77777 77777 Instructor(s): Field/Matcheva PHYSICS DEPARTMENT Final Exam PHY 2048 Name (print, last first): December 13, 2014 Signature: On my honor, I have neither given nor received unauthorized aid on this examination. YOUR TEST NUMBER IS THE 5-DIGIT NUMBER AT THE TOP OF EACH PAGE. (1) Code your test number on your answer sheet (use lines 76–80 on the answer sheet for the 5-digit number). Code your name on your answer sheet. DARKEN CIRCLES COMPLETELY. Code your UFID number on your answer sheet. (2) Print your name on this sheet and sign it also. (3) Do all scratch work anywhere on this exam that you like. Circle your answers on the test form. At the end of the test, this exam 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 completely, using a #2 pencil or blue or black ink. Do not make any stray marks or some answers may 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 form blank. (6) Hand in the answer sheet separately. Use g = 9.80 m/s2 Axis R Axis Axis Annular cylinder (or ring) about central axis Hoop about central axis R1 Solid cylinder (or disk) about central axis R2 L R 1 1 I = 2 M(R 12 + R 22 ) I = MR 2 Axis Solid cylinder (or disk) about central diameter Axis L L I = 2 MR2 Thin rod about axis through center perpendicular to length Axis Solid sphere about any diameter 2R R 1 1 Axis 2 1 I = 4 MR2 + 12 ML2 I = 12 ML2 Thin spherical shell about any diameter Axis R I = 5 MR2 Hoop about any diameter Axis Slab about perpendicular axis through center 2R b a 2 I = 3 MR2 1 I = 2 MR2 1 I = 12 M(a2 + b2) 77777 77777 PHY2048 Exam 1 Formula Sheet Vectors r r r r ˆ ˆ a = axiˆ + a y ˆj + az k b = bxiˆ + by ˆj + bz k Magnitudes: a = a x2 + a y2 + a z2 b = bx2 + by2 + bz2 r r r r r r r r Scalar Product: a ⋅ b = a xbx + a y by + a z bz Magnitude: a ⋅ b = a b cos θ (θ = angle between a and b ) r r Vector Product: a × b = (a y bz − a z by )iˆ + (a z bx − a x bz ) ˆj + (a x by − a y bx )kˆ r r r r r r Magnitude: a × b = a b sin θ (θ = smallest angle between a and b ) Motion r r r Displacement: ∆x = x(t2 ) − x(t1 ) (1 dimension) ∆r = r (t2 ) − r (t1 ) (3 dimensions) r r r r ∆x x(t2 ) − x(t1 ) ∆r r (t2 ) − r (t1 ) (1 dim) Average Velocity: vave = = vave = = t2 − t1 ∆t ∆t t2 − t1 (3 dim) Average Speed: save = (total distance)/∆t r r dx(t ) dr (t ) Instantaneous Velocity: v(t ) = (1 dim) (3 dim) v (t ) = dt dt r r r Relative Velocity: v AC = v AB + vBC (3 dim) r r r r ∆v v (t2 ) − v (t1 ) ∆v v(t2 ) − v(t1 ) (1 dim) (3 dim) Average Acceleration: aave = = = aave = ∆t ∆t t2 − t1 t2 − t1 r r r dv (t ) d 2 r (t ) dv(t ) d 2 x(t ) (1 dim) (3 dim) Instantaneous Acceleration: a (t ) = a (t ) = = = dt dt 2 dt dt 2 Equations of Motion (Constant Acceleration) v y (t ) = v y 0 + a y t v z (t ) = v z 0 + a z t vx (t ) = vx 0 + axt x(t ) = x0 + vx 0t + 12 axt 2 vx2 (t ) = vx20 + 2ax ( x(t ) − x0 ) y (t ) = y0 + v y 0t + 12 a y t 2 v y2 (t ) = v y20 + 2a y ( y (t ) − y0 ) z (t ) = z0 + v z 0t + 12 a z t 2 v z2 (t ) = v z20 + 2a z ( z (t ) − z0 ) Newton’s Law and Weight Weight (near the surface of the Earth) = W = mg (use g = 9.8 m/s2) r r Fnet = ma (m = mass) Magnitude of the Frictional Force (µs = static coefficient of friction, µk = kinetic coefficient of friction) Kinetic: f k = µ k FN (FN is the magnitude of the normal force) Static: ( f s ) max = µ s FN Uniform Circular Motion (Radius R, Tangential Speed v = Rω, Angular Velocity ω) v2 2πR 2π mv 2 Centripetal Acceleration & Force: a = Period: T = = Rω 2 F = = = mRω 2 ω R v R Projectile Motion (horizontal surface near Earth, v0 = initial speed, θ0 = initial angle with horizontal) v02 sin(2θ 0 ) Range: R = g Max Height: H= v02 sin 2 θ 0 2g Time (of flight): Quadratic Formula If: ax + bx + c = 0 2 − b ± b 2 − 4ac Then: x = 2a tf = 2v0 sin θ 0 g 77777 77777 PHY2048 Exam 2 Formula Sheet Work (W), Mechanical Energy (E), Kinetic Energy (KE), Potential Energy (U) r r