Physics C Review Mechanics

Physics C Review Mechanics Figure 1: Normal Force Kinematic Equations 1 ∆x = at2 + v0 t 2 Figure 2: Atwood’s Machine ∆v = at (v)2 − (v0 )2 = 2a(∆x) Figure 3: Draw a banked curve diagram v0 + v ∆x = ×t 2 Pulled Weights Free Body Diagrams a= F−f Σm N Normal Force f Frictional Force T = ma T Tension mg Weight Elevator F = ma In a particular direction: ΣF = (Σm)a Atwood’s Machine2 a= |(m2 − m1 )|g m1 + m2 Normal force acts upward, weight acts downward. • Accelerating upward: N = |ma| + |mg| • Constant velocity: N = |mg| • Accelerating downward: N = |mg| − |ma| Banked Curve Friction can act up the ramp (minimum velocity when friction is maximum) or down the ramp (maximum velocity when friction is maximum). videal = p rg tan θ Range s vmin = rg(tan θ − µ) µ tan θ + 1 θ represents the smaller angle from the x-axis to the direction of the projectile’s initial motion. Starting from a height of x = 0: s vmax = rg(tan θ + µ) 1 − µ tan θ xmax = Projectile Motion Circular Motion Position Centripetal (radial) ∆x = vx t (v0 )2 sin 2θ g Centripetal acceleration and force is directed towards the center. It refers to a change in direction. 1 ∆y = − gt2 + (vy )0 t 2 ac = v2 r Velocity Fc = mac = θ represents the smaller angle from the x-axis to the direction of the projectile’s initial motion. (vx )0 = v0 cos θ mv2 r Tangential Tangential acceleration is tangent to the object’s motion. It refers to a change in speed. (vy )0 = v0 sin θ at = d|v| dt ∆vx = 0 Combined ∆vy = −gt atotal = q (ac )2 + (at )2 Height θ represents the smaller angle from the x-axis to Vertical loop the direction of the projectile’s initial motion. In a vertical loop, the centripetal acceleration is Starting from a height of x = 0: caused by a normal force and gravity (weight). ymax = (v0 sin θ)2 2g Top Elastic Kinetic energy is conserved. F = ma N + mg = m × N = v2 r m1 v1 + m2 v2 = m1 v10 + m2 v20 mv2 − mg r −(v20 − v10 ) = v2 − v1 Bottom Inelastic F = ma v2 N − mg = m × r 2 mv + mg N = r Kinetic energy is not conserved. m1 v1 + m2 v2 = (m1 + m2 )v0 Center of Mass Friction Friction converts mechanical energy into heat. Static friction (at rest) is generally greater than kinematic friction (in motion). rcm = λ= fmax = µN Σm = p = mv I= Fdt = F∆t = ∆p = m∆v Z dm = λdx (Σm)vCM = Σmv = Σp dp dt Z dm M = dx L Z Momentum-Impulse F= Z Z 1 1 Σmr rdm = xλdx = Σm Σm Σm Fnet = (Σm)aCM Energy Work Collisions Z Total momentum is always conserved when there are no external forces (F = dp dt = 0). W= Fdx = ∆K Power Pavg = Fx W = t t Pinstant = dW = Fv dt Kinetic Energy Linear 1 K = mv2 2 Potential Energy dU F=− dx Z xf ∆U = − FC dx = −WC xi Z UHooke Z 1 = − FHooke dx = − −kxdx = kx2 2 Ug = mgh equilibrium point F = − du dx = 0 (extrema) stable equilibrium U is a minimum unstable equilibrium U is a maximum Angular v= dx dt = ∆x ∆t ω= dθ dt = ∆θ ∆t a= dv dt = ∆v ∆t α= dω dt = ∆ω ∆t ∆x = 12 at2 + v0 t ∆θ = 12 αt2 + ω0 t ∆v = at ∆ω = αt (v)2 − (v0 )2 = 2a(∆x) (ω)2 − (ω0 )2 = 2α(∆ω) ∆x = v0 + v ×t 2 ∆θ = ω0 + ω ×t 2 F = ma Rx W = x0 Fdx τ = Iα Rθ Wrot = θ0 τdθ W = 12 mv2 − 12 m(v0 )2 Wrot = 21 mω2 − 12 m(ω0 )2 P = Fv Prot = τω p = mv L = Iω F= dp dt τ= Figure 4: Rotational Motion Total E = K+U Ei + WNC = Ef WNC represents non-conservative work that converts mechanical energy into other forms of energy. For example, friction converts mechanical energy into heat. dL dt Torque τ = r × F = rF sin θ τ = Iα Moment of Inertia Z 2 I = Σmr = r2 dm Figure 5: Arc Length I = Icm + Mh2 (h represents the distance from the center) Values 1 2 12 ml 1 2 3 ml rod (center) Figure 6: Angular Velocity rod (end) hollow hoop/cylinder mr2 Rotational Motion 1 2 2 mr The same equations for linear motion can be mod- hollow sphere 2 mr2 3 ified for use with rotational motion (Figure 4 on 2 solid sphere 5 mr2 the previous page). solid disk/cylinder Atwood’s Machine Angular Motion θ= s r ω= v r α= at r at = r p α2 + ω4 ac = ω2 r 1 1 Krolling = Iω2 + mv2 2 2 a= |(m2 − m1 )|g m1 + m2 + 12 M Angular Momentum L = Iω L = r × p = rp sin θ = rmv sin θ τ= dL dt Total angular momentum is always conserved when there are no external torques (τ = dL dt = 0). ω = 2πf r A= (x0 )2 +  φ = arctan  v 2 0 ω −v0 ωx0  1 E = kA2 2 Figure 7: Simple Harmonic Motion Spring Fs = −kx Simple Harmonic Motion r Simple harmonic motion is the projection of uniform circular notion on to a diameter. Likewise, uniform circular motion is the combination of simple harmonic motions along the x-axis and y-axis that differ by a phase of 90◦ . Ts = 2π r ωs = m k k m amplitude (A) maximum magnitude of displacePendulum ment from equilibrium cycle one complete vibration Simple period (T ) time for one cycle s frequency (f) cycles per time T = 2π angular frequency (ω) radians per time r x = Acos(ωt + φ) v = −ωA sin(ωt + φ) a = −ω2 A cos(ωt + φ) = −ω2 x T= 2π 1 = ω f ω= 1 ω = T 2π g L Compound A cable with a moment of inertia swings back and forth. d represents the distance from the pendulum’s pivot to its center of mass. s I T = 2π mgd r f= L g ω= mgd I frictionless pivot amplitude θ massless rod Energy U= E= bob's trajectory massive bob equilibrium position −Gm1 m2 R −GMm 2r v= Figure 8: Simple Pendulum r vescape = Torsional 2πR T 2GM re For orbits around the earth, re represents the raA horizontal mass with a moment of inertia is dius of the earth. suspended from a cable and swings back and forth. r T = 2π ω= I k k I Gravity F= −Gm1 m2 R2 G ≈ 6.67 × 10−11 Nm2 kg2 Kepler’s Laws 1. All orbits are elliptical. 2. Law of Equal Areas. 2 4π 3. T 2 = GM R3 = Ks R3 , where Ks is a uniform constant for all satellites/planets orbiting a specific body