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PH1011 Tut 4: Forces, Momentum and Impulse 1) [‘A’J86/II/8]Whenabodymovesthroughafluid,aretardingforceduetoturbulencemaybe experienced.Inthecaseofasphereofradius𝑟movingwithspeedvthroughastationaryfluidof density𝜌whichisatrest,thisforceis 𝐹 = 𝑘𝜌 𝑟 ! 𝑣 ! where𝑘isaconstant.(NotethatthisformuladiffersfromStoke'slawinthelecturenotes). When spherical raindrops fall through still air, all but the smallest experience a retarding force given by the equation above. It is found that drops of a given radius approach the groundwithanapproximatelyconstantspeed,whichisindependentofthecloudinwhich theyareform. (a)Findanexpressionforthisterminalspeed𝑣! intermsoftheconstant𝑘,theradius𝑟of thedrop,itsdensity𝜌! ,thedensityofair𝜌! oftheairandtheaccelerationoffreefallg. (Youmayneglecttheupthrustduetoair).Sketchthevelocity-timegraphoftheobjectfrom themomentitisreleasedintothefluid. (b) The terminal speed of the raindrop of radius 1 mm is approximately7 𝑚𝑠 !! . In freak storms,hailstoneswithradiiaslargeas20mmmayfall.Estimatethespeedwithwhichsuch stonesstriketheground.[Takethedensityofwateras1×10! 𝑘𝑔 𝑚 !! andthedensityofice as9×10! 𝑘𝑔 𝑚 !! . 1 2) [Activity] a) Thedensityoficeis0.917g/cm3,whereasthatofseawateris1.025g/cm3.Whatpercentof an iceberg is above the surface of the water? (Discussion) If some ice and seawater are placedinameasuringcylinder,willtheseawaterlevelrise,droporstaythesamewhenall theicemelts? b) Whenacrownofmass14.7kgissubmergedinwater,anaccuratescalereadsonly13.4kg.Is thecrownmadeofgold? c) Asphericalironballbearingofdiameter4mmisplacedonapieceofStyrofoamfloatingina cylinder of oil as shown in the figure below. You may assume that the weight of the Styrofoamisnegligible.Densityofironis7.86gcm-3;densityofoilis0.83gcm-3. i) CalculatetheupthrustactingonthepieceofStyrofoam. ii) WhatisthevolumeofoildisplacedbythepieceofStyrofoamandtheironballbearing? iii) TheballbearingrollsoffthepieceofStyrofoamandfallsintotheoil.Willtheoillevelbe higher,lowerorthesameasbefore?Explainyouranswer. Ironballbearing Originaloillevel Styrofoam Oil 2 3) [S9.15] The magnitude of the net force exerted in the x direction on a 2.50-kg particle varies in time as shown in the figure below. Find (a) the impulse of the force over the 5.00-s time interval, (b) the final velocity the particle attains if it is originally at rest, (c) its final velocity if its original velocity is −2.00 ! m/s, and (d) the average force exerted on the particle for the time interval between 0 and 5.00 s. 4) [G9.32](a)Calculatetheimpulseexperiencedwhena65-kgpersonlandsonfirmgroundafter jumpingfromaheightof3.0m.(b)Estimatetheaverageforceexertedontheperson’sfeetby thegroundifthelandingisstiff-legged,andagain(c)withbentlegs.Withstifflegs,assumethe bodymoves1.0cmduringimpact,andwhenthelegsarebent,about50cm.[Hint:Theaverage netforceonherwhichisrelatedtoimpulse,isthevectorsumofgravityandtheforceexerted bytheground.] 5) A neutron collides elastically with a helium nucleus (at rest initially) whose mass is four times ! that of the neutron. The helium nucleus is observed to move off at an angle𝜃!" = 40∘ . ! ! ! Determinetheangleoftheneutron,𝜃! andthespeedsofthetwoparticles,𝑣! and𝑣!" ,after thecollision.Theneutron’sinitialspeedis6.2×10! 𝑚/𝑠. 6) [G9.59] A neon atom (m=20.0 u) makes a perfectly elastic collision with another atom at rest. Aftertheimpact,theneonatomtravelsawayata55.6°anglefromitsoriginaldirectionandthe unknown atom travels away at a–50.0° angle. What is the mass (in u) of the unknown atom? [Hint:Youcouldusethelawofsines.] 7) [G9.103]A0.25-kgskeet(claytarget)isfiredatanangleof28°tothegroundwithaspeedof25 m/s. When it reaches the maximum height, h, it is hit from below by a 15-g pellet travelling vertically upwards at a speed of 230 m/s. The pellet is embedded in the skeet. (a) How much higherh’doestheskeetgoup?(b)HowmuchextradistanceΔxdoestheskeettravelbecauseof thecollision? Answers to selected questions 1. (a) 𝑣! = !!! !"# !!!! (b) 30 m/s 2. (c)(i) 2.58×10!! 𝑁 (ii) 0.317 𝑐𝑚 ! (iii) Lower 3. (a) 𝐼 = 12.0 𝑁𝑠 𝚤; (b) 𝑣! = 4.80 ! ! 𝚤; (c) 𝑣! = 2.80 ! ! 𝚤; (d) 𝐹!"# = 2.40 𝑁 𝚤 4. (a) (magnitude) 499 Ns, (b) 301 mg = 191737 N, (c) 7 mg = 4459 N ! 5. 𝑣!" = 1.899×10! 𝑚/𝑠; 𝑣!! = 4.899×10! 𝑚/𝑠;𝜃!! = 85.6∘ 6. 39.9 u 7. (a) 8.6 m; (b) 38.4 m 3