r r r r r r dW 2 1 r→F ⋅d Kinetic Energy: KE = 2 mv Work: W = F ⋅ dr Power: P = = ⋅v F Cons tan t F ∫rr dt 2 1 r r2 r ∫ r Potential Energy: ∆U = − F ⋅ dr Work-Energy Theorem: KE f = KEi + W r r1 Work-Energy: W(external) = ∆KE + ∆U + ∆E(thermal) + ∆E(internal) Fx ( x) = − dU ( x) dx Work: W = -∆U U ( y ) = mgy Gravity Near the Surface of the Earth (y-axis up): Fy = −mg Spring Force: Fx ( x) = −kx U ( x) = 12 kx 2 Mechanical Energy: E = KE + U Isolated and Conservative System: ∆E = ∆KE + ∆U = 0 E f = Ei Linear Momentum, Angular Momentum, Torque r t r r r r dp r f r p2 Linear Momentum: p = mv F = Kinetic Energy: KE = Impulse: J = ∆p = ∫ F (t ) dt dt 2m ti N Center of Mass (COM): M tot = ∑ mi i =1 r r r dPtot Net Force: Fnet = = M tot aCOM dt r 1 rCOM = M tot N r ∑ mi ri i =1 r 1 vCOM = M tot N r ∑p i i =1 N r r r Ptot = M tot vCOM = ∑ pi i =1 N Moment of Inertia: I = ∑ mi ri 2 (discrete) I = ∫ r 2 dm (uniform) Parallel Axis: I = I COM + Mh 2 i =1 r θf r r dL Torque: τ = r × F = Work: W = ∫ τ dθ dt θi r r r r r dp Conservation of Linear Momentum: if Fnet = = 0 then p = constant and p f = pi dt r r r r r dL Conservation of Angular Momentum: if τ net = = 0 then L = constant and L f = Li dt Rotational Varables r r r Angular Momentum: L = r × p Angular Position: r d (t ) d 2 (t ) θ (t ) Angular Velocity: ω (t ) = dθ (t ) Angular Acceleration: α (t ) = ω = θ 2 dt dt Torque: τ net = Iα Angular Momentum: L = Iω Arc Length: s = Rθ Rolling Without Slipping: xCOM = Rθ L Power: P = τω 2I Tangential Acceleration: a = Rα Kinetic Energy: Erot = 12 Iω 2 = Tangential Speed: v = Rω dt 2 vCOM = Rω aCOM = Rα 2 KE = 12 MvCOM + 12 I COM ω 2 Rotational Equations of Motion (Constant Angular Acceleration α) ω (t ) = ω0 + αt θ (t ) = θ 0 + ω0t + 12 αt 2 ω 2 (t ) = ω02 + 2α (θ (t ) − θ 0 ) 77777 77777 PHY2048 Exam 3 Formula Sheet Law of Gravitation m1m2 G = 6.67 × 10 −11 Nm 2 / kg 2 2 r m1m2 2GM Potential Energy: U grav = −G Escape Speed: vescape = r R Tension & Compression (Y = Young’s Modulus, B = Bulk Modulus) Magnitude of Force: Fgrav = G Linear: F ∆L =Y A L Volume: P = F ∆V =B V A Ideal Fluids Pressure (variable force): P = dF Pressure (constant force): P = F dA A Units: 1 Pa = 1 N/m2 Rm = ρAv = constant (mass flow rate) Equation of Continuity: RV = Av = constant (volume flow rate) Bernoulli’s Equation (y-axis up): P1 + 12 ρv + ρgy1 = P2 + 12 ρv + ρgy2 = constant 2 1 2 2 Fluids at rest (y-axis up): P2 = P1 + ρg ( y1 − y2 ) Buoyancy Force: FBuoy = M fluid g Simple Harmonic Motion (SHM) (angular frequency ω = 2πf =2π/T) vmax = ωxmax x(t ) = xmax cos(ωt + φ ) amax = ω 2 xmax v(t ) = −ωxmax sin(ωt + φ ) a (t ) = −ω 2 xmax cos(ωt + φ ) = −ω 2 x(t ) ω= Ideal Spring (k = spring constant)): Fx = −kx k m E = 12 mv 2 (t ) + 12 kx 2 (t ) = constant Sinusoidal Traveling Waves (frequency f = 1/T = ω/2π, wave number k = 2π/λ) y ( x, t ) = ymax sin(Φ ) = ymax sin(kx ± ωt + φ ) (- = right moving, + = left moving) Phase: Φ = kx ± ωt Wave Speed: vwave = ω k = λ τ = λf Wave Speed (tight string): vwave = T µ n = 0,±1,±2,L ∆d = nλ n = 0,±1,±2,L Interference (Max Destructive): ∆Φ = π + 2πn n = 0,±1,±2,L ∆d = (n + 12 )λ n = 0,±1,±2,L Interference (Max Constructive): ∆Φ = 2πn Standing Waves (L = length, n = harmonic number) v nv Allowed Wavelengths & Frequencies: λn = 2 L / n f n = wave = wave n = 1,2,3L 2L λn Sound Waves (P = Power) Intensity (W/m2): I = P A Isotropic Point Source: I (r ) = Psource 4πr 2 Speed of Sound in Air (temperature T in Kelvin): vsound (T ) = v0 Speed of Sound: vsound = T T0 v0 = 331 m/s B ρ T0 = 273.15 oK Temperature (Kelvin, Centegrade, Fahrenheit): T(in oK) = T(in oC) + 273.15 T(in oF) = 1.8×T(in oC) + 32 Doppler Shift: f obs = f S vsound − vD (fS = frequency of source, vS, vD = speed of source, detector) vsound − vS Change –vD to +vD if the detector is moving opposite the direction of the propagation of the sound