2.40
2 Statics of Particles (3D)
Forces in 3D
Most real-world problems are formulated for bodies and
forces not arranged on a single plane.
Hence, for quantitative analysis we need mathematical
representation of location and orientation for bodies,
magnitude and direction of forces, and means to formulate
conditions imposed on them.
We will use 3D vectors for the representation. However,
there are multiple mutually equivalent ways to represent
vector itself and one should select the most efficient way
based on the problem.
Operations on forces in 3D are performed, almost
exclusively, by use of formal vector algebra
Force Acting on a Line Between Two points
̅ is acting along the line 𝑨 𝑩 with magnitude 𝑭 ,
If a force 𝑭
then:
What are rectangular components
̅?
of 𝑭
 Any vector can be defined as
the magnitude times the unit
vector in the same direction:
̅ = 𝑭 ∗ ̅̅̅̅
𝑭
𝒖𝑭 = 𝑭 ∗ ̅̅̅̅̅
𝒖𝑨𝑩
2.41
 Unit vector in the direction of 𝑨 𝑩 :
̅̅̅̅
𝑨𝑩
̅̅̅̅̅
𝒖𝑨𝑩 = |𝑨𝑩
̅̅̅̅|
 Need to find the vector and vector magnitude of ̅̅̅̅
𝑨𝑩
̅
̅̅̅̅
𝑨𝑩 = (𝒙𝑩 − 𝒙𝑨 )𝒊̅ + (𝒚𝑩 − 𝒚𝑨 )𝒋̅ + (𝒛𝑩 − 𝒛𝑨 )𝒌
|̅̅̅̅
𝑨𝑩| = √(𝒙𝑩 − 𝒙𝑨 )𝟐 + (𝒚𝑩 − 𝒚𝑨 )𝟐 + (𝒛𝑩 − 𝒛𝑨 )𝟐
 Expanding to force components:
̅ = 𝑭 ∗ ̅̅̅̅̅
𝑭
𝒖𝑨𝑩 =
𝑭
̅̅̅̅
∗ 𝑨𝑩
̅̅̅̅
|𝑨𝑩|
=
𝑭
√(𝒙𝑩 − 𝒙𝑨 )𝟐 + (𝒚𝑩 − 𝒚𝑨 )𝟐 + (𝒛𝑩 − 𝒛𝑨 )𝟐
Forces in 3D, Example 1:
̅]
[(𝒙𝑩 − 𝒙𝑨 )𝒊̅ + (𝒚𝑩 − 𝒚𝑨 )𝒋̅ + (𝒛𝑩 − 𝒛𝑨 )𝒌
2.42
Example 2:
Resultant of the sum of forces in 3D:
̅) =
̅ = ∑𝑭
̅ = ∑(𝑭𝒙 𝒊̅ + 𝑭𝒚 𝒋̅ + 𝑭𝒛 𝒌
𝑹
̅
(∑ 𝑭𝒙 ) 𝒊̅ + (∑ 𝑭𝒚 ) 𝒋̅ + (∑ 𝑭𝒛 ) 𝒌
Components of the resultant:
𝑹 𝒙 = ∑ 𝑭𝒙 , 𝑹 𝒚 = ∑ 𝑭𝒚 , 𝑹 𝒛 = ∑ 𝑭𝒛
Magnitude of the resultant: 𝑹𝒙 = ∑ 𝑭𝒙 , 𝑹𝒚 = ∑ 𝑭𝒚 , 𝑹𝒛 = ∑ 𝑭𝒛
Direction cosines: 𝒄𝒐𝒔(𝜶) =
𝑹𝒙
𝑹
; 𝒄𝒐𝒔(𝜷) =
𝑹𝒚
𝑹
; 𝒄𝒐𝒔(𝜸) =
𝑹𝒛
𝑹
2.43
Example:
2.44
Particle Equilibrium (3D)
Similarly to a 2D situation, the EE are expanded to 3D:
̅=0
∑ 𝐹̅ = 𝑭𝒙 𝒊̅ + 𝑭𝒚 𝒋̅ + 𝑭𝒛 𝒌
Or, if separated into components:
∑ 𝐹𝑥 = 0, ∑ 𝐹𝑦 = 0, ∑ 𝐹𝑧 = 0
Note: Three equations can be used to find 3 unknowns at
most.
Analysis of Equilibrium follows the same procedure as in
2D.
1. Draw FBD to include all acting forces.
2. Apply equations of equilibrium.
In case of 3D there are three equations per particle.
Hence, to solve them, there should be maximum 3
unknowns per particle.
In 3D elimination technique is almost never used. Instead,
use formal vector algebra.
Possible unknowns: Force components / magnitudes /
angles; some distances.
Examples: 1 force – 3 components, or magnitude and 2
angles; 3 forces: magnitudes; etc.
2.45
Example 1:
2.46
Example 2:
2.47
Example 3:
Example 4:
2.48
Example 5: