48610 Chap.8 Force Vectors and Static Equilibrium
Force Vectors and Static Equilibrium
From Week 6 the subject asks whether the structure you have drawn will actually hold. The answer comes from one idea: a body that is not accelerating has forces that cancel, both across the page and up it. Getting there requires treating force as a vector with a magnitude, a direction and a point of application, and resolving it onto whichever pair of perpendicular axes makes the question easy.
The Week 6 exercise asks for one force resolved four ways, along a member, across it, into horizontal and vertical components, and as a vector in unit vector form, because they are the same force and which decomposition is useful depends entirely on what you are about to ask.
Equilibrium of a particle, meaning a body small enough that all the forces pass through one point, is then two scalar equations, so at most two unknowns can be solved from one joint.
Counting those unknowns before starting does most of the planning. Two physical facts do the rest: a cable or cord can only pull, so a negative tension means an arrow drawn the wrong way, and a spring force comes from the extension, the current length minus the unstretched length, which is why every spring problem states the unstretched length.
The subject also publishes an explicit list of what mechanics working must contain, from arrowheads and labels to declared positive directions and units on every intermediate answer, and applies it to both the quiz and the project report.
What this chapter covers
- 01
Force as a vector: magnitude, direction and point of application
- 02
Resolving onto member axes rather than page axes
- 03
Horizontal and vertical components, and unit vector form
- 04
Why components only reconstruct a force on perpendicular axes
- 05
Equilibrium of a particle as two scalar equations
- 06
Counting unknowns before writing anything
- 07
Cables pull only, and act along their own line
- 08
Two force members and the direction you get for free
- 09
Spring force from the extension, not the length
- 10
Declaring positive directions, and reading a negative answer
- 11
The published working and free body diagram requirements
Two cable tensions from one known load
- +1Isolate the ring and declare the axes: x positive to the right, y positive upward. The three forces are the known 4.00 kN load, the horizontal tension pulling in the negative x direction, and the inclined tension pulling up and to the left at 50 degrees.
- +1Write the vertical equation first, because it contains only one unknown. The load contributes minus 4.00 sin 30, the inclined cable contributes plus its tension times sin 50, and the horizontal cable contributes nothing.
- +1Solve it. The inclined tension is 4.00 times 0.5000 divided by 0.7660, which is 2.611 kN.
- +1Now the horizontal equation, which has one unknown left. The horizontal tension equals 4.00 cos 30 minus 2.611 cos 50, that is 3.464 minus 1.678, giving 1.786 kN. Check both in the equation not used to find them, and note both are positive, so both cables are in tension as a cable must be.
Key terms
- Resultant
- The single force equivalent to a set of forces acting together. A body is in equilibrium when the resultant of all forces on it is zero, whether it is at rest or moving at constant velocity.
- Component
- The part of a force acting along a chosen direction. Components reconstruct the original force exactly only when the two chosen directions are perpendicular to each other.
- Unit vector form
- A statement of a force as its x component times i plus its y component times j, which records magnitude, direction and sign unambiguously in one expression.
- Particle equilibrium
- The condition for a body whose forces all pass through a single point: the sums of the force components in two perpendicular directions are each zero, giving two equations.
- Extension
- The current length of a spring minus its unstretched length. Spring force is stiffness multiplied by extension, so the unstretched length is given in every spring problem for a reason.
- Two force member
- A member loaded at only two points and carrying nothing in between. Its force must act along the line joining those points, so its direction is known from the geometry alone.
- Force triangle
- Three forces in equilibrium drawn head to tail as a closed triangle, solvable with the sine rule. Drawing one must be justified by writing the equilibrium condition first.
Force Vectors and Static Equilibrium FAQ
How do I decide which axes to resolve onto?
By asking what you want the answer to tell you. Resolving along and across a member hands you the tension or compression in it directly, plus the component that bends it. Resolving horizontally and vertically is better when several forces act at different angles and need adding, because weight always lies in the vertical sum.
Resolving along and across an inclined plane gives the friction and normal directions without extra work. The axes are a choice, and choosing well is most of the skill.
Why does a negative answer not mean I made a mistake?
Because a component is positive or negative only relative to the convention you declared. A negative result means the force acts opposite to the direction you assumed, which is information rather than an error, and stating it is part of the answer. The exception is a cable: a cable cannot push, so a negative tension does mean the arrow was drawn wrong or an angle was measured from the wrong line.
What is the difference between using the length and the extension of a spring?
The whole answer. Spring force is stiffness times extension, and the extension is the current length minus the unstretched length. Using the current length instead typically inflates the force by a factor of three or four and produces an answer that looks plausible if you are not checking magnitudes.
Every spring problem in this subject states the unstretched length precisely because the question is testing whether you subtract it.
How many unknowns can one free body diagram carry?
Two for a particle, because there are two equations. Three unknown member forces at a single joint cannot be solved from that joint alone, which means either a known quantity has been missed or you need to start somewhere else. Counting first turns a stuck problem into a decision about where to begin, and spotting a two force member often reduces three unknowns to two by fixing one direction.
Assessment move
Work problems with the solution covered and compare line by line rather than checking only the final number, because the subject's marking notes ask for the diagram, the labels, the declared direction and the visible substitution as much as for the answer. Build two reflexes deliberately.
First, draw the right angled triangle on the diagram and mark the angle before choosing sine or cosine, because the side adjacent to the angle takes the cosine every time. Second, before solving, count the unknowns and find the equation containing only one of them. Then check each answer in the equation you did not use to find it.
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