Mechanics: study of a system of particles interacting with each other and/or the environment.

Newtonian mechanics applicable when:

• things not too fast (compared to $c$)
  • $\frac{v}{c} \ll 1$
• things not too small (compared to $\lambda_{db}$ ^[1])
  • $\frac{L}{\lambda_{db}} \gg 1$
• gravity not too strong
  • $\frac{GM}{Lc^2} \ll 1$

1. space
   • 3-dimensional: 3 coordinates necessary to specify the position of a particle
   • euclidean: euclid’s axioms satisfied
2. time
   • universal time: time runs the same way for all bodies regardless of their properties

Units §

We describe the magnitude of physical quantities using standard units.

• Length
• Mass
• Time

Dimensions §

Combinations of powers of length, mass, time.

• $[\textrm{length}] = L$
• $[\textrm{velocity}] = LT^{-1}$
• $[\textrm{acceleration}] = LT^{-2}$
• $[\textrm{momentum}] = [\textrm{mass} \times \textrm{velocity}] = MLT^{-1}$

$F_g = \frac{G m_1 m_2}{r^2}$

$G = \frac{F_g r^2}{m_1 m_2}$

$G$ small, that’s why a large mass necessary to make $F_g$ significant

useful for:

• checking validity of equations
• finding expressions for physical qualities

not useful for dimensionless constants.

Vectors §

Vectors change^[2] under transformation of coordinate systems. Scalars do not.

Examples §

• Displacement
• Velocity

Notation §

$\vec A = A \hat a$

$|\vec A| = A$

Translation doesn’t change vectors

Physical laws can be written as relationships between geometric quantities (vectors) in Newtonian Mechanics.

Unit vectors, vector components §

Unit/base vectors:

• $\hat x$, $\hat y$, $\hat z$
• $\hat i$, $\hat j$, $\hat k$

component form:

$\vec A = A_x \hat i + A_y \hat j + A_z \hat k$

When coord system changed, unit vectors and components change such that the vector itself is unchanged.

Multiplication by a scalar §

Vector Addition §

Head-to-tail, tail-to-other-head

Subtraction: flip subtrahend, add.

Geometrically, but also component-wise.

Properties §

Vector Multiplication §

Dot product §

$\vec A \cdot \vec B = AB \cos \theta$

independent of coordinate system

projection of $\vec A$ | $\vec B$ in the direction of $\hat B$ | $\hat A$ multiplied by the magnitude of $\vec B$ | $\vec A$

Cartesian coordinates:

$\vec A = A_x \hat i + A_y \hat j + A_z \hat k$

$\vec B = B_x \hat i + B_y \hat j + B_z \hat k$

\vec A \cdot \vec B = A_x B_x + A_y B_y + A_z B_z$$ [^3] [^3]: $\hat i \times \hat i = 1$, $\hat j \times \hat j = 1$, $\hat k \times \hat k = 1$ #### Cross product $$\vec A \times \vec B = AB \sin \theta \hat c

$\hat c$ perpendicular to the plane spanned by $\vec A$ and $\vec B$ in the direction of the right hand rule.

independent of coordinate system

Magnitude of cross product: area of the parallelogram spanned by $A$ and $B$

$\vec A \times \vec B = -\vec B \times \vec A$

Cartesian coordinates:

$\vec A \times \vec B = \left|\begin{array}{ccc} \hat{i} & \hat{j} & \hat{k}\\ A_{x} & A_{y} & A_{z}\\ B_{x} & B_{y} & B_{z} \end{array}\right|$

Vector differentiation §

$\vec r (t) = x(t) \hat i$

$\frac{d \vec r (t)}{dt} = \frac{d x (t)}{dt} \hat i$

Geometrically, rate of change of $r(t + \delta) - r(t)$.

Notation: $\frac{d \vec A}{dt} = \dot{\vec A}$

Properties §

Same as properties of regular differentiation, except order matters wherever vectors/vector_derivatives are multiplied.

If $\frac{d \vec A}{dt} \perp \vec A$, then $|\vec A|$ is a constant.

If vector perpendicular to time-der, but time-der != 0, it’s just rotating with the same magnitude.

