Graphics from Slides: Here are several still images from slides of different useful equations. 1D Motion Physics is Calculus S position V velocity a acceleration I time differentiate differentiate S integrate ntegrate a V=0 a=0 particle is particle is: * momentarily at rest particle is at the origin * instantaneously at rest at constant velocity . stationary . changing directions as Minimum/ position set = 0, solve for /, plug back into S Maximum velocity set - =0, solve for f, plug back into v y, = Vit V At +-aAt- 2 Vy = Vvita, At V - = V-+ 2a Ay x = X, +V At+-a,At- Vxf = Vita,At Vxf F=V.- + 2a,Ax. Momentum and Angular Momentum . Conservation of Momentum and Angular Momentum . Elastic versus Inelastic Collisions . Kinetic Energy Considerations of Collisions . Center of Mass . Propulsion . Angular Velocity, Angular Acceleration . Relation between Translational and Rotational Kinematics . Rotational Energy Considerations . Moment of Inertia . Parallel Axis Theorem . Newton's 2nd Law for Rotating Bodies . Torque . Work and Torque Interactions for Rotating BodiesUnit 2 Equations Chapter 4: This chapter is all about properly using Newton's Three Laws F = ma \\F, = ma = -mg| FAB = -FBA Chapter 5: Circular motion equations were removed to avoid confusion (slide below) fs, mar = Us FN Ifk = UkFN | D = > CopAv | Vterm ~ V 4mg A "= mwrl_F = mat = mar | >F =0 Chapter 6: Work is a transfer of energy; it has the units of energy W=F . Ar = FArcos (@) | W = ["' F . di | Fap = -kAs Went = Ky - K. = AK |P= = dEsys dw P = F. 3 Chapter 7: The differences in the work equation come from conservative and nonconservative forces Emech = K +UIU, = mgh | Uspr = kx | Fspr = -kx | AEmech = AK + AU = 0 F = - du 1AEint = fl As | W = AK + AU + AEintEssential concepts Particle acceleration, force, interaction Basic Goals How does a particle respond to a force? How do objects interact? Newton's First Law AV =0-a=0-F =0 General Principles Newton's Second Law F = ma Newton's Third Law FAOnB = - FBon A Basic Problem Solving Strategy Linear Motion Trajectory Motion Circular Motion SF. = ma, SF. = 0 SF = ma, F, = ma = = mor r SF, =0 EF, = may F, = may SF, = ma, EF = ma, Linear & Trajectory Kinematics Circular Kinematics General Case Uniform Acceleration: T = Uniform V V, = v + alt Circular Motion: 0, =0, +wAt V = ds - - slope of position graph dt a, = constant As = v, At + ya(At ) a -=0r V = Or a - dt V- slope of velocity graph v2 = v. +2ads w = w; +aAt Av = [a dt = area under acceleration curve Uniform Motion: Nonuniform as =0 Circular Motion: 40 = W,At + Kza(At) As - [v dt - area under velocity curve v = constant As = v.At w/ = w2+2a40Equation Hub Chapter 8: Collision equations have not been given here. Just the general form. p = mulp; = p IF = |Ap = FaugAt = J |J = fy Fat | P = pa + Pu + Pc t ... - mava + movb + mcvct... Com = y E; mix; = minitmart... mitmat ... I em = fxdm| vf - vi = veln Chapter 9: Translational Motion equations and Rotational Motion equations below. la = dw d'e at dt 2 As = Aor lut = wr |at = arl I=E, mir, I = fridm| I = Iem + Md' Emech = K + U|U, = mgh | U. = -kx' | Klin = mv' | Krot = } Iw-Angular speed does not depend on radius linear angular position Ax velocity V = W = dt dt dw acceleration a = Of = dt dtPhysics 201 Exam 4 Rotational Statics and Dynamics 1. (10 points) A solid sphere with a mass of 2.00 kg and a radius of 50.0 crn is rolled on a flat surface towards the bottom of a ramp (it will be going up the ramp). When the sphere reaches the bottom of