Mechanics is the precise study of the motion of objects, the forces acting on them and more abstract concepts such as momentum and energy. You probably have an intuitive understanding of these concepts already. In this chapter we will learn how to use precise mathematical equations to support your intuition.

Mechanics is the part of physics that is most well understood. Starting from three simple principles known as Newton's laws, we can figure out pretty much everything about the motion of the objects around us. Newton's three laws are:

1. In the absence of external forces, objects will maintain their velocity and their direction of motion.
2. A force acting on an object causes an acceleration inversely proportional to the mass of the object: $\vec{F}=m\vec{a}$.
3. For each force $\vec{F}_{12}$ applied by Object 1 on Object 2,

there is an equal and opposite force $\vec{F}_{21}$ that Object 2 exerts on Object 1. This ability to express the laws of Nature as simple principles like the above is what makes Physics fascinating. Complicated phenomena can be broken down and understood in terms of simple theories.

The laws of physics are expressed in terms of mathematical equations. There are about twenty such equations (see the back of the book). In this chapter we will learn how to use each of these equations in order to solve physics problems.

To solve a physics problem is to obtain the equation of motion $x(t)$, which describes the position of the object as a function of time. Once you know $x(t)$, you can answer any question pertaining to the motion of the object. To find the initial position $x_i$ of the object, you simply plug $t=0$ into the equation of motion $x_i = x(0)$. To find the time(s) when the object reaches a distance of 20[m] from the origin, you simply solve for $t$ in $x(t)=20$[m]. Many of the problems on the mechanics final exam will be of this form so, if you know how to find $x(t)$, you will be in good shape to ace the exam.

In Chapter 2, we learned about the kinematics of objects moving in one dimension. More specifically, we showed how the process of integration over time (twice) can be used to obtain the position of a particle starting from the knowledge of its acceleration: \[ a(t) \ \ \ \overset{v_i+ \int\!dt }{\longrightarrow} \ \ \ v(t) \ \ \ \overset{x_i+ \int\!dt }{\longrightarrow} \ \ \ x(t). \] But how do we obtain the acceleration?

The first step towards finding $x(t)$ is to calculate all the forces that act on the object. Forces are the cause of acceleration so if you want to find the acceleration, you need to calculate the forces acting on the object. Newton's second law $F=ma$ states that a force acting on an object produces an acceleration inversely proportional to the mass of the object. There are many kinds of forces: the weight of an object $\vec{W}$ is a type of force, the force of friction $\vec{F}_f$ is another type of force, the tension in a rope $\vec{T}$ is yet another type of force and there are many others. Note the little arrow on top of each force, which is there to remind you that forces are vector quantities. To find the net force acting on the object you have to calculate the sum of all the forces acting on the object $\vec{F}_{net} \equiv \sum \vec{F}$.

Once you have the net force, you can use the formula $\vec{a}(t) = \frac{\vec{F}_{net}}{m}$ to find the acceleration of the object. Once you have the acceleration $a(t)$, you can compute $x(t)$ using the calculus steps we learned in Chapter 2. The entire procedure for predicting the motion of objects can be summarized as follows: \[ \frac{1}{m} \underbrace{ \left( \sum \vec{F} = \vec{F}_{net} \right) }_{\text{dynamics}} = \underbrace{ a(t) \ \ \ \overset{v_i+ \int\!dt }{\longrightarrow} \ \ \ v(t) \ \ \ \overset{x_i+ \int\!dt }{\longrightarrow} \ \ \ x(t) }_{\text{kinematics}}. \] If you understand the above equation, then you understand mechanics. The goal of this chapter is to introduce you to all the concepts that appear in this equation and the relationships between them.

Apart from dynamics and kinematics, we will discuss a number of other physics topics in this chapter.

Newton's second law can also be applied to the study of objects in rotation. Angular motion is described by the angle of rotation $\theta(t)$, the angular velocity $\omega(t)$ and the angular acceleration $\alpha(t)$. The causes of angular acceleration are angular force, which we call torque $\mathcal{T}$. Apart from this change to angular quantities, the principles behind circular motion are exactly the same as those for linear motion.

