A cable runs from the origin O(0, 0, 0) to point A(3, 4, 0) m and carries tension T = 50 N. What are the correct x and y components of this cable force?
ATx = 50·(3/4) = 37.5 N, Ty = 50·(4/3) = 66.7 N
BTx = 3 N, Ty = 4 N (the raw coordinate differences)
CTx = 50·(3/5) = 30 N, Ty = 50·(4/5) = 40 N
DTx = 50·cos(3°) = 49.9 N, Ty = 50·cos(4°) = 49.8 N
The position vector is r = 3i + 4j, with magnitude |r| = √(9 + 16) = 5. The unit vector is û = (3/5)i + (4/5)j. Cable force = T·û = 50·(3/5)i + 50·(4/5)j = 30i + 40j N. Option A uses the ratio of the two components rather than dividing by the magnitude. Option B forgets to divide by the magnitude and multiply by T. The key step is always: find the magnitude of the position vector, then divide.
Question 2 Multiple Choice
How many independent scalar equilibrium equations does a particle in 3D space provide, and what is the maximum number of unknown forces they can determine?
ATwo equations (ΣFx = 0, ΣFy = 0); up to two unknowns — same as 2D
BThree equations (ΣFx = 0, ΣFy = 0, ΣFz = 0); up to three unknowns
CSix equations (three force, three moment); up to six unknowns
DThree equations, but only two are independent because ΣF = 0 in vector form is one equation
The vector equation ΣF = 0 in 3D breaks into three independent scalar equations — one per Cartesian direction. This allows solving for at most three unknowns (e.g., three cable tensions). Six equations would apply to rigid body equilibrium in 3D, which also includes moment equations. Option D misunderstands vector equations: ΣF = 0 in 3D is equivalent to three separate scalar equations.
Question 3 True / False
If all the forces acting on a particle in a 3D problem happen to lie entirely in the x-y plane, then the ΣFz = 0 equation is automatically satisfied (0 = 0) and provides no useful information.
TTrue
FFalse
Answer: True
When all forces are coplanar in x-y, every force has zero z-component. Summing zeros gives 0 = 0, which is trivially true and imposes no constraint. This is exactly the 2D special case embedded in 3D. You can use this as a check: if your 3D setup is correct, collapsing to 2D should reproduce your 2D results exactly — including a trivial ΣFz = 0.
Question 4 True / False
In 3D particle equilibrium, you can find the unknown tension in a cable directly from the geometry without computing a unit vector, as long as you know the cable's length.
TTrue
FFalse
Answer: False
Knowing the cable length gives you the magnitude of the position vector, but you still need to divide by that magnitude to obtain the unit vector in order to decompose the tension into Cartesian components. The unit vector (and hence the direction cosines) is essential to writing the force in i, j, k form. There is no shortcut: tension × unit vector is the only systematic way to express a cable force as Cartesian components for equilibrium equations.
Question 5 Short Answer
Describe the systematic procedure for expressing a cable force in 3D Cartesian form given the coordinates of the two endpoints of the cable.
Think about your answer, then reveal below.
Model answer: Step 1: Compute the position vector r from the particle to the anchor point: r = (Bx−Ax)i + (By−Ay)j + (Bz−Az)k. Step 2: Find its magnitude: |r| = √(Δx² + Δy² + Δz²). Step 3: Divide to get the unit vector: û = r / |r|. Step 4: Multiply by the tension magnitude: F = T · û. This gives the three Cartesian components (Tx, Ty, Tz) ready to substitute into ΣFx = 0, ΣFy = 0, ΣFz = 0.
The procedure is the same every time regardless of geometry. The key insight is that the force direction is entirely encoded in the geometry of where the cable goes — you don't need angles or trigonometry separately. Once all forces in the problem are in Cartesian form, equilibrium reduces to solving a system of linear equations. Most errors in 3D statics happen in steps 1–3, not in the algebra.