Physics 4A Winter 016 Final Exa Nae: Mar, 016 Please show your work! Answers are not coplete without clear reasoning. When asked for an expression, you ust give your answer in ters of the variables given in the question and/or fundaental constants. Answer as any questions as you can. Do not forget to include appropriate units when giving a nuber as an answer. Calculators are allowed. If you detach any pages fro the test, please write your nae on every page. Constants G = 6.67 10 11 N kg g = 9.8 s (If you like you can use g = 10 s, but ake your choice clear.) x f = x i + v i t + 1 at v f = v i + at v avg = v i+v f v f = v i + a x x f = x i + v avg t ω = π T v = rω a t = rα a c = v r ( ) v f = v i + v e ln i f d Thrust = v e dt r CM = 1 M tot i ir i I = i iri I = I CM + MD T = π k T = π L g K = 1 v U g = gy U s = 1 kx Equations U G = G 1 r W = τ dθ = F ds P = τ ω = F v F x = du dx R = bv v(t) = v T (1 e bt/ ) R = 1 DρAv ( v(t) = v T tanh da = L dt M p ( T 4π = GM ) a 3 g v T t ) p = v F = dp dt τ = r F L = r p τ = dl dt I = F(t) dt L = τ dt L = Iω L = vr ω p = Mg r CM Iω F k = µ k n F s,ax = µ s n F = kx F G = G 1 ˆr r E = G 1 a 1
1. Two students are on a balcony a distance h above the street. One student throws a ball vertically downward at a speed v i ; at the sae tie, the other student throws a ball vertically upward at the sae speed. Answer the following sybolically in ters of v i, g, h, and t. (a) What is the tie interval between when the first ball strikes the ground and the second ball strikes the ground? [4pts] (b) Find the velocity of each ball as it strikes the ground. [3pts] (c) How far apart are the balls at a tie t after they are thrown and before they strike the ground? [3pts] (d) Sketch a height-versus-tie graph for both balls on the sae axes. [pts] (e) Sketch a velocity-versus-tie graph for both balls on the sae axes. [pts]
the spring constant each spring should have for the dispenser to function in this convenient way. (c) Is any 5.00. Find the work W 5 e S piece of data unnecessary for this deterination? F? d S r done by. A sall particle of ass is pulled to the top of a frictionless half-cylinder the force (of on radius the object. 4. A light spring with force constant 3.85 N/ is copressed R) by a light cord that passes over the top of the cylinder as shown. The force F is such by 8.00 that the c particle as it is held oves between at a constant a 0.50-kg speed v. block 30. Review. The graph in on the left and a 0.500-kg block on the right, both resting on a (a) horizontal What is thesurface. noral force The on spring the particle exerts when a force it is in the position functional shown? relationship [3pts] Figure P7.30 specifies a on each between the two variables u and v. (a) Find (b) block, Showtending that F = to gpush cos θ. [pts] the blocks apart. The blocks are (c) siultaneously By directly integrating released W = fro F dr, rest. findfind the work the done ineoving b u dv. the (b) particle acceleration Find a atwith constant which speed each froblock the botto starts to toove, the topgiven of the half-cylinder. [4pts] eau dv. b (c) Find e b that the coefficient of kinetic friction between each v du. a (d) What is the net work done on the particle? [pts] block and the surface is (a) 0, (b) 0.100, and (c) 0.46. 5. A sall particle of ass S is pulled to the top of a frictionless halfcylinder (of radius R) by a light cord that passes over the top of the cylinder as illustrated in Figure P7.5. (a) Assuing the particle oves at a constant speed, show that F 5 g cos u. Note: If the particle oves at constant speed, the coponent of its acceleration tangent to the cylinder ust be zero at all ties. (b) By directly integrating W 5 e F S? d r S, find the work done in oving the particle at constant speed fro the botto to the top of the half-cylinder. 