Axis. Axis. Axis. Solid cylinder (or disk) about. Annular cylinder (or ring) about central axis. Hoop about. central axis.

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1 Instructor(s): Matcheva/Yelton PHYSICS DEPATMENT PHY 2048 FINA EXAM Deceber 0th, 206 Nae (print, last first): Signature: On y honor, I have neither given nor received unauthorized aid on this exaination. YOU TEST NUMBE IS THE 5-DIGIT NUMBE AT THE TOP OF EACH PAGE. () Code your test nuber on your answer sheet (use lines on the answer sheet for the 5-digit nuber). Code your nae on your answer sheet. DAKEN CICES COMPETEY. Code your UFID nuber on your answer sheet. (2) Print your nae on this sheet and sign it also. (3) Do all scratch work anywhere on this exa that you like. Circle your answers on the test for. At the end of the test, this exa printout is to be turned in. No credit will be given without both answer sheet and printout. (4) Blacken the circle of your intended answer copletely, using a #2 pencil or blue or black ink. Do not ake any stray arks or soe answers ay be counted as incorrect. (5) The answers are rounded off. Choose the closest to exact. There is no penalty for guessing. If you believe that no listed answer is correct, leave the for blank. (6) Hand in the answer sheet separately. Use g = 9.8 /s 2 G = N 2 /kg 2 ρ water = 000 kg 3 Hoop about central axis 2 Annular cylinder (or ring) about central axis Solid cylinder (or disk) about central axis I = M 2 I = 2 M( ) I = 2 M 2 Solid cylinder (or disk) about central diaeter Thin rod about axis through center perpendicular to length 2 Solid sphere about any diaeter I = M M 2 I = 2 M 2 2 I = 5 M 2 2 Thin spherical shell about any diaeter Hoop about any diaeter a b Slab about perpendicular axis through center 2 I = 3 M 2 I = 2 M 2 I = 2 M(a 2 + b 2 )

2 PHY2048 Exa Forula Sheet Vectors a = a x î+a y ĵ +a zˆk b = bx î+b y ĵ +b zˆk Magnitudes: a = a 2 x +a 2 y +a 2 z b = b 2 x +b 2 y +b 2 z Scalar Product: a b = a x b x +a y b y +a z b z Magnitude: a b = a b cosθ (θ = angle between a and b) Vector Product: a b = (a y b z a z b y )î+(a z b x a x b z )ĵ +(a x b y a y b x )ˆk Magnitude: a b = a b sinθ (θ = angle between a and b) Displaceent: r = r(t 2 ) r(t ) Average Velocity: v ave = r t = r(t 2) r(t ) t 2 t Average Speed: s ave = (total distance)/ t Instantaneous Velocity: v = d r(t) elative Velocity: v AC = v AB + v BC Average Acceleration: a ave = v t = v(t 2) v(t ) t 2 t Instantaneous Acceleration: a = d v = d2 r 2 Motion Equations of Motion for Constant Acceleration v = v 0 + at r r 0 = v 0 t+ 2 at2 vx 2 = v2 x0 +2a x(x x 0 ) (in each of 3 di) Newton s aws F net = 0 v is a constant (Newton s First aw) F net = a (Newton s Second aw) Action = eaction (Newton s Third aw) Force due to Gravity Weight (near the surface of the Earth) = g ( use g=9.8 /s 2 ) Magnitude of the Frictional Force Static: f s µ s F N Kinetic: f k = µ k F N Unifor Circular Motion (adius, Tangential Speed v = ω, Angular Velocity ω) Centripetal Acceleration: a = v2 = ω2 Period: T = 2π v = 2π ω ange: = v2 0 sin(2θ 0 ) g If: ax 2 +bx+c = 0 Then: x = b± b 2 4ac 2a Projectile Motion Quadratic Forula

