A review

...of almost everything

Vectors

Pythagorus $$ a^2 + b^2 = c^2$$
Trig $\sin$, $\cos$, $\tan$,
$\sin^{-1}$, $\cos^{-1}$, $\tan^{-1}$ 		Needed to get the components or find resultants

Unit Vectors

A unit vector is a vector that has a magnitude of exactly 1, and points in a given direction.

If $\mathbf{\hat{i}}$ points behind you, and $\mathbf{\hat{j}}$ points straight up, then which way should $\mathbf{\hat{k}}$ point?

  1. up
  2. down
  3. left
  4. right
  5. forward
  6. backward

Dots and Crosses

Dot Product $$\mathbf{a} \cdot \mathbf{b} = a b \cos \theta$$ $\mathbf{a} \cdot \mathbf{b} = (a_x \hat{\mathbf{i}}+a_y \hat{\mathbf{j}}+a_z \hat{\mathbf{k}})\cdot(b_x \hat{\mathbf{i}}+b_y \hat{\mathbf{j}}+b_z \hat{\mathbf{k}}) $
$\mathbf{a} \cdot \mathbf{b} = a_x b_x + a_y b_y + a_z b_z$
Cross Product $|\mathbf{a} \times \mathbf{b}| = a b \sin \phi$ $\mathbf{a} \times \mathbf{b} = (a_x \hat{\mathbf{i}}+a_y \hat{\mathbf{j}}+a_z \hat{\mathbf{k}})\times(b_x \hat{\mathbf{i}}+b_y \hat{\mathbf{j}}+b_z \hat{\mathbf{k}}) $
$\mathbf{a\times b} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ a_x & a_y & a_z\\ b_x & b_y & b_z\\ \end{vmatrix}$

If you want the cross product of $\mathbf{A}$ and $\mathbf{B}$ to be as large as possible, you should orient the vectors:

  1. Perpendicular to each other
  2. Parallel to each other
  3. 180 degrees to each other
  4. 45 degrees to each other

Kinematics

The way things move.

Displacement $$\mathbf{x}$$
Velocity $$ \mathbf{v} = \frac{d\mathbf{x}}{dt}$$
Acceleration $$\mathbf{a} = \frac{d\mathbf{v}}{dt} = \frac{d^2\mathbf{x}}{dt^2}$$

True or false:

"If the slope of an object's velocity vs. time curve is zero, then the object must be at rest"

  1. True
  2. False

True or false:

"The area under a velocity vs. time curve will tell you how far an object has traveled."

  1. True
  2. True, but only for constant acceleration
  3. False

Kinematic Equations

Velocity $$\mathbf{v} = \mathbf{v}_0 + \mathbf{a}t$$
Displacement $$\mathbf{x} = \mathbf{\bar{v}} t = \frac{\mathbf{v} + \mathbf{v}_0}{2}t$$
Displacement $$\mathbf{x} = \mathbf{x}_0+ \mathbf{v}_0 t + \frac{1}{2} \mathbf{a} t^2$$
Speed $$v^2 = v_0^2 + 2a \Delta x$$

The x position vs time for a car and truck, both traveling on a straight track, is shown. What happens at point P?

  1. The two vehicles have the same velocity.
  2. The car passes the truck.
  3. The truck passes the car.
  4. They both stop accelerating.

2-D kinematics

Motions happen in 2 dimensions. Treat these dimensions separately.

x-Displacement $$\mathbf{x} = \mathbf{x}_0+ \mathbf{v}_{x0} t + \frac{1}{2} \mathbf{a_x} t^2$$
y-Displacement $$\mathbf{y} = \mathbf{y}_0+ \mathbf{v}_{y0} t + \frac{1}{2} \mathbf{a_y} t^2$$

An object is launched across a field and travels as a projectile. At the point $P$, the acceleration of the object is:

  1. Smaller than everywhere else along the trajectory.
  2. zero - it's not accelerating anywhere along the trajectory.
  3. Pointed down and equal to 9.8 meters per second.
  4. zero - but only for an instant, then it will start increasing.

What angle of launch will maximize a projectile's range? (Assume no air resistance)

  1. Around 1 or two degrees.
  2. 15 degrees.
  3. 45 degrees.
  4. 60 degrees

Forces

Newtons' Three Laws

$\sum \mathbf{F} = m \mathbf{a}$

You're driving up a hill at a constant speed of 45 mph. Summing all the forces acting on your car will yield:

  1. A net force down (i.e. straight down)
  2. A net force up (i.e. towards the sky)
  3. A net force equal to zero
  4. A net force pointed up the the hill
  5. A net force pointed down the hill

The same force is applied to two different carts on the air track - Cart 2 has a mass three times greater than Cart 1. After some amount of time, how will the velocities of the carts compare?

