University of Alabama Department of Physics and Astronomy. PH 105 LeClair Summer Problem Set 11

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Damon Stone
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1 University of Aabaa Departent of Physics and Astronoy PH 05 LeCair Suer 0 Instructions: Probe Set. Answer a questions beow. A questions have equa weight.. Due Fri June 0 at the start of ecture, or eectronicay by idnight. 3. You ay coaborate, but everyone ust turn in their own wor.. An aircraft door coses by pushing it inside the airpane first. We wi assue P = 0 outside the aircraft, and P =0.9 at inside during fight. If the seaing surface of the door is 5 c wide a around the door, and the door s outer diensions are by 0.7, what is the tota force required to open the door whie in fight? Soution: The force wi be the seaing surface area ties the pressure difference appied to it. The seaing surface one can consider to be two strips of by 0.05 and two strips of 0.6 by 0.05, giving a net area of 0.6. i The pressure difference is P =0.9 at Pa. Thus, F = PA ( Pa ) ( 0.6 ) N () For coparison, note that the gravitationa force required to ift a 000 b weight is ony about N. In other words, even with a 5 c seaing surface, there is no way you re going to open that door at atitude whie the cabin is pressurized.. Viscosity of ost fuids can be represented by an extra drag force on a body oving in a iquid. For a body of spherica shape, the drag force is reasonaby we approxiated by F drag = 6πηRv, where v is the veocity of the body and η is a paraeter of the fuid. The presence of viscosity eads to a terina veocity of a body faing in a fuid (e.g., a person faing in air). Consider a sphere of radius R and density ρ s faing through a fuid of density ρ and viscosity paraeter η. Find an expression for the terina veocity of the sphere. Soution: Terina veocity is when the the object in question reaches a constant axiu veocity, which ust be when the net force on the object is zero. Physicay, the object s speed has becoe so high, and the corresponding drag force so great that it anages to baance the object s weight and any other forces. The weight of a sphere of radius R and density ρ s is F w = 4 3 πr3 ρ s g () i If you cae up with a sighty different area, that is not a probe - the geoetry is not stricty defined in the probe, so you had a choice to either add or subtract 5 c fro the given diensions. Either way is fine, it does not change the concusion.

2 If we are in a surrounding fuid of density ρ, we ust aso account for the buoyant force, equa to the weight of the dispaced fuid. This is the sae as the expression above if we substitute ρ s ρ B = 4 3 πr3 ρg (3) The drag and buoyant forces wi act in one direction, the weight of the object opposing the. A force baance yieds, at terina veocity, 0 = F d + B F w (4) 0 = 6πηRv πr3 ρg 4 3 πr3 ρ s g (5) 6πηRv = 4 3 πr3 (ρ s ρ) g v = g 9η R (ρ s ρ) (6) (7) The fact that terina veocity depends on partice size has any interesting technoogica appications. (As a quic for-instance: 3. A penduu is fored by pivoting a ong thin rod of ass M and ength L about a point on the rod. If the pivot is a distance x fro the rod s center, for what x is the period of the penduu iniu? The oent of inertia for a thin rod about its center of ass is I= ML. Soution: In the end, we ony have a physica penduu, and we aready now that the period is given by I T = π gh where I is the oent of inertia of the rod (of ass ) about the pivot point, and h is the distance between the rod s center of ass and the pivot point. Let the pivot be a distance x fro the end of the rod, aing it a distance / x fro the center of ass. The oent of inertia is then ( ) I = I co + x = ( ) + x The distance between the center of ass and the pivot is h=/ x, so I = + h The period is thus

