# Force and motion

## 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

• final: parallel to the initial axis.

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

### rotation matrix

• two dimension:
• three dimension:

(one example for rotating about the x-axis)

### Projectile equation

Farthest distance for jumping:

$45$ degree.

### efficiency of energy conversion

$20%$

example:

### elastic collision

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

### Rocket propulsion

• vacuum

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

eliminated:

integrated:

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

• gravitational field (gravity)

expanded:

integrated:

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

flow is conserved in a tube.

$C$: the coefficient of media resistance

thermal expansion

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

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.

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