6. The force acting on a particle is F x 5 (8x 16), where F is in newtons and x is in eters. (a) Make a plot of this force versus x fro x 5 0 to x 5 3.00. (b) Fro your graph, find the net work done by this force on the particle as it oves fro x 5 0 to x 5 3.00. S F R u Figure P7.5 u ( 8 4 Section 7.5 Kinetic Energy and the Energy Theore 31. A 3.00-kg object has a veloc W (a) What is its kinetic energy is the net work done on the ob to 18.00 i^ 1 4.00 j^ /s? (Not the dot product, v 5 S v? S v.) 3. A worker pushing a 35.0-kg w speed for 1.0 along a woo Q/C by applying a constant horiz F on the crate. (a) Deterine worker now applies a force the subsequent otion of the would happen to the crate if than F. AMT 33. A 0.600-kg particle has a spee W and kinetic energy of 7.50 J a kinetic energy at, (b) its sp work done on the particle by e fro to? 4 0 3
3. A pendulu bob of ass is attached to a light string of length L. The string akes a sall angle θ i with the vertical and then the bob is released with an initial velocity v as in the diagra. (Assue there is no friction or air resistance.) θ L g v (a) What is the period of the bob s oscillation? [1pt] (b) What is the bob s initial angular speed? [1pt] (c) Assue that the bob s otion can be described by θ(t) = θ ax cos(ω 0 t + φ). Find expressions for ω 0, θ ax, and φ. [8pts] 4
4. Consider an object that rolls down an incline. θ (a) Deterine the acceleration of the center of ass of a unifor solid disk rolling down an incline aking angle θ with the horizontal. [5pts] (b) What is the iniu coefficient of friction required to aintain pure rolling otion for the disk? [4pts] (c) Consider a hollow cylinder of ass, inner radius r 1, and outer radius r and unifor density. Starting fro the expression for the oent of inertia of a unifor disk, show that the oent of inertia of the hollow cylinder is [5pts] I = 1 (r 1 + r ) (d) Find the acceleration of this hollow cylinder down the sae slope assuing it rolls without slipping. Copare the acceleration found in part (a) with that of the hollow cylinder. Is one always larger than the other, or does it depend on the ass and radius of the disk and cylinder which will be larger? [6pts] 5
oot (a strenus as shown in rce S Fg on the xerted by the ical odel of b, where S T is hilles tendon by the tibia. 00 N. ine the values of the unknowns and state the physical eaning of each. 5. A 45. unifor A unifor sign of ass sign Mof andweight width L F hangs fro a unifor horizontal bea of g and width L hangs fro ass hinged at the wall and supported by a cable as shown. Deterine S a light, horizontal bea hinged at the wall and supported tension in by the a cable [4pts] (Fig. P1.45). Deterine (a) (a) the the tension in the cable and (b) the coponents of the reaction force exerted by the wall on the bea in ters of F g, d, L, and u. (b) and the coponents of the reaction force exerted by the wall on the bea [5pts] in ters of M,, g, d, L, and θ. u T S u d Lulu and Lisa s Cafe c ut on a bea dies hanging bea is uni- L Figure P1.45 46. A 1 00-N unifor boo at f 5 658 to the vertical is supported by a cable at an angle u 5 5.08 to the horizontal as shown in Figure P1.46. The boo is pivoted at the botto, and an object of weight 5 000 N hangs fro its top. Find (a) the tension in the support cable and (b) the coponents of the reaction force exerted by the floor on the boo. u 3 4 f Figure P1.46 47. A crane of ass 1 5 3 000 kg supports a load of ass 5 10 000 kg as shown in Figure P1.47. The crane 6
6. A coet of ass oves about a star in an elliptical orbit, with its closest approach to the star being r 1 and its greatest distance r. The coet s speed at closest approach is v 1. (a) Consider the angular oentu of the coet about the star. What is the change in angular oentu as the coet oves fro its closest point to the star to its furthest point fro the star? (Argue by considering the net torque on the coet.) [3pts] (b) What is its speed when it is farthest fro the star? [3pts] (c) What is the work done on the coet by the force of gravitational attraction as the coet oves fro its closest point to the star to its furthest point fro the star? [3pts] (d) Just for this part of the question, assue the ass of the star is M. What is the period of the coet s orbit? [pts] 7