the ramp, a point halfway between the center of the sphere and the edge of the sphere has a tangential speed of 1.50 m/s. How high up the ramp will the sphere roll? 2. (10 Points) A rod of mass M and length Lis rope attached to a wall with a pin joint, around which the rod is free to rotate. A horizontal rope attached to the end of the rod holds the rod at an angle 0. to the horizontal. Express your answer in terms of M, L, 0., and g (and no other variables). rod a. What is the tension in the rope? b. Suddenly the rope breaks. As a result, the rod swings down. Immediately after the rope breaks, what is the rod's angular acceleration? c. Is angular acceleration constant as the rod is swinging? Why or why not? 3. (10 Points) A 50 g ball of clay is thrown from the left tangent to the top of a 2.0 kg, 30- cm-diameter sphere that spins at a frequency of 1 rev/s cow about its center. Assuming that the clay sticks to the sphere, how fast must you throw the clay to ensure that the spinning sphere comes to a stop?displacement Ax = X - X; Kinematics Equations (constant acceleration) Ax average velocity V. = Vyf = Vxi + a,At At Xf = X; +v .At+-a,At2 total distance 2 average speed total time ( Vy ) = (V : )2 + 20, Ax dx instantaneous velocity VE dt Free Fall instantaneous speed On Earth, when not touching anything, objects fall due to Av average acceleration Vi gravity towards the Earth's At f - t center with acceleration: instantaneous acceleration dv. d' x g = 9.80 m/$2 dt dt (neglecting air resistance) d.x de V = W = speed it angular speed It 2TT 2 71 v = 2arf = - T W = 27 f = T speed angular speed V = WrFree Body Diagrams Identify the different forces acting on a particle Represent the object as a particle at the center of a coordinate system Draw each force as a vector pointing in the correct direction If possible, identify the direction of the net force Uniform Circular Motion object moves with constant (tangential) speed Speed V, = = 2AtRf T Centripetal Acceleration a = Centripetal ForceUnit 1 Equations Chapter 1: This chapter was setting up the math. Familiarity with vectors is big here A . B = B . A = |AB| cos(0) | A . B = ArBr + Ay By + A,B, I AXB = -BX A = |AB sin(0) | Ax B = (A, B, - A,By ) i+ (A.B, - A,B.) j + (A,By, - A,B,) k Chapter 2: Gradients are for variables that dont change with time, derivatives are for variables that change with time Uf = vi + aAt laf = x; + v;At + 2at? | (v; )? = (v;)2 + 2aAx Chapter 3: Several topics didn't have equations, like reference frames and some directions of relative velocity. Its worth looking those parts over. dy Ar = (Ax)a + (Ay)y = raitryjAr = ry - rix = utt x |vang Ar dr dt y - Vi CLavg = | Range = ~ sin(20) la = "pla = = Wr At At la=Chapter 10: Uses lots of equations from Chapter 9, and the following equations. 7 =RxF|AXB= [A| |B|sin(@) | ET = 10 = 1 = d( Iw) L= DIET=LEFXP = | musin (8) | L = LF Graphics from Slides: Here are several still images from slides of different useful equations. Vf = V. + alt W . = W; + ant As = v.At + 1/za(At ) 40 = W.At + 1/za(At) v7 = V3 + 2ads w =w+2040 TABLE 9.2 Moments of Inertia of Various Bodies (a) Slender rod (b) Slender rod (c) Rectangular plate, (d) Thin rectangular plate, axis through center axis through one end axis through center axis along edge I = -ML2 1 = =ML2 1 = M(a + 62) 1 = -Ma (e) Hollow cylinder (f) Solid cylinder (g) Thin-walled hollow (h) Solid sphere (i) Thin-walled hollow cylinder sphere 1 = -M(R2 + R?) 1 = -MR 1 = MR- 1 = =MR 1 = =MR2 R2 R1 R R -R