Lecture

What does mechanics study?

An apple falling from a branch, planets orbiting the Sun, a braking car, a bridge swaying in the wind — all are examples of one common phenomenon: the mechanical motion of bodies. The oldest and most fundamental branch of physics, mechanics, describes such phenomena quantitatively and predicts them.

Mechanics is the science of the mechanical motion of bodies in nature and the interactions between them. Mechanical motion is the change in the position of a body (or its parts) in space relative to other bodies over time.

The aim of the science is to explain the mechanical side of natural phenomena and to apply the laws found in practice: to design machines and structures, to calculate the motion of vehicles and rockets, to predict the orbits of planets and artificial satellites. The main problem of mechanics is: given the initial state of a body and the forces acting on it, find its state at any later time.

Branches of mechanics and its limits of validity

• Kinematics describes motion without considering its causes (forces): where is the body, with what velocity and acceleration does it move? Its main quantities are position, displacement, velocity and acceleration.
• Dynamics relates motion to the interactions that cause it: why does the body move exactly this way? It is based on Newton's laws, e.g. $\vec F = m\vec a$.
• Statics studies the conditions for bodies to be in equilibrium under forces.

There are also fluid mechanics, elasticity theory, and the theory of oscillations and waves. Classical mechanics, created by Newton, gives correct results when the speed of a body is much less than the speed of light ($v \ll c$, $c \approx 3\cdot 10^8\ \text{m/s}$) and its size is much larger than an atom ($\sim 10^{-10}\ \text{m}$). At speeds close to the speed of light relativistic mechanics (Einstein) is used, and in the microworld quantum mechanics. Classical mechanics is contained in these theories as a limiting case.

Physical models

A real body is very complicated: it deforms, rotates, heats up. Therefore mechanics uses simplified models:

• Point particle (material point) — a body with mass whose size can be neglected compared with the distances involved. The Earth in its orbit around the Sun ($R \approx 1.5\cdot 10^{11}\ \text{m}$) can be treated as a point particle, but not when its daily rotation is studied. The same body may be a point particle in one problem and not in another.
• Rigid body — a body in which the distance between any two points does not change (deformation is neglected).
• Continuum — a body whose mass is regarded as continuously distributed over its volume (liquid, gas, elastic solid).

Frame of reference and the relativity of motion

The statement "the body is moving" is meaningful only relative to some other body. To describe motion, a frame of reference is chosen.

Frame of reference = reference body + a coordinate system attached to it + a time-measuring device (clock).

The position of a point is given by its position vector $\vec r = x\vec i + y\vec j + z\vec k$ drawn from the origin. In different frames the trajectory, displacement and velocity of the same motion are different — this is the relativity of motion. For example, an apple dropped in the carriage of a moving train traces a vertical straight line relative to the passenger, but a parabola relative to an observer on the ground. In classical mechanics the time interval, the mass and the distance between two points at the same instant are the same in all frames (absolute).

Galilean transformations and velocity addition

Let frame $K'$ move translationally relative to frame $K$ with a constant velocity $\vec u$ (the origins coincide at the initial instant). Then

where $\vec r$ and $\vec r^{\prime}$ are the position vectors of the point in $K$ and $K'$. Differentiating with respect to time gives Galileo's velocity addition rule:

i.e. the velocity of a body relative to the "fixed" frame ($\vec v$, the absolute velocity) equals the vector sum of its velocity relative to the moving frame ($\vec v^{\prime}$, the relative velocity) and the velocity of the moving frame itself ($\vec u$, the transport velocity). If the velocities lie along one line, the vector sum becomes an ordinary algebraic sum.

Example

A river is $d = 120\ \text{m}$ wide and the current speed is $u = 3\ \text{m/s}$. A boat is pointed perpendicular to the bank and moves at $v' = 4\ \text{m/s}$ relative to the water. How long does the crossing take, how far downstream is the boat carried, and what is its speed relative to the bank?

Solution. The current does not affect the motion perpendicular to the bank: $t = d/v' = 120/4 = 30\ \text{s}$. During this time the current carries the boat $s = u\,t = 3\cdot 30 = 90\ \text{m}$ downstream. The speed relative to the bank is $v = \sqrt{v'^2 + u^2} = \sqrt{16+9} = 5\ \text{m/s}$; it deviates from the perpendicular to the bank by the angle $\beta$ with $\tan\beta = u/v' = 0.75$, i.e. $\beta \approx 37^\circ$.

In the interactive model change the boat speed, the current speed and the heading angle: how does the trajectory look in the "bank" and "water" frames? At what angle must the boat be pointed to reach the point exactly opposite ($\sin\alpha = u/v'$)?