General kinematic equations §

$\dot{\vec v} (t) = \vec a (t)$

$\vec v (t) = \vec v (t_0) + \int_{t_0}^{t} \vec a(t') dt$

Uniform^[3] circular^[4] motion §

$\vec r = r\cos \theta \hat i + r\sin \theta \hat j$

$\vec r (t) = r\cos \omega t \hat i + r\sin \omega t \hat j$

$v(t) = -r \omega \sin \omega t \hat i + r \omega \cos \omega t \hat j$

$|v| = v = \sqrt{v \cdot v} = r \omega$

$r \cdot v = 0$ ($r$ is perpendicular to $v$)

Period $\zeta = \frac{C}{v} = \frac{2 \pi r}{v} = \frac{2\pi}{\omega}$

$\vec a = - r \omega^2 \cos \omega t \hat i - r \omega^2 \sin \omega t \hat j = - \omega^2 r \hat r$

$|\vec a| = \sqrt{r^2 \omega^4 \cos^2 \omega t + r^2 \omega ^4 \sin^2 \omega t} = r \omega^2 \sqrt{1} = r \omega^2$

Circular (and other) motion is sometimes better described using polar coordinates.

1. $r$: distance from origin
2. $\theta$: angle from the $x$ axis

Unit vectors point along the coordinate lines. Move from one coordinate line to the next.

$\dot{\hat r} = - \dot{\theta} \sin \theta \hat i + \dot{\theta} \cos \theta \hat j = \dot{\theta} \hat \theta$

$\dot{\hat \theta} = - \dot{\theta} \hat r$

change in $\hat r$ is along $\hat \theta$ and vice-versa

Generally:

$\vec r = r \hat r$

$\vec v = \dot{\vec r} = \dot{r} \hat r + r \dot{\hat r} = \dot{r} \hat r + r \dot{\theta} \hat \theta$

\vec a = \dot{\vec v} = (\dot\dot - r \dot \theta^2) \hat r + (2 \dot r \dot \theta + r \dot \dot) \hat \theta

UCM: $r = const$, $\theta = \omega t$, $\dot{\theta} = \omega$

$\vec r = r \hat r$

$\vec v = r \omega \hat \theta$

$\vec a = - r \omega^2 \hat r$

in polar coords, on parallel transporting, vector remains the same but components and unit vectors change.

Translate to origin: only radial components

$2 \dot{r} \dot{\theta} \hat \theta$ is non-zero only if trajectory is neither radial nor circular

Newton’s Laws §

particle: an object whose size can be neglected when describing its motion

isolated particle: a particle free from the influence of interactions with other particles

1. gravitational
2. electromagnetic

Reference frames

Newton’s laws hold only for inertial reference frames.

Newton’s first law §

Inertial reference frames exist

Inertial reference frame: frames of reference where a particle free from external forces moves with a constant velocity

An accelerated frame of reference is a non-inertial reference frame. In non-inertial frames, we introduce fictitious/pseudo forces so that Newton’s laws apply

Frames moving with a constant $v$ wrt an inertial frame are also inertial

Earth is only approximately an inertial frame

Newton’s second law §

The rate of change of momentum of a particle is equal to the external, real, unbalanced force acting on it.

$\vec F = \frac{d \vec p}{dt} = m\vec a$

Newton’s third law §

$\dot{\dot{\vec r}} = \vec g = - g \hat k$

$\dot{\dot{z}}(t) = -g$

$z(t) = z_0 + v_{z_0} t - \frac {1}{2}gt^2$

Phenomenological forces §

Normal force: A contact force exerted by a surface on a body. always $\perp$ to the surface. EM force arising from the pushback from compressed atoms.

Friction arises from contact between surfaces. Arises from attractive forces between subatomic particles.

$F_f \approx \mu N$

$\mu$ is the coefficient of static friction. usually $< 1$. dimensionless.

$0 < F_f < N$

Centripetal acceleration §

acceleration towards the center. caused by tension (resistance to stretching of atoms)

$\vec F = -T \hat r$ (T is the tension in the rope)

$\vec a = - \omega^2 l \hat r$

$\vec F = m\vec a \implies T = m \omega^2 l \hat r$

Geostationary orbit §

Uniform circular motion

$T=\frac{2\pi}{\omega} = 24h$

What is $r$ in terms of $R_E$ for the orbit to be geostationary?

$F_g$ with gravity formula. $F_g = m_sa$, $a$ in terms of $\omega$ and $r$. Equate, simplify. Plug in $\omega$ in terms of $24h$.

Whirling rope §

UCM.

What is the tension in terms of the radius?

Derive a diffeq in terms of the distance from the origin.

Element $\delta r$ of the rope.

total tension = mass of element * acceleration

Pulleys §

Drag forces §

Act to oppose motion of a solid object through a fluid (e.g. air, water). arises from the third law. EM.

fluid resistance.

