An Introduction to Basic Mechanics

1. Introduction

   1. A Definition of Mechanics

      Mechanics is a branch of physical science in which rest and motion of physical bodies are studied together with

      causes and conditions affecting rest and motion of physical bodies.

   2. Branches of Mechanics

      In the above definition there is room for:

      1. Studying physical bodies at rest and conditions for their rest, which is called statics

      2. Studying physical bodies in motion, called dynamics

      3. Studying motion or rest of fluids, naturally named fluid mechanics

      4. Studying motion or rest of solids, called rigid body mechanics

      5. Studying causes for motion (forces) of physical bodies, called kinetics

      6. Studying the motion of a rigid solid in space, therefore dealing with location, path, speed and their changed over time, which is named kinematics

      It should be added that solid mechanics (also called strength of materials) which deals with how the non-rigid solids deform when subjected to forces, can also be included as a branch of mechanics.

   3. Origins of Mechanics

      Mechanics can be traced back to the works of Aristotle (384-322 B.C.) and Archimedes (287-212 B.C.). Works of Kepler (1571-1630 A.D.) and Galileo (1564-1642 A.D.) on motion of sun and planets in the solar system paved the way for Newton's (1642–1727 A.D.) definitive formulation of laws of mechanics as we know them today.

2. Basics Principles of Vector Mechanics

   1. Vectors and Scalars

      There some physical quantities such as time and mass which can be fully described by a single value. These are called scalars.There many other physical quantities that need more than one value to be fully described. For example the displacement of an object can be described with two points: from origin to destination. To put it simply, vector is a geometric construct that the quantities which need more than one value to be defined and the number of values needed is referred to as the dimension of the vector. Geometrically, a vector can be defined with two point, and be represented by an arrow in the direction of origin to destination. Thus in a plane, vectors are two dimensional and in space three dimensional (2D and 3D vectors).

      The followings are equivalent ways to define a vector in a plane (two dimensional vector):

      • In a Cartesian coordinate system (x-y plane), as an arrow from the origin to a point A with coordinates x and y

      • As an ordered pair of numbers (x, y), in which x and y are called components

      • A size (magnitude or modulus) and a direction. The direction is normally an angle in radians, also called as argument

      In vector mechanics, to differentiate between vectors and scalars, an arrow is placed above the letter used to represent a vector, like $\overrightarrow{F}$ or using bold face $\mathbf{F}$ . Components of that vector are usually written with subscripts as $F_{\mathrm{<mi\; mathsize="150\%">x</mi>}}$  and $F_{\mathrm{<mi\; mathsize="150\%">y</mi>}}$  .

   2. Vectors Addition subtraction and Resolution

      • Geometrically, vectors are added using the parallelogram or (triangle) rule. From a point (origin) two vectors (arrows) are drawn (click and drag inside the blue box) In the interactive demo below, if two vectors are drawn (see instructions) and resultant button is pressed, the parallelogram wil be drawn (please try). For each vector a row will be added to show x and y components of that vector. These x, y values can be changed and the parallelogram will be updated.

      • The sum or resultant of a two vectors can be calculated using their x and y components:

        calculated by adding their x components to give the value for x component of the resultant vector x and then similarly adding up the y components to give the y component of the resultant vector. This means that if $\overrightarrow{R}$ is the sum of $\overrightarrow{U}$ and $\overrightarrow{V}$, then: $R_{\mathrm{<mi\; mathsize="150\%">x</mi>}}=U_{\mathrm{<mi\; mathsize="150\%">x</mi>}}+V_{\mathrm{<mi\; mathsize="150\%">x</mi>}}$   and $R_{\mathrm{<mi\; mathsize="150\%">y</mi>}}=U_{\mathrm{<mi\; mathsize="150\%">y</mi>}}+V_{\mathrm{<mi\; mathsize="150\%">y\; </mi>}}$ .

      • Given the above definition for vector addition, it is easy to define vector subtraction as: if $\overrightarrow{R}=$ $\overrightarrow{U}-$ $\overrightarrow{V}$, then: $R_{\mathrm{<mi\; mathsize="150\%">x</mi>}}=U_{\mathrm{<mi\; mathsize="150\%">x</mi>}}-V_{\mathrm{<mi\; mathsize="150\%">x</mi>}}$   and $R_{\mathrm{<mi\; mathsize="150\%">y</mi>}}=U_{\mathrm{<mi\; mathsize="150\%">y</mi>}}-V_{\mathrm{<mi\; mathsize="150\%">y\; </mi>}}$

   3. Displacement, Velocity and Acceleration Vectors

      1. Displacement

         As described above displacement is a clear example of a vector quantity.

      2. Velocity

         Average velocity is the displacement vector divided by the time. Instantaneous velocity is the time rate of change of the position vector at a certain point in time.

      3. Acceleration

         Average acceleration is the change in velocity divided by time. Instantaneous acceleration is the time rate of change of velocity at a certain instance in time.

   4. Force as a Vector

      As said above, mechanics studies the causes of motion. We know that pulling or pushing tend to cause or force an object to move. Therefore, forces exist as pulls and pushes. Of course, things can be pulled or pushed in all directions with different intensities. Therefore forces need to be defined as vectors.

   5. Newton's Laws of Motion

      1. First Law (Inertia)

         Newton's first law states that: "Every object in a state of uniform motion, will remain in that state of motion until an external force is applied to it". Here uniform motion means no change in the direction and speed of the motion, i.e. moving along a straight line (rectilinear) with a constant speed.