During a collision between two objects, there will be a sudden spike in the contact force between them which can be difficult to measure and quantify. It is therefore not possible to use Newton's law $F=ma$ to predict the accelerations that occur during collisions.

In order to predict the motion of the objects after the collision we must use a momentum calculation. An object of mass $m$ moving with velocity $\vec{v}$ has momentum $\vec{p}\equiv m\vec{v}$. The principle of conservation of momentum states that the total amount of momentum before and after the collision is conserved. Thus, if two objects with initial momenta $\vec{p}_{i1}$ and $\vec{p}_{i2}$ collide, the total momentum before the collision must be equal to the total momentum after the collision: \[ \sum \vec{p}_i = \sum \vec{p}_f \qquad \Rightarrow \qquad \vec{p}_{i1} + \vec{p}_{i2} = \vec{p}_{f1} + \vec{p}_{f2}. \] Using this equation, it is possible to calculate the final momenta $\vec{p}_{f1}$, $\vec{p}_{f2}$ of the objects after the collision.

Another way of solving physics problems is to use the concept of energy. Instead of trying to describe the entire motion of the object, we can focus only on the initial parameters and the final parameters. The law of conservation of energy states that the total energy of the system is conserved. Knowing the total initial energy of a system allows us to find the final energy, and from this calculate the final motion parameters.

In math we just deal with numbers, that is, we solve questions where the answer is just a dimensionless number like $3$, $5$ or $55.3$. The universal power of math comes precisely from this high level of abstraction. We could be solving for the number of sheep in a pen, the surface area of a sphere or the annual revenue of your startup and in all the cases the same mathematical techniques can be used despite the fact that the numbers refer to very different kinds of quantities.

In physics we use numbers too, but because we are talking about the real world the numbers always have some dimension and measurement unit attached to them. An answer in physics is a number which is either a length, a time, a velocity, an acceleration, or some other physical quantity. We must distinguish between these different types of numbers. It doesn't make sense to add a time and a mass, because the two numbers are measuring different kinds of quantities.

Here is a list of some of the kinds of quantities that will be discussed in this chapter. \[ \begin{array}{lllll} \mathbf{Dimension} & \mathbf{SI\ unit} & \mathbf{Other\ units} & \mathbf{Measured\ with} \nl \textrm{time} & [\textrm{s}] & [\mathrm{h}], [\mathrm{min}] & \textrm{clock} \nl \textrm{length} & [\textrm{m}] & [\mathrm{cm}], [\mathrm{mm}], [\mathrm{ft}], [\mathrm{in}] & \textrm{metre tape} \nl \textrm{velocity} &[\textrm{m}/\textrm{s}]& [\mathrm{km}/\mathrm{h}], [\mathrm{mi}/\mathrm{h}] & \textrm{speedometer} \nl \textrm{acceleration} &[\textrm{m}/\textrm{s}^2]& & \textrm{acceleroometer} \nl \textrm{mass} &[\textrm{kg}] & [\mathrm{g}], [\mathrm{lb}] & \textrm{scale} \nl \end{array} \]

You should always try to “check the units” in your equations. Sometimes you will be able to catch a numerical mistake because the units will not be correct. If I ask you to calculate the maximum height that a ball will reach, I expect that your answer will be a length measured in [m] and not some other kind of quantity like a velocity $[\textrm{m}/\textrm{s}]$ or an acceleration $[\textrm{m}/\textrm{s}^2]$ or a surface area $[\textrm{m}^2]$. An answer in $[\mathrm{ft}]$ would also be acceptable since this is also a length, and it can be converted to metres using $1[\mathrm{ft}]=0.3048[\mathrm{m}]$. Learn to watch out for the units/dimensions of physical quantities, and you will have an easy time in physics. They are an excellent error checking mechanism.

The units of physical quantities in this book are indicated in square brackets throughout the lessons in this book.

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We will begin our physics journey by starting from the familiar subject of kinematics which we studied in Chapter 2. Now that you know about vectors, we can study kinematics problems in two dimensions like the motion of a projectile: $\vec{r}(t)=[x(t),y(t)]$.