wave. Change –vS to +vS if the source is moving opposite the direction of the propagation of the sound wave. 77777 77777 1. A motorist drives along a straight road at a constant speed of 40 m/s. At t = 0 she passes a parked motorcycle police officer, and the officer takes off after her with acceleration a(t) = bt2 , where b is a constant and t is the time. What is the speed of the police officer (in m/s) when he reaches the motorist? (1) 160 (2) 200 (3) 240 (4) 80 (5) 100 2. A motorist drives along a straight road at a constant speed of 50 m/s. At t = 0 she passes a parked motorcycle police officer, and the officer takes off after her with acceleration a(t) = bt2 , where b is a constant and t is the time. What is the speed of the police officer (in m/s) when he reaches the motorist? (1) 200 (2) 160 (3) 240 (4) 80 (5) 100 3. A motorist drives along a straight road at a constant speed of 60 m/s. At t = 0 she passes a parked motorcycle police officer, and the officer takes off after her with acceleration a(t) = bt2 , where b is a constant and t is the time. What is the speed of the police officer (in m/s) when he reaches the motorist? (1) 240 (2) 160 (3) 200 (4) 80 (5) 120 4. A rabbit is dashing through the forest. Its position as a function of time is given by ~r(t) = (3 − 5t)î + (3t2 − 2t3 )ĵ, where position is measured in meters and time in seconds. What is the magnitude of the rabbit’s acceleration (in m/s2 ) at t = 1 s? (1) 6 (2) 18 (3) 24 (4) 2 (5) 32 5. A rabbit is dashing through the forest. Its position as a function of time is given by ~r(t) = (3 − 5t)î + (3t2 − 2t3 )ĵ, where position is measured in meters and time in seconds. What is the magnitude of the rabbit’s acceleration (in m/s2 ) at t = 2 s? (1) 18 (2) 6 (3) 24 (4) 2 (5) 32 6. A rabbit is dashing through the forest. Its position as a function of time is given by ~r(t) = (3 − 5t)î + (3t2 − 2t3 )ĵ, where position is measured in meters and time in seconds. What is the magnitude of the rabbit’s acceleration (in m/s2 ) at t = 2.5 s? (1) 24 (2) 6 (3) 18 (4) 2 7. A carnival ride near the surface of the Earth consists of the riders standing against the inside wall of a cylindrical room with radius R = 6.0 m. The room spins about the vertical cylinder axis with a constant speed. Once it is up to speed, the floor of the room falls away. If the cylindrical room completes 16 revolutions per minute, what minimum coefficient of static friction between the riders and the wall will keep them from dropping with the floor? (1) 0.582 (2) 0.460 (3) 0.372 (4) 0.288 (5) 32 ω r (5) 0.685 77777 77777 ω 8. A carnival ride near the surface of the Earth consists of the riders standing against the inside wall of a cylindrical room with radius R = 6.0 m. The room spins about the vertical cylinder axis with a constant speed. Once it is up to speed, the floor of the room falls away. If the cylindrical room completes 18 revolutions per minute, what minimum coefficient of static friction between the riders and the wall will keep them from dropping with the floor? (1) 0.460 (2) 0.582 (3) 0.372 (4) 0.288 r (5) 0.685 ω 9. A carnival ride near the surface of the Earth consists of the riders standing against the inside wall of a cylindrical room with radius R = 6.0 m. The room spins about the vertical cylinder axis with a constant speed. Once it is up to speed, the floor of the room falls away. If the cylindrical room completes 20 revolutions per minute, what minimum coefficient of static friction between the riders and the wall will keep them from dropping with the floor? (1) 0.372 (2) 0.582 (3) 0.460 (4) 0.288 10. Near the surface of the Earth, a wooden block with mass m = 4 kg is attached to a string. The string is wrapped around a frictionless pulley with a radius R = 0.5 m, and rotational inertia I as shown in the figure. The pulley and the block are initially at