Download PH1011 Tut 4: Forces, Momentum and Impulse

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 !! .
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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
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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
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