beam.

3. A force F having a magnitude of F = 100 N acts along the diagonal of the parallelepiped. Determine the moment of F about point A, using MA = rB x F.

For online solution, visit namics/Chapter%203.pd f (Ch. 4, Questions #5 & #47)

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