rest. If when the system is released and the string begins to unwind, the tension in the string is 20 N, what is I (in kg·m2 )? r (5) 0.685 R I m (1) 1.04 (2) 2.35 (3) 3.26 (4) 0.42 11. Near the surface of the Earth, a wooden block with mass m = 4 kg is attached to a string. The string is wrapped around a frictionless pulley with a radius R = 0.5 m, and rotational inertia I as shown in the figure. The pulley and the block are initially at rest. If when the system is released and the string begins to unwind, the tension in the string is 27.5 N, what is I (in kg·m2 )? (5) 4.55 R I m (1) 2.35 (2) 1.04 (3) 3.26 (4) 0.42 12. Near the surface of the Earth, a wooden block with mass m = 4 kg is attached to a string. The string is wrapped around a frictionless pulley with a radius R = 0.5 m, and rotational inertia I as shown in the figure. The pulley and the block are initially at rest. If when the system is released and the string begins to unwind, the tension in the string is 30 N, what is I (in kg·m2 )? (5) 4.55 R I m (1) 3.26 (2) 1.04 (3) 2.35 13. Near the surface of the Earth a block of mass M is released from rest at a height h on a frictionless incline as shown in the figure. The block slides down the frictionless incline to reach a flat horizontal surface with a kinetic coefficient of friction µk = 0.5. The block slides a horizontal distance d and then slides up a frictionless incline and reaches a maximum height H before sliding back down. If h = d, what is H? (1) h/2 (2) 3h/4 (3) h/4 (4) 0.42 (5) 4.55 M µk h d (4) h (5) h/3 H x-axis 77777 77777 14. Near the surface of the Earth a block of mass M is released from rest at a height h on a frictionless incline as shown in the figure. The block slides down the frictionless incline to reach a flat horizontal surface with a kinetic coefficient of friction µk = 0.25. The block slides a horizontal distance d and then slides up a frictionless incline and reaches a maximum height H before sliding back down. If h = d, what is H? (1) 3h/4 (2) h/2 (3) h/4 15. Near the surface of the Earth a block of mass M is released from rest at a height h on a frictionless incline as shown in the figure. The block slides down the frictionless incline to reach a flat horizontal surface with a kinetic coefficient of friction µk = 0.75. The block slides a horizontal distance d and then slides up a frictionless incline and reaches a maximum height H before sliding back down. If h = d, what is H? (1) h/4 (2) h/2 (3) 3h/4 M µk h x-axis d (4) h H (5) h/3 M µk h d (4) h H x-axis (5) h/3 16. Near the surface of the Earth a man whose weight at rest is 180 N stands on a scale in an elevator that starts from rest and accelerates upward with a constant acceleration. If after the elevator has travelled a distance of 10 m its speed is 4 m/s, what is his apparent weight (in N) on the scale in the elevator during his ride? (1) 194.7 (2) 213.1 (3) 238.8 (4) 165.3 (5) 146.9 17. Near the surface of the Earth a man whose weight at rest is 180 N stands on a scale in an elevator that starts from rest and accelerates upward with a constant acceleration. If after the elevator has travelled a distance of 10 m its speed is 6 m/s, what is his apparent weight (in N) on the scale in the elevator during his ride? (1) 213.1 (2) 194.7 (3) 238.8 (4) 165.3 (5) 146.9 18. Near the surface of the Earth a man whose weight at rest is 180 N stands on a scale in an elevator that starts from rest and accelerates upward with a constant acceleration. If after the elevator has travelled a distance of 10 m its speed is 8 m/s, what is his apparent weight (in N) on the scale in the elevator during his ride? (1) 238.8 (2) 194.7 (3) 213.1 (4) 165.3 (5) 146.9 19. A race car starts from rest at t = 0 and travels around a circular track of radius R with a constant angular acceleration. If the magnitude of the tangential acceleration