- A-Level Maths

# Modelling in Mechanics

The real world of mechanics is incredibly complicated, with hundreds of different factors affecting motion and stability. This would be near impossible to calculate at this stage, so to simplify it we model objects and scenarios in a number of different ways.

## Modelling Assumptions

### Particle

Has negligible dimensions

Mass acts about a single point

Rotational forces and air resistance can be ignored

### Rod

The diameter is negligible (so no thickness), only length counts

Mass acts along a line

It is rigid, so does not bend or buckle

### Lamina

An object with only area, thickness is negligible (like a sheet of paper)

Mass acts across the flat surface

### Uniform Body

The Mass is distributed evenly throughout the object

The mass is modelled to act through one point, the

**centre of mass**, in the geometrical middle of the body

### Light Object

The mass of the object is negligible

Often used for strings or pulleys

Allows us to model the tension on each end of a string as equal

### Inextensible String

A string that does not stretch when a force is applied

Allows us to model the acceleration of two connected particles as the same

### Smooth Surface

There is no friction between the surface and any object on it

### Rough Surface

There is friction between the surface and any object on it

### Wire

A rigid length of metal with negligible thickness (only modelled with length)

### Smooth and Light Pulley

The pulley has no mass

No friction in the pulley, so tension is the same either side of it

### Bead

A particle with a hole in it for string/wire to be threaded through

It can move freely along the wire or string

### Peg

An object from which a body can be hung or rested

Modelled as dimensionless and fixed

Can be either rough or smooth

### Air Resistance

Resistance forces due to motion through air

Almost always modelled as negligible

### Gravity

The force of attraction between all objects

Acceleration due to gravity is given by

*g*, which is 9.8 m/s² unless otherwise statedAssume objects are always attracted to the ground (earth)

Gravity acts uniformly and downwards

## Quantities & Units

Quantities can be either **scalar** or **vector**.

**Scalar**quantities have**only magnitude****Vector**quantities have**both magnitude and direction**

### SI Base Units

**Quantity **Unit Symbol

**Mass** kilogram kg

**Distance** metre m

**Time ** seconds s

Distance is scalar - the vector version of it is displacement

### Derived Units

**Quantity **Unit Symbol

**Velocity** metres per second m/s

**Acceleration** metres per second² m/s²

**Force **Newtons N

Velocity is the vector form of speed, as velocity has direction as well as magnitude

**Weight is not the same as mass. **Weight is a force, measured in Newtons, N. It is given by:

W = mg Weight = mass x gravity

## Representing Forces

Weight acts down from an object's **centre of gravity**. Other forces can act in any direction, and so to avoid confusion it is best to draw a force diagram:

The length of arrow should represent the magnitude of the force. Single headed arrows represent forces.

If a system is in **equilibrium **- meaning there is no net resultant force and it is motionless or at constant velocity - then all the force arrows will make a closed shape. The example above would give a triangle.