Grade 12 Physics

S6 Physics Study Doc

Unit 1: Dynamics

Kinematics Equations

$$v_2=v_1+a\Delta t$$ $$\Delta d=v_{1}t+\frac{1}{2}a\Delta t^2$$ $$\Delta d=v_{1}t+\frac{1}{2}a\Delta t^2$$ $$v_2^2=v_1^2+2a\Delta d$$

Circular Motion

$$F_{netC}=\frac{m{v}^2}{r}=\frac{G{m}_1{m}_2}{r}=\frac{m_{2}4{\pi }^{2}R}{t^2}$$

Inertia

$$I=m{r}^2$$

Torque

$$T=r\times f\times \sin \theta$$

Terminal Velocity (k = drag constant)

$$F_g=F_{drag}$$ $$F_g=mg$$ $$F_{drag}=kv$$

Unit 2: Energy and Momentum

Energy

$$Work=F\Delta d\cos \theta$$ $$E_k=\frac{1}{2}{v}^2$$ $$E_g=mg\Delta h$$ $$E_e=\frac{1}{2}k{x}^2$$ $$F=kx$$

• K = spring constant $\left(\frac{N}{m}\right)$
• x = spring extension / compression $(m^2)$ ME = PE + KE

Energy of Mass Oscillating

$$E_{total}=\frac{1}{2}m(V_{max}{)}^2$$ $$T_{total}=\frac{1}{2}k{A}^2$$

• k = coefficient
• A = amplitude

$$E_{total}=E_{elastic}+E_{kinetic}$$

If amplitude is known:

$$E_{kinetic}=\frac{1}{2}k{A}^2-\frac{1}{2}k{x}^2$$

If $V_{max}$ is known:

$$\frac{1}{2}m(V_{max}{)}^2-\frac{1}{2}k{x}^2$$

Speed of mass at any point:

$$v=\sqrt{\frac{K(A^2-x^2)}{m}}$$

Forces and Collisions

Conservative vs Nonconservative

• Conservative Force: doesn’t depend on path
• Nonconservative Force: depends on path

Elastic vs Inelastic

• Elastic collision: Kinetic Energy / Heat Conserved (pool ball)
• Inelastic collision: Kinetic Energy / Heat not Conserved
  • perfectly inelastic collision: two objects stick together 

Open vs Closed

• Open system: Momentum changes
• Closed system: Momentum doesn’t change

Momentum: How hard is it to stop something?

Momentum (kg * m/s)

$$P=mv$$

Collisions with conserved momentum:

$$p_{initial}=m_1{v}_1+m_2{v}_2$$ $$p_{final}=(m_1+m_2)v_{final}$$

Angular momentum:

$$L=lw$$

• L = angular momentum (kg m2 rad / s)
• I = moment of inertia (kg m2)
• w = angular velocity (rad / s)

Conservation of momentum (with no external force)

$$p_{initial}=p_{final}$$

Elastic collision: Heat / Kinetic energy is not generated

• Eg. knocking a billiard ball, and the cue ball stops
• $K_{bi}+K_{si}=K_{bf}+K_{sf}$

Inelastic collision: Heat / Kinetic energy is generated

• Might go into heat/sound/potential
• $K_{ai}+K_{bi}!=K_{af}+K_{bf}$
• Perfectly inelastic: Objects stick together and continue moving

Impulse: Change in momentum

$$Impulse(J)=F_{avg}\Delta t=m\Delta v$$

• Measured in Ns because $mat=mv$
• Is the area under a F-t graph

Impulse-Momentum Formula

$$\Delta p=F\Delta t$$

$Power=\frac{work}{time}$

2D and Elastic Collisions

Moving Mass

$$v_1^{\prime }=\frac{m_1-m_2}{m_1+m_2}{v}_1+\frac{2m_2}{m_1+m_1}{v}_2$$

Non-Moving Mass

$$v_2^{\prime }=\frac{2m_1}{m_1+m_2}{v}_1-\frac{m_1-m_2}{m_1+m_2}{v}_2$$

Unit 3: Gravitational, Electric, and Magnetic Fields

Kepler’s Laws

1. First Law: The orbit of every planet is an ellipse, with the sun at one of the two foci
   • Every orbit is an ellipse
   • Every ellipse has 2 foci
   • The sun is always one of the foci
2. Second Law: Law of Equal Areas
   • A line joining a planet and the sun sweeps out equal areas during equal intervals of time
   • Planet moves slowly when far from the sun!
   • Planet moves rapidly when close to the sun
3. Third Law: The square of the orbital period of the planet is proportional to the cube of the semi major axis of its orbit

$$T^2\propto r^3$$

Escape Velocity Equation:

$$c=\sqrt{\frac{2G{M}_e}{R_e}}$$

Gravitational Potential Energy = Binding Energy

• Same as activation energy of an electron
• It’s negative - “I’m this far below how much I need to leave”

Electricity

Resistance = Voltage / Current 1C = $6.24\ast 10^{18}$ $P=VI$

Kirchhoff’s Junction Rule: The algebraic sum of the currents into (or out of) any junction in the circuit is zero. 