  1. Cart 2 will be going 3 times faster
  2. Cart 1 will be going 3 times faster
  3. Cart 2 will be going 9 times faster
  4. Cart 1 will be going 9 times faster
  5. They will be going the same speed

Catalog of Forces

Pushing/pulling General, non-descript interactions
Tension Forces applied using ropes.
Gravity Attractive force between all objects with mass
Normal Interaction force - points perpendicular to plane of contact
Friction A model to explain how surfaces interact
Drag Velocity Dependent air resistance
Springs Hooke's Law
Buoyancy Force "The weight of the fluid displaced"

Gravity

Newton's Law of Gravitation $$\mathbf{F}_G = -\frac{G m_1 m_2}{r^2}\mathbf{\hat{r}}$$
little g $$g = \frac{G m_E}{R_E^2}$$
weight (near surface of the earth) $$\mathbf{F}_G = -mg\mathbf{\hat{j}}$$

An apple of mass $m_{apple}$ feels a force of attraction to the earth equal to $m_{apple} g$. What force does the earth feel due to the presence of this apple.

  1. $m_{apple} g$
  2. $m_{Earth} g$
  3. $0$
  4. Not enough information.

Friction

static friction max $$f_s^\textrm{max} = \mu_s F_N$$
kinetic friction $$f_k = \mu_k F_N$$

What are the units of $\mu_s$?

  1. Newtons (same as force)
  2. $m/s^2$
  3. kg
  4. 1/Newtons
  5. No units

True of False:

Kinetic friction force increases linearly with the velocity of the sliding object

  1. True
  2. False

Springs

Hooke's Law / Restoring Force $$\mathbf{F}_{spring} = - k \mathbf{x}$$

Circular Motion

centripetal acceleration $$a_c = \frac{v^2}{r}$$
period / speed $$v = \frac{2 \pi r}{T}$$

For an object traveling in uniform circular motion at constant speed, its velocity and acceleration vectors point:

  1. Both point inward, towards the center of the circle
  2. vel: inward / accel: outward, away from the center
  3. vel: tangential / accel: inward, towards the center
  4. vel: outward / accel: tangential
  5. vel: tangential / accel: outward, away from the center

Work & Energy

Work Non-constant Force Constant Force
$$W = \int \mathbf{F}\cdot d\mathbf{x}$$ $$W = \mathbf{F} \cdot \mathbf{d}$$
kinetic energy $$KE = \frac{1}{2}mv^2$$
gravitational potential energy $$PE = m g h$$
conservation of energy $$W_\textrm{NC} = E_f - E_i$$
Force and Potential $$\Delta U = -\int_{x_i}^{x_f} F(x)\;dx$$ $$F = -\frac{dU}{dx}$$

If you do negative work on a particle, which of the following must be true?

  1. The particle is moving in the negative direction
  2. Your force is directed in the negative direction
  3. Both the particle's velocity and the force are pointed in the negative direction
  4. The particle's kinetic energy will decrease

Power

Power: change in energy per time

Find the rate of energy change $$P = \frac{dE}{dt}$$

Impluse

Impulse Non-constant Force Constant Force
$$\mathbf{J} = \int_{t_i}^{t_f} \mathbf{F}(t) \; dt$$ $$\mathbf{J} = \mathbf{F} \Delta t$$
momentum $$\mathbf{p} = m \mathbf{v}$$
impulse-momentum $$\mathbf{J} = \Delta \mathbf{p}$$

And, momentum is always conserved.

$$\mathbf{F} = \frac{d\mathbf{p}}{dt}$$

Rotational Kinematics

Arc length $$ r\theta = s$$
angular displacement $$\theta = \theta_0 + \omega_0 t + \frac{1}{2}\alpha t^2$$
angular velocity $$\overline{\omega} = \frac{\Delta \theta}{\Delta t}$$
angular velocity $$\omega^2 = \omega_0^2 + 2\alpha \theta$$
$\boldsymbol{\omega}$ is a vector R.H.R.