of the car is equal to the magnitude of the radial acceleration (i.e., centripetal acceleration) of the car at t = 20 s, how long does it take for the race car to complete its first revolution around the track (in minutes)? (1) 1.18 (2) 1.77 (3) 2.36 (4) 1.00 (5) 3.00 20. A race car starts from rest at t = 0 and travels around a circular track of radius R with a constant angular acceleration. If the magnitude of the tangential acceleration of the car is equal to the magnitude of the radial acceleration (i.e., centripetal acceleration) of the car at t = 30 s, how long does it take for the race car to complete its first revolution around the track (in minutes)? (1) 1.77 (2) 1.18 (3) 2.36 (4) 1.00 (5) 3.00 77777 77777 21. A race car starts from rest at t = 0 and travels around a circular track of radius R with a constant angular acceleration. If the magnitude of the tangential acceleration of the car is equal to the magnitude of the radial acceleration (i.e., centripetal acceleration) of the car at t = 40 s, how long does it take for the race car to complete its first revolution around the track (in minutes)? (1) 2.36 (2) 1.18 (3) 1.77 (4) 1.00 (5) 3.00 22. A 80-N uniform plank leans at rest against a frictionless wall at an angle θ with the horizontal as shown in the figure. If θ = 53.13◦ , what is the magnitude of the force (in N) applied to the plank by the wall? θ (1) 30 (2) 25 (3) 20 (4) 80 (5) 120 23. A 80-N uniform plank leans at rest against a frictionless wall at an angle θ with the horizontal as shown in the figure. If θ = 57.99◦ , what is the magnitude of the force (in N) applied to the plank by the wall? θ (1) 25 (2) 30 (3) 20 (4) 80 (5) 120 24. A 80-N uniform plank leans at rest against a frictionless wall at an angle θ with the horizontal as shown in the figure. If θ = 63.43◦ , what is the magnitude of the force (in N) applied to the plank by the wall? θ (1) 20 (2) 30 (3) 25 (4) 80 (5) 120 25. A 0.5-kg rubber ball is dropped from rest a height H = 19.6 m above the surface of the Earth. It strikes the sidewalk below and rebounds up to a maximum height of 4.9 m. If the ball was in contact with the sidewalk for 0.2 seconds, what is the magnitude of the average force that the sidewalk exerts on the ball during the collision (in N)? (1) 73.5 (2) 58.8 (3) 49.0 (4) 38.5 (5) 82.2 26. A 0.5-kg rubber ball is dropped from rest a height H = 19.6 m above the surface of the Earth. It strikes the sidewalk below and rebounds up to a maximum height of 4.9 m. If the ball was in contact with the sidewalk for 0.25 seconds, what is the magnitude of the average force that the sidewalk exerts on the ball during the collision (in N)? (1) 58.8 (2) 73.5 (3) 49.0 (4) 38.5 (5) 82.2 27. A 0.5-kg rubber ball is dropped from rest a height H = 19.6 m above the surface of the Earth. It strikes the sidewalk below and rebounds up to a maximum height of 4.9 m. If the ball was in contact with the sidewalk for 0.3 seconds, what is the magnitude of the average force that the sidewalk exerts on the ball during the collision (in N)? (1) 49.0 (2) 73.5 (3) 58.8 (4) 38.5 (5) 82.2 77777 77777 28. A block slides along a horizontal frictionless surface with speed v. When the block reaches the point x = 0, two forces with magnitudes F1 = 3x2 N and F2 = 10 N are applied on the block as shown in the figure. What is the total work (in J) done on the box if the box travels a distance d from x = 0 to x = 10 m in the positive x direction? (1) 900 (2) 273 (3) 75 (2) 900 (3) 75 (2) 900 (3) 273 x=0 (4) 1200 x=d x axis x=d x axis x=d x axis (5) 35 v F1 F2 x=0 (4) 1200 30. A block slides along a horizontal frictionless surface with speed v. When the block reaches the point x = 0, two forces with magnitudes F1 = 3x2 N and F2 = 10 N are applied on the block as shown in the figure. What is the total work (in J) done on the box if the box