3 PHY2048 Exa 2 Forula Sheet Work (W), Mechanical Energy (E, Kinetic Energy (K)), Potential Energy (U) r2 Kinetic Energy: K= 2 v2 Work: W= F d r Constant F F d Power: P= dw = F v r Work-Energy Theore: K f = K i +W Potential Energy: U = Work-Energy: W(external) = K + U + E(theral)+ E(internal) Gravity Near the Surface of the Earth (y-axis up): F y = g Spring Force: F x (x) = kx U(x) = 2 kx2 r2 r U(y) = gy F d r Fx (x) = du(x) dx Work: W = U Mechanical Energy: E = K +U Isolated and Conservative Syste: E = K + U = 0 inear Moentu: p = v F = d p Center of Mass (COM): M tot = Net Force: Fnet = d P tot i= inear Moentu, Angular Moentu, Torque Kinetic Energy: K = p2 2 i r COM = M tot i= = M tot a COM Ptot = M tot v COM = i= tf Ipulse: J = p = F(t) i r i v COM = M tot p i i= p i Moent of Inertia: I= i ri 2 (discrete) I= r 2 d (unifor) Parallel : I=I COM +Mh 2 Angular Moentu: = r p Torque: τ = r F = d Work: W = θf i= θ i τdθ Conservation of inear Moentu: if F net = d p = 0 then p = constant and p f = p i Conservation of Angular Moentu: if τ net = d = 0 then = constant and f = i Angular Position: θ(t) Angular Velocity: ω(t) = dθ(t) otational Variables Angular Acceleration: α(t) = dω(t) t i = d2 θ(t) 2 Torque: τ net = Iα Angular Moentu: = Iω Kinetic Energy: E rot = 2 Iω2 = 2 Power: P = τω 2I Arc ength: s = θ Tangential Speed: v = ω Tangential Acceleration: a = α olling Without Slipping: x COM = θ v COM = ω a COM = α K = 2 Mv2 COM + 2 I COMω 2 otational Equations of Motion (Constant Angular Acceleration α) ω(t) = ω 0 +αt θ(t) = θ 0 +ω 0 t+ 2 αt2 ω 2 (t) = ω α(θ(t) θ 0 ) Elastic Collisions of Two Bodies, D v f = v i v 2i v 2f = v i v 2i

4 PHY2048 Exa 3 Forula Sheet aw of Gravitation Magnitude of Force: F grav = G 2 r 2 G = N 2 /kg 2 Potential Energy: U grav = G 2 r Escape Speed: v escape = Pressure (variable force): P = df da 2GM Tension & Copression (Y = Young s Modulus, B = Bulk Modulus) inear: F A = Y Volue: P = F A = B V V Ideal Fluids Pressure (constant force): P = F A Units: Pa = N/ 2 Equation of Continuity: V =Av = constant (volue flow rate) =ρav = constant (ass flow rate) Bernoulli s Equation (y-axis up): P + 2 ρv2 +ρgy = P ρv2 2 +ρgy 2 = constant Fluids at rest (y-axis up): P 2 = P +ρg(y y 2 ) Buoyancy Force: F Buoy = M fluid g Siple Haronic Motion (SHM) (angular frequency ω = 2πf = 2π/T) x(t) = x ax cos(ωt+φ) v(t) = ωx ax sin(ωt+φ) v ax = ωx ax a(t) = ω 2 x ax cos(ωt+φ) = ω 2 x(t) a ax = ω 2 x ax k inear Haronic Oscillator: ω = κ Angular Haronic Oscillator: ω = I g Siple Pendulu: ω = hg Physical Pendulu: ω = I Sinusoidal Traveling Waves (frequency f = /T = ω/2π, wave nuber k = 2π/λ) y(x,t) = y ax sin(φ) = y ax sin(kx±ωt+φ) ( = right oving, + = left oving) Phase: Φ = kx±ωt Wave Speed: v wave = ω k = λ τ T = λf Wave Speed (tight string): v wave = µ Interference (Max Constructive): Φ = 2nπ n = 0,±,±2, d = nλ n = 0,±,±2, Interference (Max Destructive): Φ = (2n+)π n = 0,±,±2, d = (n+ 2 )λ n = 0,±,±2, Standing Waves on a String ( = length, n = haronic nuber) y (x,t) = 2y ax sin(kx)cos(ωt) Allowed Wavelengths & Frequencies: λ n = 2/n f n = v wave λ n = nv wave 2 n =,2,3 Intensity (W/ 2 ): I = P A Sound Waves (P = Power) Isotropic Point Source: I(r) = P source 4πr 2 Speed of sound: v sound = B ρ Doppler Shift: f obs = f s v sound v D v sound v S (f s = frequency of source, v s, v D = speed of source, detector) Change v D to +v D if the detector is oving opposite the direction of the propagation of the sound wave. Change v s to +v s if the source is oving opposite the direction of the propagation of the sound wave.

