Physics 1A Winter 2022 Quiz 5 Practice Exam EQUATION SHEET Vector relations For a given vector magnitude: ⃗ = (Ax , Ay ) of length a =| A ⃗ |: A a2 = A2x + A2y angle with respect to x axis: tan θ = Ay /Ax ⃗ = (cAx , cAy ) ⃗+B ⃗ = (Ax + Bx , Ay + By ) cA A ⃗·B ⃗ = Ax Bx + Ay By = |A|| ⃗ B| ⃗ cos θAB Dot product: A ˆ ⃗×B ⃗ = |A|| ⃗ B| ⃗ sin θAB ⊥ Cross product: A (direction perpendicular to A and B and based on right hand rule) Average and instantaneous motion: ∆⃗r ∆t ∆⃗v = ∆t ⃗vavg = ⃗aavg ∆⃗r d⃗r = ∆t→0 ∆t dt ∆⃗v d⃗v d2⃗r ⃗a = lim = = 2 ∆t→0 ∆t dt dt ⃗v = lim Equations of motion for constant acceleration (ballistic motion): ⃗v = ⃗at + ⃗v0 1 1 ⃗r = ⃗at2 + ⃗v0 t + ⃗r0 = (⃗v + ⃗v0 )t + ⃗r0 2 2 v 2 = v02 + 2a∆r ∆r =| ⃗r − ⃗r0 | Newton’s Laws: 1. 2. 3. p⃗ is constant in the absence of net external force d⃗ p = F⃗net dt F⃗12 = −F⃗21 (equal and opposite interactions) Various Force Laws: F⃗f = −F⃗applied (opposing potential acceleration) ⃗f = −µN v̂ (opposing motion) Friction (kinetic) F Friction (static) Page 8 of 10 Physics 1A Winter 2022 Quiz 5 Practice Exam F⃗v = −bvv̂ fast speed: F⃗v = −bv 2 v̂ ⃗g = −G M1 M2 r̂ (r = distance between objects) Gravity: F r2 ⃗s = −k(x − xeq )x̂ (acts to return to equilibrium) Spring: F Viscosity: slow speed: Work-Energy 1 mv 2 ! 2 Work-energy theorem: ∆KE ≡ W = F⃗ · d⃗x = F ∆x cos θ Kinetic energy: KE = Total mechanical energy: E = KE + P E Energy conservation: for any closed system, ∆E = Wdissipative dW Power: P ≡ = F⃗ · ⃗v = F v cos θ dt Potential Energies for Conservative forces Gravity (near Earth’s surface): Gravity (large distances): Spring: PE = 1 k∆x2 2 P Eg = mgh P Eg = −GM1 M2 /r ! Impulse-Momentum p Impulse-momentum theorem: ∆⃗ Momentum conservation: ∆⃗ psys ≡ I⃗ = F⃗ dt !" # $ p⃗i = F⃗external dt ≡∆ Relative velocities for 1D elastic collisions: Center of mass: ⃗ COM = R Circular & Rotational Motion Angular rate or velocity: Angular acceleration: 1 Mtotal ! v1i − v2i = −(v1f − v2f ) mi⃗ri = i ⃗ ⃗ ∆θ dθ ω ⃗ = lim = ∆t→0 ∆t dt ⃗ ⃗ ∆ω dω α ⃗ = lim = ∆t→0 ∆t dt Page 9 of 10 1 Mtotal " ⃗rdm Rotational kinematic equations for constant acceleration same as 1D ballistic motion given above Physics 1A Winter 2022 Quiz 5 Practice Exam Direction of angular velocity is determined by the right hand rule 2 vtan Centripetal acceleration: ac = ω r = (direction inward) r Tangential motion: ∆s = R∆θ vtan = Rω atan = Rα r × F⃗ = I α ⃗ (r is measured Torque: τ = ⃗ " from center of mass) ! Moment of inertia: I = mi ri2 = dm r2 2 Some moments of inertia (all through their centers of mass): point mass ring disk MR2 MR2 ½ MR2 hollow sphere solid sphere stick 2/3 MR2 2/5 MR2 1/12 ML2 = ICOM +! M d2 ⃗ = ⃗r × p⃗ = I⃗ ⃗ = ⃗τ dt Angular Momentum: L ω Impulse: ∆L ! 1 Angular Energy: KE = Iω 2 Work: ∆KE = τ dθ 2 Moment of inertia about a position offset by d: Id Useful constants and unit conversions: g = 9.8 m/s2 1 L ≈ 1 quart ≈ 10-3 m3 G = 6.7 x 10-11 m3 kg-1 s-2 1 kWh ≈ 3.6 MJ c = 3.0 x 108 m/s Atmospheric Pressure ≈ 105 N/m2 MEarth ≈ 6.0 x 1024 kg 1 radian ≈ 57º REarth ≈ 6.4 x 106 m sin(0º) = cos(90º) = 0 1 lb ≈ 0.5 kg (on Earth) sin(30º) = cos(60º) = 0.50 1 year ≈ 3 x 107 s sin(45º) = cos(45º) = 0.71 1 mile ≈ 1600 m sin(60º) =cos(30º) = 0.87 1 mph ≈ 0.5 m/s sin(90º) = cos(0º) = 1 Page 10 of 10
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