User:Dragix/Classical Mechanics

Intermediate Mechanics is aimed mainly at physics majors, for whom this course is the first ‘serious’ mechanics course. As such, the material covered in this course will be assumed knowledge for many of your future physics courses, some of which will develop the ideas that you have met here further. Students are therefore expected to demonstrate a thorough understanding of the concepts that they encounter in this course.

In this course, we will be attempting to analyze mechanical systems (as opposed to memorising them), beginning with systems consisting of a single particle. To do this, we will use certain tools. Among these tools, several branches of mathematics, including arithmetic, algebra, trigonometry, and calculus, are indispensable. You are therefore expected to be comfortable with each of these areas in order to use them to solve physics problems.

Definition

 * Mechanics - the study of the motion of material bodies
 * 1) Kinematics - the description of possible motion of material bodies (e.g. a in terms of v and x)
 * 2) Dynamics - the study of the laws of motion which determine which of the potential motions will actually take place in any given case. Forces are introduced and described as a sum total.
 * 3) Statics - study of (system of) forces, described by components, that act on body at rest
 * Electrostatics - Electric and magnetic forces exerted by electric charges and currents upon another
 * Gravitation - Gravitational forces on another

Vectors

 * Coordinate systems
 * Cartesian $$(x, y, z)$$
 * Cylindrical $$(\rho, \phi, z)$$
 * Spherical $$(r, \theta, \phi)$$


 * Vectors
 * Dot Product (Scalar)
 * $$ \mathbf{a} \cdot \mathbf{b} = |\mathbf{a}| \, |\mathbf{b}| \cos \theta \;$$
 * $$\text{If } \mathbf{a} \perp \mathbf{b} \text{, then } \mathbf{a} \cdot \mathbf{b} = 0$$
 * Cross Product
 * $$ \mathbf{a} \times \mathbf{b} = ab \sin \theta \ \mathbf{\hat{c}}$$
 * $$\text{If } \mathbf{a} \parallel \mathbf{b} \text{, then } \mathbf{a} \times \mathbf{b} = 0.$$

{\frac{\partial}{\partial x}} & {\frac{\partial}{\partial y}} & {\frac{\partial}{\partial z}} \\ \\ F_x & F_y & F_z \end{vmatrix}$$
 * Gradient $$\nabla = \frac{\partial}{\partial x} + \frac{\partial}{\partial y} + \frac{\partial}{\partial z}$$
 * Divergence $$\nabla\cdot\mathbf{F} = \frac{\partial F_x}{\partial x} + \frac{\partial F_y}{\partial y} + \frac{\partial F_z}{\partial z}$$
 * Curl $$\nabla\times\mathbf{F} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ \\

Newtonian Mechanics

 * Newton's laws of motion
 * 1) Inertia $$\sum{F} = 0 \;$$ Decribes total force only, not its components, which is inconvenient in Statics where all $$a = 0$$
 * 2) Rate of Change of Momentum $$F = m \times a = \tfrac{d}{dt}(m \times v)= \tfrac{d}{dt}(p) \;$$ What Newton discovered was not that F = ma, but that F is most easily described this way
 * 3) Action-Reaction $$ma =-ma \;$$ Fails to hold for electromagnetic forces when the interacting bodies are far apart, rapidly accelerated, or propagated from another body with finite velocity
 * Galilean transformation: Classical physics transform between the coordinates of two reference frames which differ only by constant relative motion within the constructs of Newtonian physics
 * Inertial coordinate systems: it's when you are on the boat
 * Noninertial coordinate systems: it's when you are watching the boat
 * Principle of relativity: if these laws hold for 1 coordinate system, they hold for all coordinate systems moving uniformly with respect to the first

Problem Solving
The main Equations of Motion - These all have constant acceleration

$$v = v_0 + at \;$$

$$r = r_0 + v_0t + \tfrac{1}{2}at^2 \;$$

$$v^2 = v_0^2 + 2a(r - r_0) \;$$

Examples of Elementary Physics Problems and Equations Associated with them


 * Particle in a Straight Line
 * Kinematics example
 * $$ x = x_0 + v_0t + \tfrac{1}{2}at^2 \;$$


 * Projectile Motion
 * Dynamics example
 * $$y = x \tan \theta - \frac{g}{2v_0^2 \cos^2 \theta} x^2 \;$$


 * Pulley
 * Dynamics example
 * $$T = \frac{2m_1m_2}{m_1+m_2} g$$


 * Block on an Incline
 * Statics example, introduces friction, which is nonconservative and inelastic
 * $$F_{fric} \leq \mu N \;$$


 * Centripetal Motion
 * $$F_c = ma_c = m \tfrac{v^2}{r}$$


 * Moon's Orbit about Earth
 * Gravitation example
 * $$F = \frac{Gm_1m_2}{r^2}$$
 * $$g = \frac{GM}{R^2}$$
 * The fact that g is proportional to m, instead of something else like q, is totally a coincidence. This is a big mystery in physics, thats why Gravitation has its own little subdivision in physics and other forces don't

Motion of Particle in 1-Dimension
Forces in 1st dimension go by $$F(x, v, t)$$


 * Momentum and Energy Theorems
 * Momentum
 * $$ p = mv \;$$
 * Kinetic energy
 * $$T = \tfrac{1}{2}mv^2 = \tfrac{p^2}{2m}$$
 * Impulse
 * From Newtons 2nd Law $$F = \frac{dp}{dt}$$
 * $$I = p_2 - p_1 = \int_{t_1}^{t_2}Fdt \;$$
 * Work Energy Theorem
 * $$W = T_2 - T_1 = \int_{x_1}^{x_2} F dx \;$$
 * where $$W = F \cdot d\cos\theta$$
 * Power
 * $$P = T_2 - T_1 = \int_{t_1}^{t_2} Fv dt = \frac{W}{t} \;$$