travels a distance d from x = 0 to x = 5 m in the positive x direction? (1) 75 F2 (4) 1200 29. A block slides along a horizontal frictionless surface with speed v. When the block reaches the point x = 0, two forces with magnitudes F1 = 3x2 N and F2 = 10 N are applied on the block as shown in the figure. What is the total work (in J) done on the box if the box travels a distance d from x = 0 to x = 7 m in the positive x direction? (1) 273 v F1 (5) 35 v F1 F2 x=0 (5) 35 31. Two stars with masses M1 and M2 orbit with uniform circular motion around their common center of mass. If M1 = 3 × 1030 kg and M2 = 2M1 , and the distance between the stars is 1 × 1010 km, what is the period of their orbit (in years)? (1) 257 (2) 472 (3) 727 (4) 315 (5) 578 32. Two stars with masses M1 and M2 orbit with uniform circular motion around their common center of mass. If M1 = 3 × 1030 kg and M2 = 2M1 , and the distance between the stars is 1.5 × 1010 km, what is the period of their orbit (in years)? (1) 472 (2) 257 (3) 727 (4) 315 (5) 578 33. Two stars with masses M1 and M2 orbit with uniform circular motion around their common center of mass. If M1 = 3 × 1030 kg and M2 = 2M1 , and the distance between the stars is 2 × 1010 km, what is the period of their orbit (in years)? (1) 727 (2) 257 (3) 472 (4) 315 (5) 890 34. Planet Roton, with a mass of 7 × 1024 kg and a radius of 1,500 km, gravitationally attracts a meteorite that is initially at rest relative to the planet, at a distance great enough to take as infinite. The meteorite falls toward the planet. Assuming the planet is airless, what is the speed (in km/s) of the meteorite relative to the planet when it reaches the planet’s surface? (1) 25.0 (2) 17.6 (3) 14.4 (4) 31.2 (5) 11.6 35. Planet Roton, with a mass of 7 × 1024 kg and a radius of 3,000 km, gravitationally attracts a meteorite that is initially at rest relative to the planet, at a distance great enough to take as infinite. The meteorite falls toward the planet. Assuming the planet is airless, what is the speed (in km/s) of the meteorite relative to the planet when it reaches the planet’s surface? (1) 17.6 (2) 25.0 (3) 14.4 (4) 31.2 (5) 11.6 77777 77777 36. Planet Roton, with a mass of 7 × 1024 kg and a radius of 4,500 km, gravitationally attracts a meteorite that is initially at rest relative to the planet, at a distance great enough to take as infinite. The meteorite falls toward the planet. Assuming the planet is airless, what is the speed (in km/s) of the meteorite relative to the planet when it reaches the planet’s surface? (1) 14.4 (2) 25.0 (3) 17.6 (4) 31.2 37. A block of mass M = 4 kg is at rest on a horizontal frictionless surface and is connected to an ideal spring as shown in the figure. A 2-gram bullet traveling horizontally at 290 m/s strikes the block and becomes embedded in the block. If the bullet-block system comes to rest after compressing the spring a distance of 4 cm, what is the period (in s) of the subsequent simple harmonic motion of the system? (1) 1.73 (2) 2.60 (3) 3.47 (2) 1.73 (3) 3.47 (1) 3.47 (2) 1.73 (3) 2.60 (4) 0.87 40. A cubical metal box with sides of mass M and length L has a square lid also with mass M and length L. The lid is not attached to the box, however, the lid and the box form an airtight seal. Near the surface of the Earth, the lid is held at rest by a steel cable, as shown in the figure. The pressure outside the box is the atmospheric pressure, Pout = Patm = 101 kPa. The box is partially evacuated to an inside pressure Pin = 95 kPa. If L = 0.2 m, what is the maximum mass M (in kg) of the sides of the cubical metal box such that the box remains at rest and does not fall? (1) 4.90 (2) 8.98 (3) 13.06 (4) 2.65 41. A cubical metal box with sides of mass M and length L has a square lid also with mass M and length L. The lid is not attached to the box, however, the lid