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Common Prefixes Value Name Symbol 10−9 nano n −6 10 micro µ 10−3 milli m −2 10 centi c −1 10 deci d 103 kilo k 6 10 mega M 109 giga G Common Length 1 in = 2.540 cm 1 m = 39.37 in 1 m = 3.281 f t 1 f t = 0.3048 m 1 km = 0.6214 mile 1 mile = 1.609 km 1 mile = 1609 m Conversions Force 1 N = 0.2248 lb Energy 1 cal = 4.184 J 1 kW h = 3.60 × 106 J Power 1 hp = 746 W On Earth: A mass of 1 kg equals a weight of 2.20 pounds ~ = (A, θ) : magnitude and angle (CCW) with +X axis. Polar: A ~ = (Ax , Ay , Az ) or A ~ = Ax î + Ay ĵ + Az k̂ Cartesian: A ~·B ~ = |A||B| cos φ (angle between the vectors). Scalar Product: A ~ =A ~ × B. ~ Note A ~×B ~ = −B ~ × A. ~ Vector Product: C Cx = Ay Bz −Az By Cy = Az Bx −Ax Bz Cz = Ax By −Ay Bx Velocity average: vavg = ∆x/∆t Acceleration average: aavg = ∆v/∆t instantaneous: v = dx/dt. instantaneous: a = dv/dt. Straight-line (i.e. 1-D) motion with constant acceleration. v = vo + at vavg = vo + 21 at vavg = ∆x/∆t 1 2 x = xo + vo t + 2 at x = xo + vavg t vavg = 12 (vo + v) v 2 = vo2 + 2a(x − x0 ) Freely-falling bodies: ~a = ~g , where g = 9.80 m/s2 . Generic Equations and definitions (even when ~a varies) ~v = d~r/dt ~vavg = ∆~r/∆t ~aavg = ∆~v /∆t ~r = xî + y ĵ + z k̂ ~r = ~ro + ~vavg t vx = dx/dt vy = dy/dt vz = dz/dt ax = dvx /dt ay = dvy /dt az = dvz /dt Vector Equations of Motion (constant ~a ) vx = vox + ax t vy = voy + ay t vz = voz + az t 1 1 2 2 x = xo + vox t + 2 ax t y = yo + voy t + 2 ay t z = zo + voz t + 12 az t2 ~v = ~vo + ~at ~r = ~ro + ~vo t + 21 ~at2 v 2 = vo2 + 2ax ∆x + 2ay ∆y + 2az ∆z 2 2 2 + 2ax ∆x + 2ay ∆y + 2az ∆z vx2 = vox vy2 = voy vz2 = voz Projectile Motion: ~a = ~g with g = |~g | = +9.80 m/s2 . If +Y points upward (ax = 0, ay = −g) then: x = xo + vox t vx = vox y = yo + voy t − 21 gt2 vy = voy − gt If launched at origin with ~vo = (vo , θ) : vox = vo cos θ voy = vo sin θ g 1 2 2 y = (tan θ)x − ( 2v2 cos x = (vo cos θ)t y = (vo sin θ)t − 2 gt 2 θ )x o 2 Apogee: h = (vo2 sin θ)/(2g) above the launch point. tA = (vo sin θ)/(g) Rmax = vo2 /g at θ = 45o If yf inal = yinitial : Range R = (vo2 sin 2θ)/(g) Uniform Circular Motion: Radial acceleration: ar = v 2 /r Period: T = 2πr/v or Total flight time: t = 2tA v = 2πr/T Newton’s First Law: ΣF~ = 0 implies ~a = 0 (and vice versa). Newton’s Second Law: ΣF~i = m~a. (I.e., ~a = (ΣF~i )/m) Newton’s Third Law: F~1 on 2 = −F~2 on 1 (interaction) P P Equilibrium: At every point Fx = 0 and Fy = 0 separately. (Motion ok, just no acceleration.) Friction: fk = µk n fs ≤ fs,max = µs n Direction opposes motion. Work: W = F~ · d~ Kinetic Energy: K = 21 mv 2 Average Power: Pavg = ∆W/∆t Work and Power R 1-D Varying Force: W = if Fx dx P Work - Kinetic Energy Theorem: K = Ko + Wi Power: P = dW/dt = F~ · ~v Potential Energy : Ug = mgh Us = 12 kd2 (careful with h and d) Mechanical Energy E = K + Ug + Us (K = Kcm + Krot if object rotates) Conservation of Energy (K + Ug + Us )2 = (K + Ug + Us )1 + ΣWother Linear Momentum of a moving object: p~ = m~v P Generalized Newton’s Law: F~ = d~p/dt Impulse: J~ = ∆~p = p~2 − Rp~1 Constant force: J~ = F~ ∆t t f ~ Time-varying force: J~ = ti F~ dt = F~avg ∆t | F~avg = J/∆t or F~avg = ∆~p/∆t. P P P P Collisions : Total momentum is conserved: p~i = p~f ie mi~vi = mf ~vf Inelastic Collision : some mechanical energy is lost (most common case) Perfectly inelastic : objects stick together, move off as single object. Max loss of energy. Elastic collisions: the momentum and the total kinetic energy are both conserved (rare). Trig: sin (2θ) = 2 sin θ cos θ Quadratic formula: if Ax2 + Bx + C = 0 then x = √ −B± B 2 −4AC 2A Analogies between Rotational and Linear Motion Rotational Motion About a Fixed Axis Linear Motion Angle θ (radians) Position x (meters) Angular speed ω = dθ/dt (rad/sec) Linear speed v = dx/dt (m/s) Avg Angular speed ωavg = ∆θ/∆t (rad/sec) Avg Linear speed vavg = ∆x/∆t (m/s) Angular acceleration α = dω/dt (rad/sec2 ) Linear acceleration a = dv/dt (m/s2 ) Rotational kinetic energy Krot = 12 Iω 2 Kinetic energy Ktrans = 12 mv 2 Rotational work (constant τ ) W = τ θ W = F~ · d~ Rotational power (constant τ ) P = τ ω P = F~ · ~v For CONSTANT α For CONSTANT a ω = ωo + αt v = vo + at 1 2 θ = θo + ωo t + 2 αt x = xo + vo t + 21 at2 θ = θo + 12 (ωo + ω)t x = xo + 12 (vo + v)t ω 2 = ωo2 + 2α(θ − θo ) v 2 = vo2 + 2a(x − xo ) Total kinetic energy: K = Ktrans + Krot Conversion: 360o = 2π radians Rigid body rotation: s = rθ vt = rω at = rα. 2 2 ar = v /r = ω r ω = 2π/T T = 2π/ω f = 1/T Center of mass: Xcm = (Σmi xi )/(Σmi ) Ycm = (Σmi yi )/(Σmi ) Moment of Inertia Collection of particles: I = Parallel axis: I = Icm + M d2 Torque: force acting at a point P on an object that (may) rotate about a point O ~τ = ~r × F~ Σ~τ = I~ α, with I computed about the rotation axis through O. If all forces are in the XY plane, rotation will be about the Z axis: Στ = Iα (all computed about Z). Note: Fg acts at CM of object. Static Equilibrium: • ΣF~ = 0 for every object • Σ~τ = 0 about any axis P mi ri2 Zcm = (Σmi zi )/(Σmi ) Solid object: I = r2 dm R