The Ultimate AP Physics C Mechanics Review The year is almost over and here is what you need to know for the Mechanics portion of the final and the AP Physics C Mechanics exam. Hey, it’s only 5 pages; enjoy! I. One Dimensional Motion A. Equations of Motion: Assuming a = constant Be able to use (algebraically) these equations; Δx = v0Δt + ½ aΔt2 vf = v0 + aΔt where Displacement = x Velocity = v 2 2 2 Δx = vfΔt - ½ aΔt vf = v0 + 2aΔx Acceleration = a Δx = ½ (v0 + vf) Δt B. Graphing: Understand the relationships between the three graphs of motion; x vs t (area under v vs t graph) v vs t (slope of x vs t, or area under a vs t graph) a vs t (slope of v vs t) C. Understand the difference between average velocity and instantaneous velocity. vave = Δx/Δt vinst = dx/dt which is the slope of the tangent to the x vs t graph at a point. D. Similarly: aave = Δv/Δt ainst = dv/dt which is the slope of the tangent to the v vs t graph at a point. II. Acceleration due to Gravity A. Acceleration due to gravity g = ag = 9.8 m/s2, as measured in our labs. For objects in free-fall (with no air resistance) g = -9.8 m/s2 if the upward direction is chosen to be positive. This means that every second an object gains 9.8 m/s of speed if it is moving downward, or loses 9.8 m/s of speed if it is moving upward. What is the acceleration of an object tossed straight up at its highest point? B. Vectors 1. Definition: something that has magnitude and direction 2. Vector addition: the head to tail method 3. Vector subtraction: vf-v0 = vf + (-v0) C. Motion in two dimensions on Earth’s surface 1. Equations of motion in x and y 2. Projectile Motion: Independence of motions that are at right angles to each other. The acceleration of projectile is g, straight down, independent of its horizontal motion. Range Eq: x =(vo2/g) sin 2 where the object lands at the same height it was fired from (yf = 0) Projectile Eq: y = tanx – (gx2)/(2vo2cos2) parabolic motion, yf y0 3. Graphing the parabolic paths of objects in y vs x D. Introduction to relative motion 1. Moving observer vs “stationary” observer: No preferred reference frame. It is impossible to say who is “really” moving 2. An observer can feel acceleration but not velocity. 1 III. Newton’s Laws and Other Force Stuff A. Newton’s Laws Newton’s 1st Law (Swiped from Galileo, but really just a special case of the 2nd Law): If FNET = 0, then a = 0. (v = constant) If a = 0, then FNET = 0. Newton’s 2nd Law (Physics bedrock, arguably the most important equation in science): FNET = m a FNET Vector relationship! Newton’s 3rd Law (Important but tricky, even Newton himself made mistakes applying this law: review the Lawn Mower problem): Fon A by B = -Fon B by A F on A by B A F on B by A B Note: Third Law forces NEVER EVER EVER EVER cancel because they act on two different objects! B. Hooke’s Law for springs (Named after Robert (Bob) Hooke, Newton’s sworn enemy): Spring constant F = -k x Force exerted by spring is in opposite direction from displacement. Many things besides springs obey this law (molecular bonds, compression waves etc.) C. Difference between Mass and Weight: Mass m is universal: same everywhere. Units: kg Weight depends on the local force of gravity. Units: N True Weight = Fgrav = mg Apparent Weight = FN (what the scale reads.) “Weightlessness” occurs when FN is 0 (free fall: a = g) D. Friction 1. Kinetic friction fk = k FN 2. Static friction fs max = s FN Example: skidding, sliding Example: rolling, turning, no relative motion between surfaces IV. Circular Motion and Rotation Uniform Circular Motion 1. centripetal acceleration ac = v2/R 2. centripetal force Fc = m ac Don’t put on FBD! Any force that points toward the center of a circle can be called a centripetal force: the centripetal force is not an additional force! 3. v = (2 R)/ T where T is the period, or time for one complete circle 2 V. Work and Energy A. Work: W = F.d = |F| |d| cos where θ is is the angle between F and d when placed tail to tail Note W = area under F vs d graph Or W = ∫ F dx F Can be + or -. Unit: Joule J = Nm W=AREA F θ d B. Potential Energy “stored ability to do work” d 1. Ugrav = mgh (on earth’s surface) 2 2. Uspring = ½ kx 3. F = - dU/dx (the force is the negative slope of the tangent to the U vs. t graph.) C. Kinetic Energy: K = ½ mv2 (Always +, or 0) D. Conservation of Energy 1. The Work/Kinetic Energy Theorem: Wtot = K always true! 2. If there is no funny business (Wnc = 0) then mechanical energy E is conserved: E0 = EF In other words: Ko + U0 = KF + UF Conservative forces do no work for any round trip! You can associate potential energy U with conservative forces. E. Three Step problem solving program 1. Draw picture: Label initial and final points, and ref line for gravitational potential energy. 