1Chapter 6 – Force and Motion III. Drag forces and terminal speed.II. Uniform circular motion.III. Non-Uniform circular motion.I. Drag force and terminal speed-Fluid: anything that can flow. Example: gas, liquid.-Drag force: D- Appears when there is a relative velocity between a fluid and a body.- Opposes the relative motion of a body in a fluid. - Points in the direction in which the fluid flows.Assumptions:* Fluid = air.* Body is blunt (baseball).* Fast relative motion  turbulent air.2)3.6(212AvCDρ=)4.6(2ACFvgtρ=-Terminal speed: vt02102=−→=→=−ggFAvCaifmaFDρC = drag coefficient (0.4-1).ρ = air density (mass/volume). A= effective body’s cross sectional area  area perpendicular to v- Reached when the acceleration of an object that experiences a vertical movement through the air becomes zero  Fg=DII. Uniform circular motion-Centripetal acceleration:)6.6(2RvmF =)5.6(2rva =A centripetal force accelerates a body by changing the direction of the body’s velocity without changing its speed.v, a = ctes, but direction changes during motion.-Centripetal force:a, F are directed toward the center of curvatureof the particle’s path.3III. Non-Uniform circular motion22traaa +=- A particle moves with varying speed in a circular path.- The acceleration has two components: - Radial  ar= v2/R- Tangential  at= dv/dt- atcauses the change in the speed of the particle.- In uniform circular motion, v = constant  at= 0  a = arrrvdtvdaaartˆˆ2−=+=θ∑∑∑+=trFFF49. A puck of mass m slides on a frictionless table while attached to a hanging cylinder of mass M by a cord through a hole in the table. What speed keeps the cylinder at rest?MgmgTTN0=→=→caMgTMFor33E. Calculate the drag force on a missile 53cm in diameter cruising with a speed of 250m/s at low altitude, where the density of air is 1.2kg/m3.Assume C=0.75( )kNsmmmkgAvCD 2.6/250)2/53.0()/2.1(75.05.0212232=⋅⋅⋅⋅==πρ32. The terminal speed of a ski diver is 160 km/h in the spread eagle position and 310 km/h in the nose-dive position. Assuming that the diver’s drag coefficient C does not change from one point to another, find the ratio of the effective cross sectional area A in the slower position to that of thefaster position.7.322/310/1602=→==→=DEEDDgEggtAAAAACFACFhkmhkmACFvρρρmMgrvrvmMgrvmTmFor =→=→=→22411P.A worker wishes to pile a cone of sand onto a circular area in his yard. The radius of the circle isR, and no sand is to spill into the surrounding area. If µsis the static coefficient of friction betweeneach layer of sand along the slope and the sand beneath it (along which it might slip), show thatthe greatest volume of sand that can be stored in this manner is π µs R3/3. (The volume of a coneis Ah/3, where A is the base area and h is the cone’s height).- To pile the most sand without extending the radius, sand is added to make theheight “h” as great as possible.- Eventually, the sides become so steep that sand at the surface begins to slip.- Goal: find the greatest height (greatest slope) for which the sand does not slide.Cross section of sand’s coneRhθNmgxyfFgxFgyStatic friction  grain does not moveθθsincosmgFfmgFNgxgy====θµθµµθtancossinmax,≥→==≤=ssssgxmgNfmgFIf grain does not slideThe surface of the cone has the greatest slope and the height of the cone ismaximum if :33)(3tan32RRRhAVRhRhssconessπµµπµθµ==⋅==→==21.Block B weighs 711N. The coefficient of static friction between the block and the table is 0.25; assume that the cord between B and the knot is horizontal. Find the maximum weight of block Afor which the system will be stationary.FgANFgBT1fT3T1T2T3NfstationarySystemssµ=→max,NNTfTgmNBBlocksB75.17771125.001max,1=⋅=→=−=→322222130sin25.20530cos75.17730cosTTTNNTTTTKnotyx====→==→NNTgmTABlockA62.10225.2055.030sin23=⋅===→23P. Two blocks of weights 3.6N and 7.2N, are connected by a massless string and slide down a 30ºinclined plane. The coefficient of kinetic friction between the lighter block and the plane is 0.10;that between the heavier block and the plane is 0.20. Assuming that the lighter block leads, find(a) the magnitude of the acceleration of the blocks and (b) the tension in the string. (c) Describethe motion if, instead, the heavier block leads.ABFgAFgBNANBTTfk,Bfk,AMovementBlock A Block BNANBfkAfkBTFgxAFgyAFgxBFgyBTLight block A leads5Light block A leadsHeavy block B leadsaTaNTNamfTFNNNfNgmFNBBlockaTaTNNamTfFNNNfNgmFNABlockBkBgxBBkBkBBgyBBAkAgxAAkAkAAgyAA73.035.273.025.130sin)2.7(25.1)23.6)(2.0(23.630cos37.049.137.0312.030sin)6.3(312.0)12.3)(1.0(12.330cos=+→=−+→=−+======→=−→=−−→=−−======→µµNTsma2.0/49.32==( )NWWWWTkAkBBABA2.0cos =−+=θµµThe above set of equations is not valid in this circumstance  aA≠ aBThe blocks move independentlyfrom each other.Reversing the blocks is equivalent to switching the labels. This would give T~(µkA-µkB)<0 impossible!!!NNNfNNNfkkss8.22)60(38.033)60(55.0max,======µµ74. A block weighing 22N is held against a vertical wall by a horizontal force F of magnitude 60N. Thecoefficient of static friction between the wall and the block is 0.55 and the coefficient of kineticfriction between them is 0.38. A second P acting parallel to the wall is applied to the block. For thefollowing magnitudes and directions of P, determine whether the block moves, the direction of motion, and the magnitude and direction of the frictional force acting on the block: (a) 34N up(b) 12N up, (c) 48N up, (d) 62N up, (e) 10N down, (f) 18N down.F=60N(a) P=34N, upmg=22NNN=F=60NP22NfWithout P, the block is at rest movenotdoesBlockNffdownNffNNaffassumeweIfmafmgPss→=<=→=−=→==−−331222340max,(b) P=12N, upP22NfmovingNotNffupNNNfmamgfPs→=<=−===−+331012220max,(c) P=48N, upP22NfmovingNotNffdownNNNfmamgfPs→=<=−===−−332622480max,(d) P=62N, upP22NfdownNffwithmamgfPwrongAssumptionupmovesBlockNffupNNNfmgfPks8.22(*)33402262(*)0max,===−−→→→=>=−=→=−−6(e) P=10N, downP22NfmovingNotNffupNNNfmamgPfs→=<=+===−−333212220max,(f) P=18N, downP22NfupNffmovesNffupNNNfmamgPfks8.22334022180max,==→=>=+===−−28. Blocks A and B have weights of 44N and 22N, respectively. (a) Determine the minimum weight of block C to keep A from sliding if µsbetween A and the table is 0.2. (b) Block C suddenly is lifted of A. What is the acceleration of block