1.1 Scope of mechanics, idealisations (particle, rigid body, continuum) and SI units
After this lesson you should be able to:
- Say what statics, dynamics and the mechanics of deformable bodies each deal with.
- Choose between the particle, rigid-body and continuum idealisations for a given problem and justify the choice.
- Work in SI base and derived units, convert loads between mass and force, and check equations for dimensional consistency.
Theory
Engineering mechanics is the study of forces and the motion (or absence of motion) they cause. Statics deals with bodies at rest or moving at constant velocity, where the net force and net moment are zero. Dynamics deals with accelerating bodies. The mechanics of deformable bodies, taught as strength of materials and structural analysis, asks how forces stretch, bend and twist members. Every later subject in this textbook, from beam design to soil pressure on a wall, rests on the ideas of this chapter.
Real objects are complicated, so mechanics replaces them with models that keep only what matters for the question asked. A particle is a body whose size does not affect the answer: all forces are taken to act at one point and rotation is ignored. A rigid body has size and shape, so forces can turn it, but the distances between its points never change. A continuum is a deformable body treated as matter spread continuously through its volume, so that stress and strain can be defined at every point even though real materials are made of grains, crystals or fibres.
The same object can need different models. A loaded truck crossing a 40 m bridge can be a single moving load when we want the largest bending moment in a long girder, a set of axle loads when we design a short deck slab, and a vibrating mass-spring system when we study ride comfort. The model is a decision, and it should be stated at the start of every calculation.
Newton's laws are the foundation. A body stays at rest or in uniform straight-line motion unless a net force acts on it; the net force equals mass times acceleration; and the forces two bodies exert on each other are equal, opposite and collinear. Statics is the special case of the second law with zero acceleration, which is why equilibrium equations are simply sums of forces and moments set to zero.
Units carry physical meaning. The SI system has seven base units, of which mechanics uses the metre, the kilogram and the second. The newton is derived: 1 N is the force that gives 1 kg an acceleration of 1 m/s², so 1 N = 1 kg·m/s². Weight is a force, W = m g, where the standard acceleration of gravity is defined as 9.806 65 m/s²; engineering calculations normally use 9.81 m/s². Civil engineers work mostly in kN, kN·m, kN/m (line loads), kN/m² (area loads) and N/mm² (stresses), where 1 N/mm² = 1 MPa = 1 000 kN/m².
Derivation
- Start from Newton's second law for a particle: ΣF = m a.
- For a body at rest or moving at constant velocity, a = 0, so ΣF = 0. In a plane this gives two scalar equations, ΣFx = 0 and ΣFy = 0.
- For a rigid body, the forces may also cause rotation. Summing the moments of all forces about any point O and setting the angular acceleration to zero gives ΣM_O = 0, a third equation in two dimensions.
- Dimensional check of W = m g: [kg]·[m/s²] = kg·m/s² = N. Every term of a valid equation must reduce to the same dimensions; a mismatch signals an error before any number is substituted.
W = m g; W in N, m in kg, g = 9.81 m/s² (standard value 9.806 65 m/s²)1 N = 1 kg·m/s²; 1 kN = 1 000 N1 MPa = 1 N/mm² = 10⁶ N/m² = 1 000 kN/m²Self-weight of a prismatic member per metre: w = γ A; w in kN/m, γ in kN/m³, A in m²Equilibrium (2D): ΣFx = 0, ΣFy = 0, ΣM_O = 0
- Newtonian mechanics: speeds far below the speed of light and bodies far larger than atoms.
- A particle model is valid only when the body's size and rotation do not affect the result asked for.
- A rigid-body model gives external reactions correctly for statically determinate structures, but it cannot give deflections or the forces in statically indeterminate structures; those need the continuum model.
- The continuum model fails at scales comparable to the grain size of the material (for example, individual aggregate particles in concrete).
Codes and standards
Indian Standards are cited by number and clause only; no clause text, tables or figures are reproduced. Other standards are listed for comparison, not ranked.
- No design code governs the content of this lesson; it is pure mechanics.
- Units: the International System of Units as defined in the BIPM SI Brochure (9th edition, 2019) and its US version NIST Special Publication 330 (2019 edition); the ISO 80000 series covers quantities and units internationally.
- In Indian practice, the dead loads (unit weights of materials) and imposed loads that feed these calculations come from IS 875 (Part 1):1987 and IS 875 (Part 2):1987. Other countries use EN 1991-1-1 and ASCE 7 for the same purpose; they are listed for comparison, not ranked.
Worked examples
Problem. A precast concrete beam is 300 mm wide, 600 mm deep and 8 m long. Take the unit weight of reinforced concrete as 25 kN/m³. Find its self-weight per metre, its total weight and its mass.
- Cross-section area A = 0.300 × 0.600 = 0.180 m².
