ENG1011 Chap.1 Units, Dimensions and Resultant Forces
Units, Dimensions and Resultant Forces
Units, Dimensions and Resultant Forces is the Week 1 starting point of Engineering Methods. Every later topic in the unit multiplies and divides physical quantities, so the first habit is to carry SI units through each line and to check that the terms of an equation share the same dimensions.
A moment equation with one term missing its distance, or a stress mixing metres with millimetres, fails that check before any number is substituted. The second half of the chapter treats forces as vectors. Two forces acting at one point can be added with the parallelogram rule, which in numbers becomes the cosine rule using the angle between the two arrows.
Three or more forces are easier to handle by resolving each into x and y components, summing the components, and rebuilding the magnitude and direction of the resultant.
The unit asks for both the graphical idea and the calculation, and its auto-marked tests expect plain numerical answers without units, so the working has to carry the units instead.
In the Content Test this material appears as short numerical items: an angle or length from geometry, and a resultant found by resolving forces.
The practice test solutions show both a cosine rule route and a component route for the same answer, which is a useful model: solve once, then confirm with the other method. Week 1 also sets the habit of writing working on paper even when the answer box is auto-marked, because the written working is what supports a method mark if a typed value is slightly off.
What this chapter covers
- 01
SI base units and derived units for force, moment, stress and stiffness
- 02
Dimensional homogeneity as a check on any equation
- 03
Working in newtons and millimetres so stresses come out in megapascals
- 04
Forces as vectors with magnitude, direction and point of application
- 05
The parallelogram rule and the cosine rule for two forces
- 06
Converting angles measured from different reference lines
- 07
Resolving several forces into x and y components
- 08
Rebuilding the resultant's magnitude and direction from component sums
Worked example · free
Resultant of two forces at sixty degrees
- 1The included angle between the two forces is 60 degrees, so the cosine rule gives FR squared = 250 squared + 400 squared + 2(250)(400)cos 60 = 62,500 + 160,000 + 100,000 = 322,500.
- 1Take the square root: FR = 567.9 N.
- 1Direction from components: Rx = 250 + 400 cos 60 = 450 N and Ry = 400 sin 60 = 346.4 N, so the angle is the inverse tangent of 346.4/450, which is 37.6 degrees above the x axis.
Key terms
- Dimensional homogeneity
- The property of an equation in which every added term has the same physical dimensions.
- Newton
- The SI unit of force, equal to one kilogram metre per second squared.
- Included angle
- The angle between two force vectors drawn tail to tail from the same point.
- Force component
- The part of a force acting along one chosen axis, found by multiplying its magnitude by a sine or cosine.
- Parallelogram rule
- A graphical method that adds two forces by completing the parallelogram on them and taking its diagonal.
Units, Dimensions and Resultant Forces FAQ
Why does a dimension check matter in a statics question?
Every equilibrium equation adds terms together, and terms can only be added when they share dimensions. A moment sum in which one term lacks a distance, or a stress built from mixed metres and millimetres, is wrong before any arithmetic, so a ten second check catches it early.
Should I use the cosine rule or components for a resultant?
The cosine rule is quickest for exactly two forces when the angle between them is known. With three or more forces, or when the direction is also needed, resolving into x and y components is faster and leaves a clearer trail for the marker.
Do I include units in an online answer box?
No. The unit's practice solutions ask for plain numbers such as 5.29 rather than 5.29 m, and the test instructions reject units, commas and scientific notation. Keep the units in your written working and type only the number.
How should angles given from different references be handled?
Convert every angle to one reference before using it. A rope described as 60 degrees below the horizontal is 30 degrees from the vertical, so combining it with a rope 25 degrees from the vertical on the other side gives an included angle of 55 degrees.
Exam move
Practise this chapter in two short drills. First, take any formula you meet later in the unit, such as a deflection or buckling formula, and confirm its dimensions in one line. Second, set yourself three forces at awkward angles and find the resultant twice, once with the cosine rule in pairs and once by components; agreement between the two is your check.
Always sketch the resultant on the axes so its quadrant matches the signs of your component sums. Finish each practice set by redoing one resultant with a scaled sketch, head to tail, to confirm both the size and the direction by eye.
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