understanding physics 01: derivation of fundamental linear motion equations (cont'd)

i finally got around to doing some of the work this morning (note: since i don't know how to insert the integral symbol, i'll use an ampersand instead):

• v = dx/dt, vdt = dx, &vdt = &dx -> vt = x (assuming x0 is zero, v is avg velocity)
• a = dv/dt, adt = dv, &adt = &dv -> at + v0 = v
• distance (w/o vf): x2 - x1 = v2t - v1t, v2t = (1/2)at^2 -> x2 = x1 -v1t + (1/2)at^2
• velocity (w/o t): x = vt, v = [(v + v1)/2], t = (v - v0)/a -> x = [(v + v0)/2][(v - v0)/a] = [v^2 - (v)(v0) + (v)(v0) - v0^2]/2a = (v^2 - v0^2)/2a, 2ax = v^2 - v0^2, v^2 = v0^2 + 2ax

understanding physics 01: derivation of fundamental linear motion equations

i'm tutoring someone in physics again and i was thinking about how to explain why conservation of energy is so fundamental in physics but specifically in classical mechanics while driving to work this morning. one of the reasons i love physics more than other sciences is that it conceptual understanding doesn't require excessive blind memorization of symbols, constants, tables, taxonomic ranks, etc. If you can remember Newton's laws, understand that energy is conserved, know how work relates to force, distance and direction, you pretty much just need two equations and basic calculus to solve any problem through intermediate mechanics: F=ma and v=dx/dt. there are numerous equations you could memorize to solve problems involving linear and angular motion, but all of them are derived from v=dx/dt (i guess you also need to know that there are 2pi radians in a circle for angular motion problems). i remember that at the start of every exam, i would derive the equations for distance, velocity, acceleration and time and, for whatever reason, this would always calm me down. i've decided to see if i can still do it and will post my notes when i finish or give up

classical mechanics problem: box on an incline

for some reason i'm in a physics mood so i thought i'd share a classic problem that anyone that took elementary mechanics would come across at some point and how to solve it .

Q: a box of mass m sits motionless on an incline. what is the maximum angle Θ of the incline before the box begins to slide down?

A: all good little physics students know the first step is to draw a diagram and then add forces:


where Θ is the angle of the incline, μ is the coefficient of friction and N is the normal force

now lets disect the horizontal and vertical forces on the box:

ΣFx = horizontal component of gravity - friction = mgsin(Θ) - μN = 0 (box is stationary)

-> mgsin(Θ) = μN

ΣFy = vertical component of gravity - normal force = mgcos(Θ) - N = 0 (box is stationary)

-> mgcos(Θ) = N

we have two equations and three unknowns (Θ, μ, and N). the second equation gives us a N in terms of Θ, so let's plug it into the first equation and then solve for Θ:

mgsin(Θ) = μ(mgcos(Θ))

cancel out mg on both sides and get μ by itself:

μ = sin(Θ)/cos(Θ) = tan(Θ)

bring tan to the other side and we are done:

Θ = arctan(μ)

the solution tells us that the largest angle of incline before the box moves is dependent on the coefficient of friction of the incline.

mercy i loves me some classic mechanics :)