2. If Wnc = 0 Set Eo = Ef , otherwise use Wnc = E 3. Solve for missing variable F. Pave = W/t: Power is the rate at which work is done. Units: Watt W=J/s Pinst = F v (Pinst = Fv cos θ if F and v are not collinear) VI. Center of Mass and Momentum A. For systems of particles, xcm = (1/M) Σmixi For extended objects, xcm = (1/M) ∫x dm B. Momentum and Impulse ideas come from Newt’s 2nd Law (where else?) F = ma = mΔv/Δt So FΔt = Δ(mv) , define Momentum p = mv Then the Impulse FΔt = Δp Graphically, the Impulse is the area under the F vs t graph F Or p = ∫ F dt AREA =p Can be + or -. Unit: N∙s = kg∙m/s t Note that we can also write F = Δp/Δt Which means the instantaneous force F = dp/dt Which means you can get the force by finding the slope of the p vs t graph C. Impulse Applications Safety : crash helmets, freeway barricades, airbags etc. The idea is to minimize F by maximizing t 3 t D. Conservation of Momentum:p0 = pf if Fext Δt= 0 m1v10 + m2v2o = m1v1f + m2v2f p1f p2f E. Collisions in 1 and 2 dimensions always conserve momentum 1. if elastic, then conserve K too. 2. if inelastic: don’t conserve K VII. Rotation A. Equations of Motion θ = ω0Δt + ½ αΔt2 θ = ωfΔt - ½ αΔt2 p1 0 Is momentum conserved in this collision? How would you test this graphically? p2 Assuming α = constant 0 ω = ω0 + αΔt where Angular Displacement = θ Angular Velocity = ω 2 2 ωf = ω0 + 2αθ Angular Acceleration = α θ = ½ (ω0 + ωf) Δt B. Connection between linear and angular variables θ = s/r ω = v/r α = aT/r (ar = ω2r) r θ s C. Rotational Inertia I: For discrete masses: I = Σmr2 For continuous masses: I = ∫r2 dm Parallel Axis Theorem I = Icm + Mh2 dm r D. Torque Def: τ = r x F = |r| |F| sin τ=Iα θ r F τ E. Energy If τext = 0, then Eo = Ef. When doing energy problems you need to account for both the translational and rotational kinetic energy: K = ½ mv2 + ½ Icmω2 Work (Kinetic) Energy Theorem W = ΔK Work W = τ θ Power P = τ ω r F. Angular Momentum For a discrete mass: l = r x p For an extended mass: L = I ω Note that τ = dL/dt l θ p l=rxp Conservation of Angular Momentum: if τext Δt = 0, Lo = Lf 4 VIII. Simple Harmonic Motion SHM A. Def of SHM: anytime the restoring force is directly proportional to displacement x from equilibrium: Frestore = -kx B. Reference Circle: If you have SHM, then the projection of uniform circular motion onto a diameter will represent the motion. It is much easier to analyze uniform circular motion than linear motion with nonconstant acceleration. The motion equation for SHM is x = A sin(ωt), where ω is the angular frequency: ω = 2π/T Where T is the period (the time it takes for one complete cycle.) Note that T = 1/f, where f is the frequency. C. If SHM, then the period T of oscillation will be independent of the amplitude of the oscillation and you can use the reference circle to show that T 2 m k D. Applications: i. Mass on a spring (see equation for T above) ii. Pendulum T 2 l g iii. Motion through a hole in the Earth (T = 84 minutes, remember?) VIII. Universal Law of Gravity and Orbits A. Kepler’s Laws 1st Law: Planets travel in elliptical orbits with sun at one of the foci 2nd Law: Planets sweep out equal area in equal time 3rd Law: T2/Rave3 = Constant B. Newton’s Universal Law of Gravity Rave mm Fgrav G 1 2 2 r C. Orbits: Elliptical orbits are equivalent to circular orbits where we let the radius of the circle r be equal to the semi-major axis (Rave) of the ellipse D. Set Fg = mac to find: 1. Mass of orbited body (orbiting mass always drops out) 2. Period T of orbit 3. Radius of orbit (including geosynchronous orbits) 4. Speed of orbiting object E. Universal Gravitational Potential Energy U grav G r Fg m1m2 r F. Escape Speed: apply conservation of energy E0=Ef, set Ef = 0, vesc = SQRT(2GM/r) G. Black holes: set the escape speed equal to the speed of light c. H. Tides caused by differential gravitational pulls. 5 r