Depends on:

• size/shape of the object
• speed of the object relative to the fluid
• microscopic properties of the fluid

$F_{\textrm{drag}} = -f(v) \hat v$

• at low speeds, $f(v) \propto v$

  laminar flow.

• at high speeds, $f(v) \propto v^2$

  turbulent flow.

Simple harmonic motion §

• massless spring
• neglect friction
• $x$ is measured from equilibrium position

Hooke’s law.

$\vec F = -kx \hat x$ (linear restoring force)

if $x>0$, spring is stretched. $\vec F$ pulls back towards $x = 0$

if $x < 0$, spring is compressed. $\vec F$ pushes towards $x = 0$

$\dot{\dot{x}}+\frac{k}{m} x = 0$

$x = B\cos(\omega_0 t) + C\sin(\omega_0 t) = A\cos(\omega_0t+\phi)$

$\omega_0^2 = \frac{k}{m}$

projection of uniform circular motion

Gravity §

force on $m_1$ due to $m_2$: $\vec F_{21} = -\frac{Gm_1 m_2}{r^2} \hat r $

shell theorem:

• outside the shell: $\vec F_{m} = -\frac{GM m}{r^2} \hat r$
• inside the shell: $\vec F_m = 0$ (coz only the stuff outside the shell affects)

ball theorem:

• outside: $-\frac{GMm}{r^2}\hat r$
• inside (only the mass inside affects you): $-\frac{GMm}{r^2}\left(\frac{r^3}{R^3}\right) \hat r$

Momentum §

Per Newton’s 3rd law, $\vec f_{12} = - \vec f_{21}$

m_1 r_1 … = f_12 and m_2 r_2 … = f_21

For particles interacting with each other but isolated from everything else,

total momentum $\vec {\mathbb{P}}$ vector conserved for isolated systems. $\frac{dP}{dt} = 0$

Not isolated: $\frac{dP}{dt} = F_{ext}$

Center of mass: $R = \frac{\sum m_i r_i}{\sum m_i}$

$M \vec R'' = \frac{d\vec P}{dt} = \vec F_{ext}$

The center of mass moves as if:

• the entire mass of the object is concentrated at it
• all the external forces are acted upon it

There may be motion around the center of mass

$R = \frac{\int dm' \vec r'}{\int dm'} = \frac{\int dv' p \vec r'}{\int dv' p}$

2 particle system:

• motion of CM
• relative motion ($r_1 - r_2$)

Impulse §

$\int_0^t \vec F_{\mathrm{ext}} dt$

$\int_0^t \vec F_{\mathrm{ext}} dt = \int_0^t \vec dP dt = \vec P(t) - \vec P(0)$

Rockets §

$M\frac{d\vec v}{dt} - \vec u \frac{dM}{dt} = \vec F_{\mathrm{ext}}$

$\vec u$ is the speed of the expelled mass wrt the rocket (not wrt the inertial frame)

Energy §

Conservative forces §

$\vec F (\vec r)$ is said to be conservative if $\int_{r_a}^{r_b} \vec F(\vec r)\cdot d\vec r$ is independent of the path for arb. $r_a$, $r_b$.

$F(r)$ is conservative iff $\oint_c F(r)\cdot dr = 0$

Conservation of mechanical energy §

potential energy: $u(\vec r) = - \int_o^{\vec r} \vec F (\vec r)\cdot d\vec r$

possible to define iff $\vec F(\vec r)$ is conservative ($u$ is a function of $\vec r$ only).

mechanical energy $K + U = E$ is constant as the system evolves. $E$ is conserved.

only differences in $E$ are meaningful

In general, for a conservative force $\vec F (\vec r)$ mvoing some particle from 0 to $\vec r$ with an applied force $- \vec F (\vec r)$:

$\int_0^{\vec r} \left(\vec F (\vec r\right) \cdot d\vec r = - \int_0^{\vec r} \vec F (\vec r)\cdot d\vec r = u(\vec r)$

$\vec F = -\nabla u(\vec r)$

$\int_{\vec a}^{\vec b} \vec F \cdot d\vec r = - \int_{\vec a}^{\vec b} \nabla u \cdot d\vec r = u(\vec b) - u(\vec a)$

Angular Momentum §

$\vec L = \vec r \times \vec p$

$\tau = \vec r \times \vec F$

$\vec \tau = \frac{d\vec L}{dt}$

$\frac{d\vec L}{dt} - \tau_i + \tau_e$

$L_z = I \omega$

$I = \int \rho^2 dm$