Note: This Is Still an Early-Stage Draft of This Document

Problem Solving

Note: this is still an early-stage draft of this document.

Overview of a step-by-step procedure:

Get a clear picture (mental or sketch) of the problem. Usually, this is a physical process, so imagine watching something happen over time, or relate it to things you observed in class or in real life. In some cases (force problems, torque problems), it’s required to draw a new sketch of your own, since it’s nearly impossible to solve correctly without such a sketch. Physics is not just a special kind of algebra.
If possible, try to get a ballpark estimate of what your final answer should be. You may get good at this after doing many physics problems, or by having a good feel for the way that sort of physical system works.
Translate the information in the problem into variables, with the appropriate letters and units. Make sure that you list prominently what you’re solving for. Note: some of the key information might not be given as numbers, but instead as words in the text of the problem. Also: be careful about minus signs, and vectors vs. scalars.
Try to decide which equations are true and useful for this problem. The best way to get good at this is by doing a lot of problems, though a solid understanding of the meaning of each equation is very useful no matter how many or how few problems you’ve done.
Choose an equation to try, and make sure you understand exactly what each of the variables in that equation means. Without that sort of clear understanding, you’ll make mistakes more often than not.
Look at what you know and what you’re trying to find, to make sure that the equation you chose is indeed both true and useful for the problem you’re trying to solve. You may need to use multiple equations, so it would be helpful to map out your strategy for getting your final answer.
Plug in your numbers, verifying again that the variable in the equation really does have the same physical meaning as the number you’re using. This includes vector vs. scalar distinctions, being careful with minus signs, being careful with units, etc.
Run through the algebra, using units in all steps.
Get an answer.
Look at your answer: are the units correct? Does the size of your answer make sense (especially given your guess in step 2)? If not, check your work. It is actually more common to make an error in the concept stage (steps 1-6) than in the algebra (8-9).

Note that only steps 7, 8, and 9 involve calculation. All of the other steps are important, and are usually worth credit where partial credit is available. Those other steps are not a waste of time, they are necessary steps toward understanding processes and getting correct answers. Physics is a way of understanding physical systems, not just a process of plug-and-chug through equations.

The main equations:

Equation / Is only true when... / May be useful when it is true, and... / Other
The 4 kinematic equations.
e.g. Δx = vixΔt + ½ ax(Δt)2
Δx = ½(vix + vfx) Δt
vfx = vix + axΔt
vfx2 = vix2 + 2 axΔx
Δy = viyΔt + ½ ay(Δt)2
etc. / 1. you are dealing with a purely linear problem, or you are dealing with one component of 2- or 3-dimensional motion, AND
2. the acceleration in the direction you are considering is constant. / 1. You have some “initial” situation and some “final” situation that you care about.
2. You have enough information to get any 3 of the 5 variables (e.g. Δx, vix, vfx, ax, Δt) for either direction.
3. The net force is constant (therefore acceleration is constant too). / You must split this up into components.
The 4 rotational kinematic equations.
e.g. Δθ = ωiΔt + ½ α(Δt)2 / the angular acceleration of your system is constant. / The net torque is constant (therefore angular acceleration is constant too). / You may use whatever angular units you want, as long as you stay consistent with them.
Newton’s Laws
I. If ΣF = 0 then a = 0
II. ΣF = ma or Fnet = dp/dt
III. FAB = – FBA / are always true, as long as you are completely clear about what object(s) the forces are acting on. / You must draw free body diagrams.
You must split this up into components.
Newton’s Laws for Torque
Στ = Iα or τnet = dL/dt
τAB = – τBA / 1. are always true, as long as you are clear about what object(s) the torques are acting on, AND
2. α must use radians as the angular unit / You must draw free body diagrams.
Work and Energy:
Wnc + Ei = Ef / is always true. / 1. You have some way of knowing what Wnc is (it may be zero, but you need to be sure).
2. You have some “initial” situation and some “final” situation that you care about.
3. You have changes in speed and changes in height.
4. You have a mass and a spring.
5. You have something rolling without slipping. / 1. Wnc is zero if you have an elastic collision (and for no other collisions).
2. Wnc is zero for cases of rolling without slipping (as long as there are no applied forces)
Conservation of Momentum
Ptot_i = Ptot_f / 1. ...the net force on your system of objects is zero OR
2. ...the net force times the time between “initial” and “final” is sufficiently small. This will be true any time something is called a “collision” and you have linear types of motion (not spinning) OR
3. ...the net force on your system along a certain direction is zero, and you only apply this equation for the component of momentum along that direction. / 1. You have different objects pushing/pulling/hitting on a frictionless surface.
2. You see the word “collision.” / You must split this up into components.
Conservation of Angular Momentum
Ltot_i = Ltot_f / 1. ...the net torque on your system of objects is zero OR
2. ...the net torque times the time between “initial” and “final” is sufficiently small. This will be true any time something is called a “collision” and you have spinning happening. / 1. You have different objects pushing/pulling/hitting with a frictionless axle.
2. You see the word “collision” and you clearly have some rotation at some time.

