Modelling in Mechanics Mechanics Year 1 ¡ Chapter 8 ¡ Side 1 of 2
Mathematical models
A mathematical model is a simplified description of a real situation, used to make predictions.
A modelling assumption simplifies the situation so that the mathematics can be done (for example, ignoring air resistance). A model is improved, or refined, by changing its assumptions.
Modelling assumptions
| Word | What it means | What it means for the working |
|---|---|---|
| Particle | Its dimensions are negligible, so its mass acts at a single point. | Ignore rotation (spin) and air resistance; all the forces act at one point. |
| Rod | Long, thin and rigid: every dimension except its length is negligible. | It does not bend; its mass is along a line. |
| Lamina | A flat object whose thickness is negligible. | Its mass is spread over a flat area. |
| Uniform | Its mass is spread evenly (non-uniform: not evenly). | Its weight acts at its geometrical centre (a rod: its midpoint). |
| Light | Its mass is negligible (a string, a rod, a pulley). | Its weight is ignored; the tension is the same throughout a light string. |
| Inextensible | It does not stretch. | Particles joined by it have the same magnitude of acceleration. |
| Smooth surface | There is no friction. | The only force from the surface is the normal reaction. |
| Rough surface | Objects in contact with it experience friction. | Friction acts along the surface, opposing the motion. |
| Smooth pulley | No friction between the pulley and the string. | The tension is the same on both sides of the pulley. |
| Bead | A particle with a hole in it, threaded on a wire (a rigid, thin length of metal) or a string. | It moves along the wire (no friction if the wire is smooth). |
| Peg | A fixed support from which a body can hang or on which it can rest. | It has no dimensions; it can be rough or smooth. |
| Air resistance negligible | The resistance of the air is ignored. | The only force on a falling object is its weight, so its acceleration is $g$. |
| Gravity | The acceleration due to gravity is constant. | $g = 9.8\text{ m s}^{-2}$ (a question may say $10\text{ m s}^{-2}$). |
| Equilibrium | The resultant force is zero. | At rest, or moving with constant velocity. |
Using a model
- Substitute into the model: e.g. the diver $h = 10 + 4t - 4.9t^2$: when $t = 1$, $h = 9.1$ m.
- Interpret each number: put $t = 0$; here $10$ is the height of the board (m) and $4$ is the speed (m sâťÂš) she leaves it with, upwards.
- Solve: she reaches the water when $h = 0$; reject the negative solution of the quadratic.
- Greatest value of a quadratic: on its line of symmetry (or complete the square).
- A model is only valid for a range of values, e.g. $0 \le t \le 1.89$, until something changes (it lands, stops or enters the water). A negative height, or a car âmoving backwardsâ, shows it is no longer valid.
Quantities, Units and Vectors Mechanics Year 1 ¡ Chapter 8 ¡ Side 2 of 2
SI units
| Quantity | SI unit |
|---|---|
| mass | kilogram (kg) ¡ base unit |
| length, displacement | metre (m) ¡ base unit |
| time | second (s) ¡ base unit |
| speed, velocity | $\text{m s}^{-1}$ |
| acceleration | $\text{m s}^{-2}$ |
| force, weight | newton (N), $1\text{ N} = 1\text{ kg m s}^{-2}$ |
One newton is the force that gives a mass of $1$ kg an acceleration of $1\text{ m s}^{-2}$.
Weight is the force due to gravity: $W = mg$. A mass of $5$ kg has weight $5 \times 9.8 = 49$ N. Mass is in kg; weight is in N.
Scalars and vectors
A scalar has magnitude (size) only. A vector has magnitude and direction.
| Scalar | Vector |
|---|---|
| distance | displacement |
| speed | velocity |
| mass | weight (it always acts downwards, but that is still a direction) |
| time | acceleration, force |
Speed is the magnitude of velocity; distance is the magnitude of displacement.
Vectors in $\mathbf{i}$, $\mathbf{j}$ form
$\mathbf{i}$ and $\mathbf{j}$ are unit vectors (magnitude $1$) in the directions of the $x$ and $y$ axes, perpendicular to each other.
Magnitude of $p\mathbf{i} + q\mathbf{j}$: $\sqrt{p^2 + q^2}$ by Pythagorasâ theorem. For a velocity this is the speed: $(3\mathbf{i} - 4\mathbf{j})\text{ m s}^{-1}$ has speed $\sqrt{3^2 + (-4)^2} = 5\text{ m s}^{-1}$.
Resultant force: add the forces, e.g. $(2\mathbf{i} + 5\mathbf{j}) + (4\mathbf{i} - 3\mathbf{j}) = (6\mathbf{i} + 2\mathbf{j})$ N.
Constant velocity: displacement $=$ velocity $\times$ time; position $\mathbf{r} = \mathbf{r}_0 + \mathbf{v}t$.
From speed and direction: speed $v$ at $\theta$ above $\mathbf{i}$ gives $v\cos\theta\,\mathbf{i} + v\sin\theta\,\mathbf{j}$; put in the signs for the quadrant.
The angle with $\mathbf{i}$, $\mathbf{j}$, $-\mathbf{i}$ or $-\mathbf{j}$: always draw a diagram. âThe angle with $\mathbf{i}$â means with the positive $\mathbf{i}$ direction, and the answer is between $0^\circ$ and $180^\circ$.
$\alpha = \tan^{-1}\left(\dfrac{4}{3}\right) = 53.1^\circ$, the angle with $-\mathbf{i}$.
The angle with $\mathbf{i}$ is $\theta = 180^\circ - 53.1^\circ = 126.9^\circ$.
With $\mathbf{j}$ or $-\mathbf{j}$, use $\tan^{-1}\left(\dfrac{|p|}{|q|}\right)$ in the same way.
Improving a model
- Compare the predictions with real data. Is the answer reasonable?
- Find the assumption that causes the difference, and change it:
- include air resistance (a ball goes less high, a skydiver reaches a steady speed);
- model a surface as rough, not smooth (include friction);
- model an object as a rigid body, not a particle (spin, turning);
- model a rod as non-uniform, or a string as extensible.
- Say how the change moves the prediction (higher, lower, longerâŚ).
- A refined model is more realistic; the mathematics is usually harder.