3 ions ured, reperope ring. tion, With ured ues ding and each and eastaue e for aue Probes 483 T = π + h = π gh gh + h 70. Consider a daped osciator as iustrated g in Figures 5. and 5.. Assue the ass is 375 g, the spring constant We wish to is find 00xN/, such that and T is b a axiu, 0.00 N s/. which(a) eans How dt/dx=0. Noting that dt/dx= dt/dh, ong does it taes for the apitude to drop to haf its initia vaue? (b) What If? How ong does dt it tae dx = dt for dh = the echanica energy to drop to haf its initia vaue? 0 (8) (c) Show that, in genera, the fractiona rate d π + h at which the apitude decreases in a daped haronic osciator = 0 is (9) dh gh haf the fractiona rate at which the echanica energy decreases. ( ) ( π gh + ) ( ) / + h 7. A boc of ass is connected to two springs of = force 0 constants and as shown in Figures P5.7a and P5.7b. (0) g gh In each case, the boc oves = on a frictioness gh + tabe g = 0 after it is dispaced fro equiibriu and reeased. Show that in () the two cases the boc exhibits sipe haronic h = otion () with periods h = 0.9 (3) 3 ( (a) T A quic second derivative test ) or a pot of dt/dh verifies that this is indeed a iniu, not a axiu. The iniu period is therefore (b) T T in = T = π + = π h= 3 g 3 3g.6 s 4. A boc of ass is connected to two springs of force constants and as shown beow. The boc oves on a frictioness tabe after it is dispaced fro equiibriu and reeased. Deterine the period of sipe haronic otion. (Hint: what is the tota force on the boc if it is dispaced by an aount x? (a) y to M as is is s is by is ition f the s (b) Soution: Say we dispace the boc to the right by an aount x. Both springs wi try to bring Figure P5.7 the boc bac toward equiibriu - one wi pu, one wi push, but both wi act in the sae direction. That eans the net force is 7. A obsteran s buoy is a soid wooden cyinder of radius r and ass F net = M. It is x weighted x = at (one + end ) x = so a that it foats upright in ca sea water, having density. A passing shar (4) tugs on the sac rope ooring the buoy to a obster trap, puing the buoy down a distance x fro its equiibriu position and reeasing it. Show that the buoy wi execute sipe haronic otion if the resistive effects of the

