A review of all of the main concepts discussed in PHYS 118 at UNC, which was also a decent bit of review from when I studied for the SAT physics subject test (arguably the only test I ever did really well on).
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Dot product:
mag(A) * mag(B) * cos(angle between)
Cross product:
Mag is mag(A) * mag(B) * sin(angle between). Right hand rule is convention.
Newton's first law:
Objects remain at rest or remain in motion unless acted on by a net force. Inertia (tendency for objects to remain at constant velocity.)
Newton's second law:
F=ma.
Newton's third law:
To every action, there is an equal and opposite reaction.
Work:
Product of force times displacement. Measured in joules (kg*m^2/s^2). Essentially dot product of force * displacement. Work is the transferring of energy into a force acting over a distance. Specifically _for an object_, net work W = delta KE (work = change in kinetic energy). You can think about this as applying a force over a distance to a ball, which will give it some velocity (Kinetic Energy). Also, for an object, net work W = -delta PE (work = negative change in potential energy). You can think of this as dropping a ball, it had potential energy which was decreased when gravity did work on it (applied its force over a distance). Good to remember that the energy is constant, so any change in potential energy must be explained by some corresponding change in kinetic energy and vice versa.
Pulley:
If holding up a mass with a pulley, the weight is distributed between the two sides of the pulley, so the tension in each is just T = mg/2 for each. For strange movement problems, imagine shortening the string, for the setup that I hate, the string gets shortened half as much so the falling block accelerates twice the other’s acceleration.
Energy:
Is the ability to do work. Be sure to consider the energy of the entire system. Delta(KE) = - Delta(Potential Energy) only applies for the entire system.
Power:
The rate at which work is done. So deltaW/deltat or deltaE/deltat.
Spring:
F = -kx. Potential energy is 1/2 k x^2. Kinetic energy is maximized at the equilibrium position. It’s going to be whatever the maximum displacement was.
Momentum:
F = change in momentum / change in time. Impulse: change in momentum = force * time. So F * delta (t) = delta( mv ). Impulse = integral(F*t).
Collisions:
Kinetic energy is not conserved in inelastic. Momentum is conserved in both.
\frac(-b \pm \sqrt{b^2 - 4ac}}{2a}
Static and kinetic friction: be sure to think of the value [0,1] as the fraction of its weight that must be exerted to get it to start/continue moving. As you increase weight, the resistive friction force will change.
Intuition about impulse: imagine a puck coming at you at a velocity, and you want to know how much force to apply to it to get it to rest in a certain amount of time. Obviously this is going to be mv = F*t, because think about how as you apply a force it, F, it's going to decelerate at a = F/m. So if you want to get a puck of a certain mass to get to rest in 1 second, then you're going to need to decelerate at a=F/m, which means you need v / F/m to be = t, and v/ F/m = mv/F.
Force decreases momentum points, just like acceleration decreases velocity points.
When you push off a wall, the equal and opposite reaction is that you exert a force on the wall of F, and the wall exerts a force on you of F. So you realize that, when looking at just you as the object, you are just having the force of the wall exerted on you (equal to F), and so you'll accelerate backward.
Modern Physics:
Special Relativity postulates:
The laws of physics are the same in all inertial reference frames.
The speed of light is constant in every reference frame.
Consequences:
Absence of simultaneity:
If you’re observing two flashes of light that are coming at an object between the flashes and will hit the object at the same time (in the object’s frame), and you’re moving relative to the object, then you’re going to see the emission of the lights as not simultaneous.
Time Dilation:
Time slows down for objects traveling close to light speed. Both feel like time is moving at the same pace, but if they each have a stopwatch and measure a time interval, if the person moving measures t_0, then the person still will measure t = t_0 / sqrt(1-(v^2/c^2)). Similarly, t = gamma*t_0 (gamma is this factor). From the person moving, the person still will appear to experience time more slowly. And from the person still, the person moving will appear to experience time more slowly.
Length Contraction:
If you’re standing still, and you measure the length of a train, and then measure that train again when moving at half the speed of light, the train appears shorter. The rest/proper length, l_0, contracts to length L by L = l_0 sqrt(1-(v^2/c^2)).
Velocity Addition:
If an object is traveling with speed v_0 in one frame and another frame is traveling relative to that frame at speed u, then the speed of the object in the other frame, v, is related by v = (u+v_0) / (1+ (u*v_0/c^2)).