Kirchhoff’s Loop Rule: The sum of the voltage changes across the circuit elements forming any closed loop is zero.

Resistivity and Electric Fields

$e^{-}=1.6\ast 10^{-19}C$ Resistivity = 1 / conductivity

$$R=p\frac{L}{A}$$

• R = resistance (ohms)
• p = resistivity (ohm * m)
• L = length (m)
• A = cross-sectional area (m^2)

Coulomb’s Law: Electrical Force between point charges

$$F_e=\frac{k{q}_1{q}_2}{r^2}$$

• F = electrostatic force (N)
• k = 9.0 * 10
• q1, q2 = charge 1, charge 2 (C)
• r = distance between charges (m)

Electric Field around a point charge

$$\vec{E}=\frac{kq}{r^2}=\frac{\overrightarrow{F_e}}{q}$$

• E = electric field (N/c)
• k = 9.0 * 10
• q = charge of the point charge (C)
• r = distance from the point charge

Electric Fields

1. Field lines must be tangent to the direction of the field at any point
2. Greater line density = Greater field magnitude
3. Go from + to -
4. Electric Field Lines Never Cross

Magnetic Fields

1. Go from N to S
2. Inside a bar magnet, go from S to N

Unit 4: The Wave Nature of Light

Wavelength / Energy Equations

Wavelength

$$\Delta \lambda =\frac{h}{mc}(1-\cos \theta )$$ $$\Delta {\lambda }_{total}\le N\Delta \lambda$$ $$\lambda =\lambda +\Delta \lambda$$

Energy

$$E^{\prime }=hf=\frac{hc}{\lambda }$$ $$E=Pt$$ $$\Delta E=E^{\prime }-E$$

Light and Snell’s Law

$\frac{sin{\theta }_1}{sin{\theta }_2}=\frac{n_2}{n_1}=\frac{v_1}{v_2}$ Wave goes from deep → shallow water

• Speed (v) slows down
• Wavelength ($\lambda$) gets smaller / shorter
• Angle ($\theta$) gets smaller
• Frequency (f) doesn’t change

Thin Film Interference (Rainbows)

1. Soap Film Rainbows
2. Eyeglasses
3. Oil spills on water

How to calculate path difference (air → oil → water)

• wavelength in each medium
• indexes of refraction

Double Slit Equations

$$\Delta x=\frac{\lambda L}{d}$$

l = screen to slit, d = between slit

$$sin{\theta }_n=\frac{\left(n-\frac{1}{2}\right)\lambda }{d}$$ $$sin{\theta }_m=\frac{n\lambda }{d}$$

Single Slit Equations

$$\Delta y=\frac{\lambda L}{w}$$

w = width of slit

$$sin{\theta }_n=\frac{n\lambda }{w}$$ $$sin{\theta }_n=\frac{\left(m+\frac{1}{2}\right)\lambda }{w}$$

Sample Question:

• Single slit = use w equations
• First fringe = n = 1
• Dark fringe = use sin theta equation

$$sin{\theta }_n=\frac{(1)(6.75\ast 10^{-5}nm)}{1.8\ast 10^{-4}}$$

Unit 5: Revolutions in Modern Physics

Special Relativity

Proper Length & Proper Time

• ”Proper” measurement comes from whoever is at rest with respect to both the start and the end time
• Ex: Spaceship moving away from Earth
  • Observer = proper length
  • Spaceship = proper time

Time Dilation Equation:

$$t=\frac{t_0}{\sqrt{1-\frac{v^2}{c^2}}}$$

Length Concentration Equation:

$$L=L_0\sqrt{1-\frac{v^2}{c^2}}$$

Ex:

$$d=40ly\ast \sqrt{1-(\frac{1\ast 10^8}{3\ast 10^8}{)}^2}=37.7ly$$ $$t=\frac{d}{v=\frac{37.7ly}{\frac{1}{3}c}}=113yrs$$

Alternatively, do it in reverse (find time, then use time dilation equation)

$$t=\frac{d}{v}=\frac{40}{\frac{1}{3}}=120yrs$$ $$120=\frac{t_0}{\sqrt{1-(\frac{1}{3}}{)}^2}\to t_0=113yrs$$