4 with a force constant of 50.0 N/. At tie t 0 the pension, but you can ode it as a singe spring supp partice has its axiu speed of 0.0 /s and is oving a boc. You can estiate the force constant by th to the eft. (a) Deterine the partice s equation of about how far the spring copresses when a big bi otion, specifying its position as a function of tie. down on the seat. A otorcycist traveing at hi (b) Where in the otion is the potentia energy three speed ust be particuary carefu of washboard b ties the inetic energy? (c) Find the ength of a sipe that are a certain distance apart. What is the order o penduu This with is the exacty sae theperiod. sae as(d) what Find wethe woud iniu find for a singe spring, except the spring constant has nitude of their separation distance? State the qua tie interva becoe required + for rather partice than just to. ove Thefro soution x ust 0 be you tae as data and the vaues you easure or es to x.00. T = π for the. 6. A horizonta pan of ass ω = π and ength L is pivoted at one (5) end. The pan s other end is supported A boc of ass M is connected to a spring of by a spring of force and osciates in sipe haronic otion on a ho constant (Fig P5.6). The oent of inertia of the pan about the pivot is L ta, frictioness trac (Fig. P5.66). The force const 3. The pan is dispaced by a sa ange fro its horizonta equiibriu position and reeased. the spring is and the equiibriu ength is. A (a) Show 5. that Ait horizonta oves with pan sipe ofharonic ass and otion ength with Lan is pivoted that one a end. portions The of pan s the spring other osciate end is in phase an anguar frequency supported by a spring. (b) of force Evauate constant the frequency. The oent if of the inertia veocity of the of pan a segent about the dx pivot is proportiona is to th the ass I is = 5.00 tance x fro the fixed end; that is, v 3 L g. and Thethe pan spring is dispaced has a force by constant a sa ange of x (x/ )v. θ fro horizonta equiibriu and reeased. note that the ass of a segent of the 00 N/. Find the anguar frequency ω of sipe haronic otion. is (Hint: d consider (/ )dx. the Find torques (a) about the the inetic energy o pivot point.) syste when the boc has a speed v and (b) the p of osciation. 3/ Pivot θ x dx M v Figure P5.6 Figure P5.66 Soution: The presence of a pivot point even abeed as such iediatey suggests the use of torque to sove this probe. First, we need to find the copression of the spring at equiibriu, 6. Review probe. A partice of ass 4.00 g is attached to i.e., θ = 0. Since the pan has non-zero ass, even without 67. an A anguar ba of dispaceent ass connected the spring to two rubber ba a spring with a force constant of 00 N/. It is osciating on a horizonta ust befrictioness copressedsurface by soewith aount an apitude at equiibriu. of Once we ength have L, found each the under equiibriu tension position, T, as in Figure P5.6 ba is dispaced by a sa distance y perpendicuar.00. A we 6.00-g can worry object about is dropped the torques verticay whenon a sa top of anguar the dispaceent θ is appied. ength of the rubber bands. Assuing that the t 4.00-g object as it passes through its equiibriu point. does not change, show that (a) the restoring The two objects Let countercocwise stic together. rotations (a) By how be defined uch as does positive, the and etisthe (T/L)y equiibriu and position (b) the of syste the spring exhibits sipe har apitude of the vibrating syste change as a resut of the correspond to the tip of the pan being at vertica position coision? (b) By how uch does the period change? otion x o reative with an to its anguar unstretched frequency ength.. (c) By how The uch sudoes of the the torques energy about change? the(d) pivot Account point at forequiibriu (θ = 0) is given by considering the the change weight in energy. of the pan acting about its center of ass and the restoring force of the spring. The pan 63. A sipe ay penduu be treated with as a ength a point of ass.3a distance and a L/ ass fro of the pivot point, whie the spring force y acts at 6.74 g is agiven distance initia L. At equiibriu, speed of.06 the/s net at torque its equiibriu position. Assue it undergoes sipe haronic ust be zero. L L ( ) otion, and deterine its (a) L τ = g period, + x (b) tota energy, and o L = 0 (6) (c) axiu anguar dispaceent. Figure P Review probe. xone o = end g of a ight spring with force constant 00 N/ is attached to a vertica wa. A ight string is 68. When a boc of ass M, connected to the en (7) tied to the other end of the horizonta spring. The string Now that we have the equiibriu position, we can find the torques spring of when ass the s pan 7.40 aes g and an force ange constant, is s changes fro horizonta to vertica as it passes over a soid sipe haronic otion, the period of its otion is puey of θdiaeter with the4.00 vertica. c. The Since puey the pan is free cannot to turn change on a its ength (we assue), the aount that the fixed sooth springaxe. stretches The vertica shoud correspond section of to the the string arc ength supports a 00-g object. The string does not sip at its contact that the tip of the pan oves through, Lθ, with the puey. Find the frequency of osciation of the object if the ass of the puey is (a) negigibe, (b) 50 g, A two-part experient is conducted with the u and (c) 750 g. bocs of various asses suspended verticay fro M ( s/3) T/L

5 if the ange is reativey sa. ii The spring is therefore dispaced by an aount Lθ x o fro its unstretched ength. This gives us the spring s restoring force. Since the spring is attached to the pan, the spring force aways acts perpendicuary to the ength of the pan at distance L, and the torque is easiy found. The pan s weight sti acts at a distance L/ fro the center of ass, but now at an ange 90 θ reative to the axis of the pan. The overa torque is then τ = g ( L ) sin (90 θ) (Lθ x o ) L = gl cos θ L θ + x o L Since the ange θ is sa, we ay approxiate cos θ, and we ay aso ae use of our earier expression for x o. Finay, out of equiibriu the torques ust give the oent of inertia ties the anguar acceeration. τ = gl cos θ L θ + x o L gl L θ + gl = L θ = Iα (8) L θ = I d θ dt (9) Noting that I= 3 L for a thin pan, we can put the ast equation in the desired for for sipe haronic otion in ters of nown quantities: d θ dt = 3 θ 3 ω = (0) () Note that the ength of the pan does not atter at a. ii Sa enough such that we don t have to worry about the spring bending to the eft, for one.

OSCILLATIONS. dt x = (1) Where = k m

OSCILLATIONS. dt x = (1) Where = k m OSCILLATIONS Periodic Motion. Any otion, which repeats itsef at reguar interva of tie, is caed a periodic otion. Eg: 1) Rotation of earth around sun. 2) Vibrations of a sipe penduu. 3) Rotation of eectron