The angular velocity of the earth about its own axis points in the direction:

  1. From the center of the earth to the north pole
  2. From the center of the earth to the south pole
  3. Eastward (from NY towards the Atlantic)
  4. Westward (from NY towards NJ)
  5. The earth has no angular velocity: $\omega = 0$.

The angular acceleration of the earth about its own axis points in the direction:

  1. From the center of the earth to the north pole
  2. From the center of the earth to the south pole
  3. Eastward (from NY towards the Atlantic)
  4. Westward (from NY towards NJ)
  5. The earth has no angular acceleration: $\alpha = 0$.

Rotational Dynamics

Torque $$\tau = \mathbf{r} \times \mathbf{F}$$ $$\tau = r F \sin \phi$$
Moment of Inertia $$I_\textrm{point-mass} = m r^2$$ $$I_\textrm{parallel axis} = I_\textrm{C.o.M} + mh^2$$
Rotational 2nd law $$\boldsymbol{\tau} = I \boldsymbol{\alpha}$$
Kinetic energy of rotation $$KE_\textrm{Rot} = \frac{1}{2}I \omega^2$$

And, angular momentum is always conserved.

$$\mathbf{L} = I \boldsymbol{\omega}$$ $$\boldsymbol{\tau} = \frac{d\mathbf{L}}{dt}$$

An object is rotating. If both its moment of inertia $I$ and its angular velocity $\omega$ are doubled, what will happen to the rotational kinetic energy?

  1. Decreases by a factor of 8
  2. Decreases by a factor of 4
  3. Decreases by a factor of 2
  4. Stays the same
  5. Increases by a factor of 2
  6. Increases by a factor of 4
  7. Increases by a factor of 8

Oscillations

General Restoring Force $$\mathbf{a} = \frac{d^2\mathbf{x}}{dt} = \ddot{\mathbf{x}} = -\cal{O} \mathbf{x}$$ Solution: $x(t) = A \cos (\omega t + \phi)$
Mass on a spring $$\omega = 2 \pi f = \frac{2 \pi}{T} = \sqrt{\frac{k}{m}}$$
Potential Energy $$U_\textrm{sp} = \frac{1}{2}k x^2$$
Physical Pendulum $$T = 2 \pi \sqrt{ \frac{I}{mgL}}$$
Simple Pendulum $$T = 2 \pi \sqrt{ \frac{L}{g}}$$

Two identical spring/mass systems are shown. Both springs have the same $k$, and the masses are equal. The only difference is one is oriented vertically with respect to gravity, the other is horizontal. How will their frequencies of oscillations compare?

  1. System A will oscillate faster ($f_A > f_B$)
  2. System B will oscillate faster ($f_B > f_A$)
  3. The will have the same oscillation frequency ($f_A = f_B$)

On a regular swing set at the playground, and adult (big $m$) and a child (small $m$) are both swinging on the swings. Can they swing with the same frequency?

  1. No
  2. Yes, as long as the lengths of the chains holding the swings are different
  3. Yes, as long as the lengths of the chains holding the swings are the same

Simple Harmonic Motion

Position $$x(t) = A \cos (\omega t)$$
Velocity $$v(t) = - A \omega \sin (\omega t)$$
Acceleration $$a(t) = - A \omega^2 \cos (\omega t)$$
Damping $$x(t) = e^{-t/\tau} A \cos (\omega t + \phi)$$

Fluids

density $$\rho = \frac{m}{V}$$
Pressure $$P = \frac{F}{A}$$
Change in pressure due to depth $$P_2 = P_1 + \rho g h$$
Archimedes' Principle. $$F_B = W_\textrm{fluid}$$

How will the lifting force caused by buoyancy on a ballon depend on its height above the ground?

  1. It will get weaker the higher up the balloon is
  2. It won't change at all even if the balloon reaches very high altitudes
  3. It will get stronger the higher up the balloon is

Moving Fluids

Continuity $$ \rho_1 A_1 v_1 = \rho_2 A_2 v_2$$
Bernoulli $$P_1 + \frac{1}{2} \rho v_1^2 + \rho g y_1 = P_2 + \frac{1}{2} \rho v_2^2 + \rho g y_2 $$

Temperature and Heat

Thermal Contraction $$\Delta L = \alpha L_0 \Delta T$$
Specific Heat $$Q = cm \Delta T$$
Phase changes $$Q = mL_F \; \textrm{or} \; mL_V$$
Thermal Conductivity $$Q = \frac{ k A \Delta T \; t }{L}$$