Topic 7: Determinacy and Special Conditions

• ∑F_x = 0

• ∑F_Y = 0

• ∑M = 0

what are the general rules for equilibrium

P_B = P_A

what is the equation that relates two forces that are self-reacting and are in alignment with each other (but opposite directions)

• orthogonal components of all forces must sum to zero

• use the two equilibrium equations:

  • ∑F_x = 0

  • ∑F_Y = 0

how would you solve for forces where there are three or more forces acting concurrently at a point

• all three equilibrium equations will be required

  • ∑F_x = 0 (1)

  • ∑F_Y = 0 (2)

  • ∑M = 0 (3)

  → use (3) to solve one reaction force then (2) for the other reaction force

how to solve for forces in statical determinacy

use equations:

• ∑F_x = 0

• ∑F_Y = 0

then express one force in terms of the other to solve and use simultaneous equations to solve

how to solve for forces where the magnitudes of two forces are unknown

use equations:

• ∑F_x = 0 → Tcosθ = x

• ∑F_Y = 0 → Tsinθ = y

sinθ/cosθ = tanθ:

• therefore: Tsinθ/Tcosθ = tanθ = y/x

• θ = tan^-1 (y/x)

• magnitude of T can now be determined

how to solve for a force with unknown magnitude and angle

use the equations:

• ∑F_x = 0

• ∑F_Y = 0

→ square both equations and add them together

→ then rearrange to factor out sin²θ + cos²θ

• use the identity sin²θ + cos²θ = 1 to simplify

• solve for the unknown force

how to solve for a force where the angle of the other force is unknown