gravitational field of the Earth. Find the angular oentu of the projectile about the origin (a) when the projectile is at the origin, (b) when it is at the highest point of its trajectory, and (c) just before it hits the 7. A projectile of ground. ass is (d) launched What torque with an causes initial its velocity angular v oentu shown. to change? The projectile oves in the gravitational field of the i aking an angle θ with the horizontal as Earth. y S v 1 v xi i Figure S 19. The position vector of a v i a function of tie is u M O x where S r is in eters and R S v angular oentu of th a function of tie. Figure P11.15 (a) What are the coponents of the velocity of the projectile as a function0. ofa tie? 5.00-kg [pts] particle starts Q/C Its velocity as a function (b) Find the 16. tie-of-flight Review. A conical of thependulu projectile. consists (Show a derivation.) [3pts] S of a bob of ass in otion in a circular of path the in projectile, a horizontal R. (Show plane a as derivation.) [3pts] S v 5 6t (c) Find the range u where S v is in eters per (d) Find the angular shown in oentu Figure P11.16. of theduring projectile the about the origin as a function (a) Find of its position as a tie, L(t), otion, between the itssupporting launch andwire landing. of length [5pts] its otion qualitatively., aintains a constant angle u with (e) Check your function of tie, (d) the the expression vertical. Show in thethat previous the agnitude part is correct for the instant just before it hits the ground. ticle as a function of tie of the angular Do thisoentu by pluggingof your the bob tie-of-flight into your expression for origin exerted on the p L(t) and confiring about the vertical that itdashed is equal line to is the angular Figure oentu P11.16 about the origin (f) the angular oent just as it will strike the ground using your range result. [4pts] tion of tie, (g) the kine L 5 a g, 3 sin 4 u 1/ (f) What torque causes its angular oentub cos u to change? [1pt] function of tie, and (h syste of the particle as 17. A particle of ass oves in a circle of radius R at a S constant speed v as shown in Figure P11.17. The otion 1. A ball having ass Q/C begins at point Q at tie t 5 0. Deterine the angular tened at the end of a fla S oentu of the particle about the axis perpendicular to the page through point P as a function of tie. of a tall building at poin that is connected to th shown in Figure P11.1 y length of the flagpole is S v it akes an angle u with axis. The ball becoes R and starts to fall with acc tion g j^. (a) Deterin x P Q angular oentu o ball about point P as a f physical reason does the (c) What is the rate of ch Figure P11.17 Probles 17 and 3. tu of the ball about po 8
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All objects listed here have ass M. Moents of Inertia Thin rod, length L, axis through CM perpendicular to rod: I = 1 1 ML Thin rod, length L, axis through endpoint of rod, perpendicular to rod: I = 1 3 ML Solid sphere, radius R, axis through CM: I = 5 MR Cylinder or disc, radius R, axis through CM: I = 1 MR Thin ring, radius R, axis through CM: I = MR Trigonoetric Identities sin θ + cos θ = 1 sin(θ) = sin(θ) cos(θ). cos(θ) = cos θ sin θ sin(α ± β) = sin α cos β ± cos α sin β cos(α ± β) = cos α cos β sin α sin β cos α cos β = 1 [cos(α β) + cos(α + β)] sin α sin β = 1 [cos(α β) cos(α + β)] sin α cos β = 1 [sin(α + β) + sin(α β)] sin ( θ + π ) = cos θ cos ( θ + π ) = sin θ sec θ := 1 cos θ csc θ := 1 sin θ cot θ := 1 tan θ 1