Lorentz Transformation:
Say you have a moving frame (primed) with a relativistic velocity, v, with respect to another (non-primed) frame. Then the time an event occurs t’ and location x’ in terms of the stationary frame is:
x’ = (x-vt)/sqrt(1-v^2/c^2).
t’ = (t-(vx)/c^2)/sqrt(1-v^2/c^2)
The inverse is just changing the numerator’s minuses to pluses:
x = (x’ + vt’)/sqrt(1-v^2/c^2)
t = (t’+(vx’/c^2))/sqrt(1-v^2/c^2)
Momentum of particle with relativistic speed:
p = mv/sqrt(1-v^2/c^2)
Energy - mass conversion (rest mass):
E = mc^2
Kinetic energy with relativistic speed:
KE = (1/sqrt(1-(v^2/c^2)) -1)*mc^2. Note that this is infinity when v=c. So infinite energy is required to accelerate massive object to speed of light. This is just your relativistic total mass-energy minus your rest mass, so obviously you’ll just get the kinetic from this.
Total energy of a particle:
E = (1/sqrt(1-v^2/c^2)*mc^2) or = sqrt(p^2c^2 +(m_0)^2c^4), that is E^2 = m^2c^4 = p^2c^2 +(m_0)^2c^4.
From the book:
Space-time interval is constant for all reference frames: s^2 = (c*Delta(t))^2 - (Delta(x))^2.
Circular Motion:
An object going over a hill: think about how, to keep changing velocity while going over the hill, a certain force toward the center of the circle is required, so essentially the centripetal force is stealing away from gravity’s force, because gravity’s force is being redirected to keep the object moving along the path, this means the normal force is decreasing, and is exactly zero when the centripetal acceleration is equal to the force of gravity on the mass.
Kinetic energy of rolling/rotating object, K = 1/2 mv_{cm}^2 + 1/2 * I *v^2/r^2, where v^2/r^2 is just omega (rotational velocity). where v_cm is the velocity of center of mass. Rolling masses lose to etecfy to friction.
Moment of inertia defines, with respect to an axis, how much torque is needed to rotate the object at a certain acceleration about that axis.
Parallel axis theorem: I = I_cm + md^2, where d is distance to the parallel axis.
Torque = I * a. Where I is moment of inertia and a is angular acceleration. This is equivalent to f=ma.
Moment of inertia calculation: I = mr^2, for all masses in system. You essentially do integral of radius^2 * infinitesimal vertical/horizontal mass element (whatever is orthogonal to the radius, essentially) times the density (m/L) from 0 to the length of the object.
With masses on opposite sides of an axis of rotation (essentially, along different radii), add their moments of inertia separately.
For a door rotating on its hinges, the I_about hinge = m*L^3 / 3.
Torque = F*r*sin(theta). Direction of torque is given by right hand rule. Object will not rotate without some net torque. Also change of the angular momentum over change in time.
Angular frequency: how many full rotations occur per time. Period is the inverse. Should know the formula f=omega/2pi.
Angular velocity (omega): change in angle over change in time.
Instantaneous linear velocity: omega*r.
T = 2*pi*sqrt(I/mg*L_cm)
(rotational equivalents): omega^2 = omega_0^2 + 2alpha*(psi-psi_0)
Direction of angular velocity is given by right hand rule. Curl fingers along direction of spin.
Moments of Inertia:
Hollow cylinder:
1/2 M (m_o^2 + m_i^2).
Uniform sphere:
2/5 M R^2
Uniform ring:
MR^2
Uniform disk/cylinder:
1/2MR^2
Uniform Rod about center:
1/12 ML^2
Uniform Rod about end:
1/3 ML^2
Angular momentum: L = I*omega. Angular momentum always points with angular velocity.
Gravitation:
Force of Gravity = G(m_1 * m_2) / r^2
G = 6.67*10^-11.
Acceleration on planet: a = G * (m_planet)/r^2
Waves we need to know (Chapter 14?):
v = wavelength * frequency.
Speed of wave on string: v=sqrt(Ftension/(total mass/total length))
Check this out: Doppler shift: frequency = f_0 ( c+v_r / c+v_s). Where v_r positive when moving towards source. v_s positive when moving away from receiver.
Waves: (note on exam)
Destructive interference: Happens when one is positive amplitude and one is negative. the waves actually aren’t destroyed, they will continue along exactly as they did before they met.
Constructive interference: Happens when both are in the same direction. They will momentarily add together and then continue to pass through each other.
Standing waves: if you are oscillating at a multiple of one half the wavelength of the wave, then you’ll have a standing wave. Distance between nodes and antinodes is wavelength/2.