5 . Student a has ass of 90 kg and student b has a ass of 60 kg. They sit in identical, ass-less office chairs facing each other as shown in the figure. Student a places his bare feet on the knees of student b as shown. Student a then suddenly pushes outward with his feet, causing both chairs ove. During the push, which of the following are true? () Each student exerts the sae aount of force on the other. (2) Neither exerts a force on the other. (3) Student a exerts a force on student b, but b does not exert a force on a. (4) Each student exerts a force on the other, but b exerts the larger force. (5) Each student exerts a force on the other, but a exerts the larger force. a b 2. After the push is over the two ove apart (they are on wheels with no frictional losses). What is then the ratio of the kinetic energy of a to that of b? () 2/3 (2) 3/2 (3) (4) 0 (5) 4/9 3. A boy in a hot air balloon leans over the side and drops a stone. At the oent he drops it, the balloon has (taking up to be positive), v = 2.0 /s and a = 3.0/s 2. What speed (relative to the ground and in /s), is the stone travelling second later? () 7.8 (2) 6.8 (3) 2.8 (4) 9.8 (5).8 4. You drive 20 k north at 60 k/hr, then 50 k west at 50 k/hr. What, in k/hr, is the agnitude of your average velocity? () 43 (2) 57 (3) 70 (4) 55 (5) A potential energyfunction of a point ass is given by the expressionu(x) = 2x 2 +4x, where x is the position coordinate in eters, and U is easured in Joules. Only conservative forces are acting. If a ass is placed at x = 0 and released, how far (in eters) does it travel before it first reaches its axiu speed? () (2) 2 (3) 3 (4) 4 (5) A unifor solid cylinder, a unifor solid sphere, a thin annular cylinder, and a thin spherical shell are rolling without slipping along a horizontal surface. All objects have the sae kinetic energy and the sae ass. Which one has the highest value of the center-of-ass velocity? () unifor solid sphere (2) thin spherical shell (3) unifor solid cylinder (4) thin annular cylinder (5) they all have the sae center-of-ass velocity 7. Find the rotational inertia of the coposite object in the figure. The axis of rotation is shown with a dashed line. The two thin rods are unifor and each has a length and ass M. The hoop has a radius and a ass M. The hoop, the rods, and the axis of rotation are all in the sae plane. () (7/2)M 2 (2) (/2)M 2 (3) (29/6)M 2 (4) (5/6)M 2 (5) (3/2)M 2

