World in Physics

一直认为物理是理科中最强大的学科，本学期取得了好成绩，再此想要好好地告别。量子与相对论带给了我思考的全新视角，希望我能借此更加理解这个世界。本文待完成。

Force and motion

Formulas

$$KE = \int F \mathrm{d}s = \int ma \mathrm{d}s = \int m\frac{\mathrm{d}v}{\mathrm{d}t} \mathrm{d}s = \int_u^v mv' \mathrm{d}v' = \frac{1}{2}mv^2$$ $$P = \frac{\mathrm{d}E}{\mathrm{d}t} = \frac{\frac{1}{2}mv^2}{t}$$ $$<g(r)> = \frac{1}{R}\int_0^R \frac{GMr}{R^3}\mathrm{d}r$$ $$\vec{L}=\vec{r}\times \vec{p} = mvr = mr^2\dot{\theta}=I\dot{\theta}$$ $$I = \int r^2\mathrm{d}m$$ $$\vec{\tau}=\vec{r}\times \vec{F}=\vec{r}\times \frac{\mathrm{d}\vec{p}}{\mathrm{d}t}=\frac{\mathrm{d}\vec{L}}{\mathrm{d}t}=I\omega$$ $${\dot{\theta}}^2=2\ddot{\theta}\theta$$ $$W=\tau \theta (Fs)=\frac{1}{2}I{\dot{\theta}}^2(\frac{1}{2}m{v}^2)$$ $$g=\frac{GM}{R^2}$$ $$g(r)=\frac{GMr}{R^3}$$ pressure at the centre of the earth: $\rho \overline{g}R$ $$\overline{g}=\frac{1}{R}\int_0^R \frac{GMr}{R^3} \mathrm{d}r=\frac{1}{R}\frac{GMr^2}{2R^3}|_0^R=\frac{GM}{2R^2}$$

theorems

statics

conditions for equilibrium:

• resultant force is $0$

• no rotation: net torque must be $0$

that is, $F=0$ and $\tau{cw} =\tau{ccw}$.

Parallel-axis theorem

• initial: rotating about CM.

• final: parallel to the initial axis.

$I{parallel-axis}=I{CM}+md^2$

rotation matrix

• two dimension:

$$\begin{matrix} \cos{\alpha} & \sin{\alpha} \\ -\sin{\alpha} & \cos{\alpha} \\ \end{matrix}$$

• three dimension:

(one example for rotating about the x-axis)

$$\begin{matrix} 1 & 0 & 0 \\ 0 &\cos{\alpha} & \sin{\alpha} \\ 0 & -\sin{\alpha} & \cos{\alpha} \\ \end{matrix}$$

Projectile equation

Farthest distance for jumping:

$45$ degree.

efficiency of energy conversion

$20%$

method of dimensional analysis

example:

$$LT^{-2}=L^n(LT^{-1})^{m}$$

the change in gravitational potential energy

$$U=-GMm(\frac{1}{r_f}-\frac{1}{r_i})=GMm(\frac{r_f-r_i}{r_f r_i})$$

elastic collision

one-dimension elastic collision.
conservation of momentum and conservation of kinetic energy.

$$\left\{ \begin{array} m_1u_1+m_2u_2=m_1v_1+m_2v_2, & \\ \frac{1}{2}m_1{u_1}^2+\frac{1}{2}m_2{u_2}^2=\frac{1}{2}m_1{v_1}^2+\frac{1}{2}m_2{v_2}^2, & \end{array} \right.$$ $$v_1=\frac{u_1(m_1-m_2)+2m_2u_2}{m_1+m_2}$$ $$v_2=\frac{u_2(m_2-m_1)+2m_1u_1}{m_1+m_2}$$

Rocket propulsion

• vacuum

$$[(m-\mathrm{d} m_g)(v+\mathrm{d} v)+\mathrm{d} m_g(v-u)]-mv=0$$

u: the constant velocity of the ejected burned gas relative to the rocket.

eliminated:

$$m\mathrm{d} v=-u\mathrm{d} m$$

integrated:

$$\int_{v_{i}}^{v} \mathrm{d} v=-u\int_{m_i}^{m} \frac{1}{m} \mathrm{d} m$$ $$v-v_i=uln(\frac{m_i}{m})$$

the change of velocity of a rocket relates to the change of the rocket’s mass.

$$m(v)=m_iexp\frac{-(v-v_i)}{u}$$

• gravitational field (gravity)

$$(m-\mathrm{d} m_g)(v+\mathrm{d} v)+\mathrm{d} m_g(v-u)-mv=-mg\mathrm{d} t$$

expanded:

$$m\mathrm{d} v+u\mathrm{d} m=-mg\mathrm{d} t$$

integrated:

$$v-v_i=uln(\frac{m_i}{m})-g\delta t$$

the rocket will run faster if more mass of the burned fuel gas is expelled at a shorter time interval.

Fluid and thermodynamics

Formulas

Bernoulli’s equation

$$P+\frac{1}{2}\rho{v}^2+\rho gh = constant$$ $$Q=\frac{\mathrm{d}V}{\mathrm{d}t}=Sv$$

flow is conserved in a tube.

$$E=PV=Fh$$ change of thermal energy $$P=\frac{E}{V}=\rho gh$$ $$F_{buoyancy}=\rho gV$$ $$F_{fluid friction}=cAv^2$$

$C$: the coefficient of media resistance

thermal expansion

$$\delta{L}=\alpha L\delta{T}$$

$\frac{\mathrm{d} L}{\mathrm{d} T}=\alpha L$,$\frac{\mathrm{d} A}{\mathrm{d} T}=2\alpha A$,$\frac{\mathrm{d} V}{\mathrm{d} T}=3\alpha V$

heat capacity

$C$: the amount of energy needed to raise the temperature of that sample by $1$ degree.

heat transfer $Q=C\delta T$

$c$: specific heat capacity. the heat capacity per unit mass.

$c=\frac{Q}{m\delta T}$,$Q=cm\delta T$

heat flow(heat transfer by conduction)
the amount of heat conducted per second: $P_c=\frac{K_cA}{L}(T_1-T_2)$
A: cross section area L: length K: coefficient of thermal conductivity

average kinetic energy of a partical: $\frac{1}{2}mv_{rms}^2=\frac{3}{2}k_BT$
rms: root mean square

examples

• final temperature of mixture at thermal equilibrium

Let T be the final temperature of the system at thermal equilibrium. The total heat transfer is just 0.

$$m_ic_i(T-90)+m_wc_w(T-12)+m_mc_m(T-24)=0$$

substituting the values gives the T.

Theroems

how liquid wets container

attractive force between surface molecules and wall: adhesion

attractive force between surface molecules and liquid: cohesion

curved upward glass-water>water-water

curved downward glass-mercury<mercury-mercury

capillary effect

rise up: ad force is stronger against the weight of liquid

depress: ad is weaker than co

Wave

Optics

Electricity and magnetism

Relativity and quantum physics