Supporting equations

Equation / Is only true when... / Is often used when/with... / Other
Centripetal Acceleration
ac = ar = v2/r = rω2 / 1. ... you have circular motion, so that you can define a radius AND
2. ... ω is in radians / 1. you have a circular orbit
2. Newton’s Laws
3. rotational kinematics / “Centripetal acceleration” and “radial acceleration” mean the same thing.
Tangential Acceleration
at = d|v|/dt / you have circular motion / something going around in a circle is changing speed / If the speed |v| isn’t changing, then at is zero.
Rotation and Translation
Δs = rΔθ
v = rω
at = rα / 1. you have a thread unwinding without slipping from a pulley OR
2. you have an object rolling without slipping on a surface / 1. rotational kinematics
2. energy conservation
3. angular momentum / The angular unit MUST be radians.
Uniform Circular Motion
v = (2πr)/T / you have uniform circular motion (that is, motion around a circle at constant speed) / 1. you have a circular orbit
2. the words “uniform circular motion” are used / In uniform circular motion, α and at are zero, ac (or ar) is constant but nonzero.
Friction:
fs ≤ μsN fk = μkN / 1. you have any 2 surfaces in contact
2. N is the normal force between the two surfaces being considered.
3. fs for static friction, only when no slipping going on.
4. fk for kinetic friction, only when there is slipping. / 1. Newton’s Laws
2. Work and energy / Be careful about direction of the force: kinetic friction opposes the actual slipping, static friction opposes the tendency to slip.
Spring Forces:
|Fspring| = k |x| / 1. you have a Hooke’s Law spring
2. |x| is the magnitude of the distance the spring has been stretched from its equilibrium position. / 1. Newton’s Laws
2. Spring potential energy / 1. k is called the spring constant. It has units N/m.
2. You must draw a free body diagram.
3. The direction of the force from the spring onto the object pulling it is opposite to the direction the spring was stretched from its equilibrium postion.
Newton’s Law of Gravitation:
/ 1. you have two point masses, or two spherical masses.
2. r12 is the distance between the centers of mass of the 2 objects (not the radius of either object). / 1. Newton’s Laws
2. Centripetal Acceleration / 1. Gravity is always an attractive force.
2. If you have more than 2 masses, you must calculate the strength of each gravitational force separately, then do a vector sum.
Torque:
a. τ = r×F
b. τ = rF sinθ = Fd = Fl / a) r is the vector starting from your chosen pivot point, to the location where the force F starts.
b) d or l mean the same thing: the shortest distance between the pivot point, and the line along which the force acts. / 1. Newton’s Laws for Torque
2. Rotational work and energy / 1. θ is the angle between the vectors r and F. Make a side sketch to be sure you have the correct angle.
2. Be sure you remember how to apply the right hand rule.
Moment of Inertia:
a) I = Σmiri2
b) ICM = KMR2
c) I = ICM + Md2 / a) only for one or more point masses where ri is the radius of the circle that that mass makes. Any object with a size much smaller than the radius of the circle it sweeps out may be considered a point mass.
b) only for a shape about the center of mass. The equation at left is a shorthand for the moments of inertia given in the figure in the text. K is some constant, M is the total mass of the shape, and R is the radius of the shape. e.g. for a solid cylinder, I = ½MR2.
c) only for an axis of rotation other than the center of mass, where you know the moment of inertia of that shape for an axis through the center of mass and parallel to your actual axis. (“Parallel axis theorem”). d is the distance from the actual axis of rotation to the center of mass of the shape. / 1. Torque
2. Rotational Kinetic Energy
3. Angular Momentum / 1. The moment of inertia of multiple objects is just the sum of the moments of inertia of each alone.
2. Be careful about the difference between the radius of an object, and the radius of the circle an object sweeps out.
Wave Speed:

Angular Frequency (SHM):

Definitions equations

Equation / May be useful when... / Is often used with... / Other
vavg = Δx/Δt v=dr/dt
aavg = Δv/Δt a=dv/dt
ωavg = Δθ/Δt ω = dθ/dt
αavg = Δω/Δt α = dω/dt
,
p = mv
L = r×p L = Iω
Ktr = ½mv2
Krot = ½Iω2
Ug = mgh
Uspr = ½kx2
W = τ Δθ
P = τω

ω = (2π rad)f = (2π rad)/T
k = (2π rad)/λ

Other notes:

Know the trig. functions, for splitting up vectors into their components. Draw triangles if necessary.

Know how to take dot products and cross products, when given magnitudes and directions, and when given vector components or unit vector notation.

Know how to apply the right hand rule, to get the direction of the result of a cross product.