      2. Second Law (Momentum)

         Newton's first law says that external force is the cause of changes in state of motion of objects, and the secod law quantifies their relationship.

         It says that: "The external force applied to a body is equal to the time rate of change in its motion (acceleration) multiplied by its mass".

         Newton's second law is perhaps best expressed mathematically as the formula: $\overrightarrow{F}=m\overrightarrow{a}$ , where $\overrightarrow{F}$ is the force vector, $m$ is the mass of the object and $\overrightarrow{a}$ is the acceleration vector.

      3. Third Law (reaction)

         Newton's third law of motion states that: "to every action, there is an equally opposed reaction". It means that if a body exerts a force (action) on another body, the acted upon body will also exert a force (reaction) on the actor body with the same size, but in opposite direction.

3. An Introduction to Particle Mechanics

   1. Physical Bodies as Particles

      In many cases in mechanics, the geometry and dimensions of the body is not of concern and the shape and size of the body cam be ignored, i.e. as if it were a mere particle. These are cases where motion of the parts of the body relative to each other are of no significance. An example would be studying motion of a car on a road.

      This of course has nothing to do with the field of particle physics which studies elementary and sub-atomic particles.

   2. Free-Body Diagram

      Free-Body Diagram (FBD) is graphic tool for solving problems is vector mechanics. FBD is a simple sketch of the isolated body, on which force vectors are shown as arrows. Usually a coordinate system (xy directions) are chosen and the force vectors are resolved into their components along chosen x and y directions. This reduces additions of force vectors to simply adding up the x and then y components. The relevant law of motion (Newton's first law in case of equilibrium or uniform motion and Newton's second law in other cases) will then be used to calculate the unknowns. FBD can be applied to non-particle (e.g. rigid bodies) as well. A FBD can be drawn using the interactive particle mechanics tool below.

   3. Statics (equilibrium) of a Particle

      Statics was introduced earlier. Newton's first law implies that the there should be no resultant external force acting on a body at rest, because there is no change in the state of motion. This means that the forces acting on the body should have a sum of zero: $\sum {\overrightarrow{F}}_i=\overrightarrow{0}$ or, ${\overrightarrow{F}}_1+{\overrightarrow{F}}_2+{\overrightarrow{F}}_3+\mathrm{....}=\overrightarrow{0}$ .

   4. Kinetics of a Particle

      Kinetics of a particle is basically the application of Newton's second law to determine how the resultant of forces acting on a particle causes changes in its state of motion.

4. An Introduction to Rigid Body Mechanics

   In contrast to the particle mechanics, when the body has a fixed shape and motion of parts of a body relative to its other parts are important, we have a case of rigid body mechanics. For a fixed shape or rigid body, the distance between any two points is fixed, and therefore the only relative motion between the points on a rigid body is rotation. Hence, when the rotation of an object around one of its point is important, the rigid body mechanics needs to be applied. In a plane, any motion of a rigid body at any given time is a combination of: (1) a rotation around one of its points called (instantaneous) centre of rotation, plus (2) a translation which is the common motion of all points of the body.

   1. Moment of a Force (Torque)

      If an object has an axis or a pivot, around which it can rotate, the application of a force, the body tends to rotate around its pivot. Common examples are lever and seesaw. This rotation effect depends on the (perpendicular) distance of the force vector from the pivot, the farther the distance the stronger this rotation effect. In general, the product of a force and its normal distance from a point is called the moment or Torque of that force. In vector terms: $\overrightarrow{\tau }=\overrightarrow{r}\times \overrightarrow{F}$ where $\overrightarrow{\tau }$ is the torque (moment) vector, $\overrightarrow{F}$ the force vector and $\overrightarrow{r}$ a displacement vector from (e.g. the point of application of) $\overrightarrow{F}$ to the pivot. Please try the interactice rigid body demo below.

   2. Couple

      When two parallel forces of the same size but opposite directions are applied to an object, the only effect they will have is a rotation. This is because: (a) the sum of forces is zero and therefore Newton's first law guarantees that no translation motion occurs, and (b) for any point on the object there will be a torque of: $\tau =Fd$, where $F$ is the magnitude (size) of the force vector and $d$ is the normal distance between the two force vectors.

   3. Statics of a Rigid Body

      For equilibrium of a rigid body, in addition to the condition of static equilibrium for a particle, there needs to be an equilibrium of moments (torques) too: $\sum \overrightarrow{\tau }=\overrightarrow{0}$ , which for a rigid body in plane, will be reduced to: $\sum \tau =0$ .

      This means that the sum of moments around each point (axis) must be zero.

Interactive Rigid Body Demo

In the light yellow box below, there is a simple interactive tool to help understand rotation and torque. It is similar to the particle mechanics interactive tool in the blue box above. The difference here is the pivot button. When clicked, it will place a pivot at the centre of the rectangle marked with a small blue circle. You can then change the location of the pivot, by clicking on any point on the rectangle. When you chose the location of the pivot, you need to click the pivot button again (which is now turned red to show it needs to be clicked again) and the button will return to normal colour. Now that the pivot is added to the rectangle, if you click of the rectangle and move (drag) the mouse, forces will be drawn and added to the rectangle. After you click start button the simulation will start and if there is a pivot on the rectangle it can only rotate around the pivot of forces produce a torque. If there is no pivot, the rectangle can be displaced according to Newton's second law.

Please make sure to place pivot before applying the forces, otherwise the pivot will be ignored.