 * Conservation of Energy
 * $$\int_{x_1}^{x_2} F dx = -V(x) + V(x_0) \;$$
 * $$T + V(x) = T_0 + V(x_0) \;$$


 * Force dependent on time
 * Given an applied force F(t) dependent only on time t, not displacement and not velocity
 * $$mv - mv_0 = \int_{t_0}^{t} F(t) dt$$


 * Frictional forces
 * Damping forces F(v) are dependent only on velocity v, not time and not displacement
 * $$\int_{v_0}^{v} \frac{dv}{F(v)}= \frac{t-t_0}{m}$$
 * $$F = \mp bv^{(n)}$$


 * Conservative Forces and Potentials
 * Conservative forces F(x) are dependent only on displacement x, not time and not velocity
 * Conditions for being conservative:
 * $$\triangledown \times F = 0 \;$$
 * $$W = \oint_{C} \vec{F} \cdot d\vec{r} = 0 \;$$
 * $$F = - \triangledown U \;$$


 * Falling Objects
 * Equation for the vertical motion of a falling object
 * $$ y = y_0 + v_0t + \tfrac{1}{2}gt^2 \;$$
 * Force of a falling body
 * $$F = -mg - F_{fric}$$
 * where $$F_{fric} = bv^{(n)}$$

Motion of Particle in n-Dimensions
2 Dimensional forces go by F(x, y) and 3 Dimensional forces go by F(x, y, z)


 * Momentum and Energy Theorems
 * Momentum
 * $$ \mathbf{p} = m\mathbf{v} \;$$
 * Kinetic energy
 * $$\frac{dT}{dt} = \tfrac{d}{dt}\tfrac{1}{2}m\mathbf{v}^2 = \mathbf{F} \cdot \mathbf{v}$$
 * Impulse
 * From Newtons 2nd Law $$\mathbf{F} = \tfrac{d^2\mathbf{r}}{dt^2} = \tfrac{d\mathbf{p}}{dt}$$
 * $$I = \mathbf{p}_2 - \mathbf{p}_1 = \int_{t_1}^{t_2}\mathbf{F}dt \;$$
 * Work Energy Theorem
 * $$W = T_2 - T_1 = \int_{r_1}^{r_2} \mathbf{F} d\mathbf{r} \;$$
 * where $$W = F \cdot d\cos\theta$$
 * Power
 * $$P = T_2 - T_1 = \int_{t_1}^{t_2} \mathbf{F} \cdot \mathbf{v} dt = \frac{W}{t} \;$$


 * Conservative forces in 3D
 * Conservation of energy and the work-energy theorem
 * Planar kinematics
 * Three-dimensional kinematics
 * Conservation laws: linear momentum, angular momentum, energy

System of Particles

 * Central forces, center-of-mass coordinates
 * Properties of central forces, specific examples
 * Inverse-square central force, Kepler’s laws, classification of orbits
 * Systems of particles
 * Specific examples, variable mass systems

Rigid Body
is a special kind of system of particle

Harmonic Oscillator
$$x = A \cos(\omega_0 t + \theta)$$

$$\gamma = \tfrac{b}{2m}$$ Damping coefficient

$$\omega_0 = \sqrt{\tfrac{k}{m}}$$

$$\omega_1 = \sqrt{\omega_0^2 - \gamma^2}$$

The Harmonic Oscillator Equation
$$\frac{\mathrm{d}^2x}{\mathrm{d}t^2} + \frac{b}{m} \frac{\mathrm{d}x}{\mathrm{d}t} + {\omega_0}^2x = A_0 \cos(\omega t) $$

$$\Rightarrow m\frac{\mathrm{d}^2x}{\mathrm{d}t^2} + b \frac{\mathrm{d}x}{\mathrm{d}t} + kx = 0 $$

$$\Rightarrow m\frac{\mathrm{d}^2x}{\mathrm{d}t^2} = - b \frac{\mathrm{d}x}{\mathrm{d}t} - kx $$

$$\Rightarrow \mathrm{F} = - k x - b v $$

It's the equation of a particle with linear restoring force and friction proportional to velocity

1 Dimension
Solve for the equation. Try

$$x = e^{\rho t} $$

$$v = \rho e^{\rho t} $$

$$a = \rho^2 e^{\rho t} $$

$$\Rightarrow m\rho^2 + b\rho + k = \rho$$

$$\Rightarrow \rho = -\frac{b}{2m} \pm \sqrt{\frac{b^2}{4m^2}-\frac{k}{m}}$$

$$\Rightarrow \rho = -\gamma \pm \sqrt{\gamma^2 -\omega_0^2}$$

$$\Rightarrow \rho = -\gamma \pm i\omega_1$$


 * $$\omega_0^2 < \gamma^2$$ Damped
 * $$\omega_0^2 > \gamma^2$$ Undamped

$$\tfrac{\omega_0}{2\pi}$$ is the natural frequency of undamped oscillator


 * $$\omega_0^2 = \gamma^2$$ Critically damped


 * Damped Energy balance in damped oscillating systems
 * Linear and nonlinear oscillating systems
 * Forced oscillator, transient and driven response
 * Mechanical resonance, phase and amplitude response
 * Principle of superposition

Gravitation

 * Classical gravitation
 * $$\mathbf{F} = \frac{Gm_1m_2}{r^2}$$ Gravitational force
 * $$\mathbf{g} = \tfrac{\mathbf{F}}{m}$$ Gravitation intensity/field


 * At Earth's surface
 * $$g = \frac{GM}{R^2}$$
 * $$U = \frac{GMm}{R+h}$$ where h is the height from the earth's surface


 * Center of gravity
 * Gravitational fields and potentials
 * Fields from general mass distributions