- Self-weight per metre w = γ A = 25 × 0.180 = 4.5 kN/m.
- Total weight W = w L = 4.5 × 8 = 36 kN.
- Mass m = W / g = 36 000 / 9.81 = 3 669.7 kg, about 3.67 tonnes.
Answer. 4.5 kN/m; 36 kN; about 3 670 kg
Problem. A crane must lift a 2.5 tonne bundle of reinforcement. A site engineer also needs to know the load on a 4 m × 6 m slab panel that will carry an imposed load of 2.5 kN/m² during construction. Find both forces in kN.
- Weight of the bundle W = m g = 2 500 × 9.81 = 24 525 N = 24.5 kN.
- Slab panel area = 4 × 6 = 24 m².
- Total imposed load = 2.5 × 24 = 60 kN.
- Note the trap: "2.5 tonne" is a mass and "2.5 kN/m²" is a pressure; the numbers are equal but the quantities are not.
Answer. Crane load 24.5 kN; slab panel load 60 kN
Step-by-step procedure
- Read the question and decide what is asked: a force, a reaction, a motion or a deformation.
- Choose the idealisation (particle, rigid body, continuum) and write it down with the reason.
- List all given quantities with their units, converting to N or kN, m and s before substituting.
- Write the governing equation in symbols and check its dimensions.
- Substitute, compute and round only at the end; state the answer with units and a sanity check.
Site notes
- Site documents mix units: crane charts in tonnes, drawings in mm, loads in kN. Convert everything to one system before calculating.
- A "tonne" on a crane chart is a mass. Multiply by about 9.81 to get kN; using 10 is a quick, slightly conservative check.
- When unit weights are needed, take them from the relevant code (IS 875 (Part 1) in India) rather than from memory, and record the value used in the calculation sheet.
- Treating kilograms as a force: a 50 kg bag of cement weighs about 490 N, not 50 N.
- Mixing mm and m in the same formula, giving answers out by factors of 1 000 or 10⁶.
- Using a particle model when the problem involves moments, such as a beam on two supports.
- Forgetting that 1 MPa equals 1 000 kN/m², not 1 kN/m².
- Rounding intermediate values too early and losing accuracy in the final answer.
A quick check for weight: 1 tonne of mass weighs about 10 kN (9.81 kN exactly with g = 9.81 m/s²). A cubic metre of reinforced concrete weighs about 25 kN, of brick masonry about 19–20 kN and of water 9.81 kN. If a self-weight comes out wildly different, look for a unit slip.
Summary
- Mechanics is split into statics, dynamics and the mechanics of deformable bodies.
- Particle, rigid body and continuum are models chosen to suit the question.
- Equilibrium is Newton's second law with zero acceleration: ΣF = 0 and ΣM = 0.
- The newton is a derived unit, 1 kg·m/s²; weight W = m g.
- Check units and dimensions in every equation.
Practice questions
Tap a question to see the answer.
1What is the weight of a 2 tonne precast unit in kN? (g = 9.81 m/s²)
W = 2 000 × 9.81 = 19 620 N = 19.62 kN.
2Express 1 MPa in kN/m².
1 MPa = 10⁶ N/m² = 1 000 kN/m².
3A vehicle travels at 36 km/h. Express this in m/s.
36 × 1 000 / 3 600 = 10 m/s.
4A load hangs from a crane hook by a single cable. Which idealisation suits finding the cable tension, and why?
A particle: all forces (weight and cable tension) pass through one point and rotation does not matter.
5A 230 mm thick brick wall is 3 m high. Taking a unit weight of 20 kN/m³, find the line load it puts on the beam below.
w = 0.23 × 3 × 20 = 13.8 kN/m.
6Show that F = m a is dimensionally consistent when F is in newtons.
kg × m/s² = kg·m/s², which is the definition of 1 N, so both sides have the same dimensions.
References
Standard texts, listed in no order of merit. Our explanations, examples and diagrams are original; nothing is copied from these books.
- F. P. Beer, E. R. Johnston, D. F. Mazurek, P. J. Cornwell, B. P. Self, Vector Mechanics for Engineers: Statics and Dynamics, McGraw Hill, 12th edition, 2019.
- R. C. Hibbeler, Engineering Mechanics: Statics, Pearson, 15th edition, 2022 (published 2021).
- J. L. Meriam, L. G. Kraige, J. N. Bolton, Engineering Mechanics: Statics, Wiley, 9th edition, 2018.
- S. Timoshenko, D. H. Young, J. V. Rao, S. Pati, Engineering Mechanics (in SI units), McGraw Hill Education (India), 5th edition, 2013.
- BIPM, The International System of Units (SI Brochure), 9th edition, 2019; US version NIST Special Publication 330, 2019 edition.