6 8. A long bea with 5 kg is supported at its ends by two ropes A and B of equal length. A box is hanging 20 c fro the left end A of the bea. If the tension in the left rope is 30N, what is the ass of the box in kg? A B () 0.7 kg (2) 2.0 kg (3).4 kg (4) 2.8 kg (5).7 kg 9. A projectile with a ass is shot directly away fro Earth s surface. Neglect the rotation of Earth. What is the least initial echanical energy required at launch if the projectile is to escape Earth? M is the ass of the Earth, is the radius of the Earth and U( ) = 0. () 0 (2) GM/ 2 (3) GM/ (4) GM/ 2 (5) GM/ 0. In 993 the spacecraft Galileo sent an iage of asteroid 243 Ida and a tiny orbiting oon (now known as Dactyl), the first confired exaple of an asteroid-oon syste. In the iage, the oon, which is.5 k wide, is 00 k fro the center of the asteroid. Assue the oon s orbit is circular with a period of 27 h. What is the ass of the asteroid? () kg (2) kg (3) kg (4) kg (5) kg. Certain neutron stars (extreely dense stars) are believed to be rotating at about rev/s. If such a star has a radius of 20 k, what ust be its iniu ass so that aterial on its surface reains in its place on the star s surface during the rapid rotation? () kg (2) kg (3) kg (4) kg (5) kg 2. In the figure water stands at depth D = 35 behind the vertical upstrea face of a da of wih W = 34. Find the net horizontal force on the da fro the gauge pressure of the water. (ρ water = 000 kg ) 3 () N (2) N (3) N (4) N (5) N W D O 3. A hollow spherical iron shell floats alost copletely suberged in water. The outer diaeter is 60.0 c, and the density of iron is 7.87 g/c 3. Find the inner diaeter of the shell. (ρ water = 000 kg =.000 g 3 c ) 3 () 57 c (2) 43 c (3) 36 c (4) 25 c (5) 30 c 4. A block of ass M = 5.4 kg at rest on a horizontal frictionless table, is attached to a rigid support by a spring of constant k = 6000N/. A bullet of ass = 9.5g and velocity v of agnitude 630 /s strikes and is ebedded in the block. Assuing the copression of the spring is negligible until the bullet is ebedded, deterine the aplitude of the resulting siple haronic oscillation? () 3.3 c (2) 2.5 c (3) 4.6 c (4) 5.2 c (5) 6.0 c v M k

7 5. You are at rest and holding a siple pendulu with a period of s. You are then in an elevator and observe that the period of the sae pendulu is now 0.8 s. You can conclude that: () the elevator is accelerating upward (2) the elevator is at rest (3) the elevator is oving upwards (4) the elevator is oving downwards (5) the elevator is accelerating downwards 6. You have been abducted by aliens and find yourself on a strange planet. Fortunately, you have a unifor eter stick with you. You observe that, if the stick oscillates around one end as a physical pendulu, it has a period of 0.85 s. What is the acceleration of gravity on this planet? () 36/s 2 (2) 0/s 2 (3) 2/s 2 (4) 20/s 2 (5) 28/s 2 7. A transverse wave on a cable is described by the function y(x,t) = 2.3cos(4.7x+2t π/2) where distance is easured in eters and tie in seconds. If the tension in the cable is 25 N, what is the linear ass density of the cable? () 3.8 kg/ (2) 9.8 kg/ (3) 63 kg/ (4) 56 kg/ (5) 42 kg/ 8. What is the lowest frequency for standing waves on a wire that is 0.0 long, has a ass of 00 g, and is stretched under a tension of 250 N? () 8 Hz (2) 6 Hz (3) 32 Hz (4) 24 Hz (5) 4 Hz 9. A stationary otion detector sends sound waves of frequency 50 khz toward a truck approaching at a speed of 45 /s. What frequency will the detector easure for waves reflected back to it fro the truck? The speed of sound in air is V sound = 340/s. () 95 khz (2) 5 khz (3) 50 khz (4) 70 khz (5) 30 khz 20. Sound waves A and B, both of wavelength λ are initially in phase and traveling rightward, as indicated by the two rays. Wave A is reflected fro four surfaces but ends up traveling in its original direction. Wave B ends in that direction after reflecting fro two surfaces. et distance in the figure be expressed as = qλ. What is the sallest value of q > 0 that put A and B exactly out of phase with each other after all the reflections? () 0.5 (2).0 (3) 0.25 (4).25 (5).5 A B