notes/mechanical/mmme1028_statistics_and_dynamics.md

---
author: Alvie Rahman
date: \today
tags:
- uni
- nottingham
- mmme1028
- maths
- statics
- dynamics
title: MMME1028 // Statics and Dynamics
---

# Lecture L1.1, L1.2

### Lecture L1.1 Exercises

Can be found [here](./lecture_exercises/mmme1028_l1.1_exercises_2021-09-30.pdf).

### Lecture L1.2 Exercises

Can be found here [here](./lecture_exercises/mmme1028_l1.2_exercises_2021-10-04.pdf)

## Newton's Laws

1.  Remains at constant velocity unless acted on by external force

2.  Sum of forces on body is equal to mass of body multiplied by
    acceleration

    > 1st Law is a special case of 2nd

3.  When one body exerts a force on another, 2nd body exerts force
    simultaneously of equal magnitude and opposite direction

## Equilibrium

-   Body is in equilibrium if sum of all forces and moments acting on
    body are 0

### Example

Determine force $F$ and $x$ so that the body is in equilibrium.

![](./images/vimscrot-2021-10-04T09:14:41,378027532+01:00.png)

1.  Check horizontal equilibrium

    $\sum{F_x} = 0$

2.  Check vertical equilibrium

    $\sum{F_y} = 8 - 8 + F = 0$

    $F = 2$

3.  Take moments about any point

    $\sum{M(A)} = 8\times{}2 - F(2+x) = 0$

    $F(2+x) = 16)$

    $x = 6$

## Free Body Diagrams

A free body diagram is a diagram of a single (free) body which shows all
the external forces acting on the body.

Where there are several bodies or subcomponents interacting as a complex
system, each body is drawn separately:

![](./images/vimscrot-2021-10-04T09:23:03,892292648+01:00.png)

## Friction

-   Arises between rough surfaces and always acts at right angles to the
    normal reaction force ($R$) in the direction to resist motion.
-   The maximum value of friction $F$ is $F_{max} = \mu{}R$, where
    $\mu{}$ is the friction coefficient
-   $F_{max}$ is also known as the point of slip

## Reactions at Supports

There are three kinds of supports frequently encountered in engineering
problems:

![](./images/vimscrot-2021-10-04T09:41:56,080077960+01:00.png)

## Principle of Force Transmissibility

A force can be move dalong line of action without affecting equilibrium
of the body which it acts on:

![](./images/vimscrot-2021-10-04T09:43:04,689667620+01:00.png)

This principle can be useful in determining moments.

## Two-Force Bodies

-   If a body has only 2 forces, then the forces must be collinear,
    equal, and opposite:

    ![](./images/vimscrot-2021-10-04T09:44:05,581697277+01:00.png)

    > The forces must be collinear so a moment is not created

## Three-Force Bodies

-   If a body in equilibrium has only three forces acting on it, then
    the lines of actions must go through one point:

    ![](./images/vimscrot-2021-10-04T09:55:59,773394306+01:00.png)

    > This is also to not create a moment

-   The forces must form a closed triangle ($\sum{F} = 0$)

## Naming Conventions

| Term                 | Meaning                                                  |
|----------------------|----------------------------------------------------------|
| light                | no mass                                                  |
| heavy                | body has mass                                            |
| smooth               | there is no friction                                     |
| rough                | contact has friction                                     |
| at the point of slip | one tangential reaction is $F_{max}$                     |
| roller               | a support only creating normal reaction                  |
| rigid pin            | a support only providing normal and tangential reactions |
| built-in             | a support proviting two reaction components and a moment |

## Tips to Solve (Difficult) Problems

1. Make good quality clear and big sketches
2. Label all forces, dimensions, relevant points
3. Explain and show your thought process---write complete equations
4. Follow standard conventions in equations and sketches
5. Solve everything symbolically (algebraicly) until the end
6. Check your answers make sense
7. Don't forget the units