and the box form an airtight seal. Near the surface of the Earth, the lid is held at rest by a steel cable, as shown in the figure. The pressure outside the box is the atmospheric pressure, Pout = Patm = 101 kPa. The box is partially evacuated to an inside pressure Pin = 90 kPa. If L = 0.2 m, what is the maximum mass M (in kg) of the sides of the cubical metal box such that the box remains at rest and does not fall? (1) 8.98 (2) 4.90 (3) 13.06 M (5) 4.95 (4) 2.65 Ideal spring v M (4) 0.87 39. A block of mass M = 4 kg is at rest on a horizontal frictionless surface and is connected to an ideal spring as shown in the figure. A 2-gram bullet traveling horizontally at 290 m/s strikes the block and becomes embedded in the block. If the bullet-block system comes to rest after compressing the spring a distance of 8 cm, what is the period (in s) of the subsequent simple harmonic motion of the system? Ideal spring v (4) 0.87 38. A block of mass M = 4 kg is at rest on a horizontal frictionless surface and is connected to an ideal spring as shown in the figure. A 2-gram bullet traveling horizontally at 290 m/s strikes the block and becomes embedded in the block. If the bullet-block system comes to rest after compressing the spring a distance of 6 cm, what is the period (in s) of the subsequent simple harmonic motion of the system? (1) 2.60 (5) 11.6 (5) 4.95 Ideal spring v M (5) 4.95 Pout Pin L L (5) 18.89 Pout Pin L (5) 18.89 L 77777 77777 42. A cubical metal box with sides of mass M and length L has a square lid also with mass M and length L. The lid is not attached to the box, however, the lid and the box form an airtight seal. Near the surface of the Earth, the lid is held at rest by a steel cable, as shown in the figure. The pressure outside the box is the atmospheric pressure, Pout = Patm = 101 kPa. The box is partially evacuated to an inside pressure Pin = 85 kPa. If L = 0.2 m, what is the maximum mass M (in kg) of the sides of the cubical metal box such that the box remains at rest and does not fall? (1) 13.06 (2) 4.90 (3) 8.98 (4) 2.65 Pout Pin L L (5) 18.89 43. Stan and Ollie are standing next to a train track. Stan puts his ear to the steel track to hear the train coming. When the train is 750 m away he hears the sound of the train whistle through the track 2.1 s before Ollie hears it through the air. If the speed of sound in steel is 5790 m/s, what is the temperature of the air (in ◦ C)? (1) 9.0 (2) 23.3 (3) 37.9 (4) 18.2 (5) 15.5 44. Stan and Ollie are standing next to a train track. Stan puts his ear to the steel track to hear the train coming. When the train is 770 m away he hears the sound of the train whistle through the track 2.1 s before Ollie hears it through the air. If the speed of sound in steel is 5790 m/s, what is the temperature of the air (in ◦ C)? (1) 23.3 (2) 9.0 (3) 37.9 (4) 18.2 (5) 15.5 45. Stan and Ollie are standing next to a train track. Stan puts his ear to the steel track to hear the train coming. When the train is 790 m away he hears the sound of the train whistle through the track 2.1 s before Ollie hears it through the air. If the speed of sound in steel is 5790 m/s, what is the temperature of the air (in ◦ C)? (1) 37.9 (2) 9.0 (3) 23.3 (4) 18.2 (5) 15.5 46. A large cargo container has a square base with an area of 4 m2 and height H = 6 m. When empty, it floats on the water (ρwater = 1, 000 kg/m3 ) with 4 meters above the surface of the water and 2 m below the surface as shown in the figure. The cargo container is being loaded with small 50-kg boxes. What is the maximum number of boxes the cargo container can hold without sinking? (1) 320 (2) 400 (3) 500 (4) 240 4m 2m (5) 300 47. A large cargo container has a square base with an area of 4 m2 and height H = 6 m. When empty, it floats on the water (ρwater = 1, 000 kg/m3 ) with 4 meters above the surface of the water and 2 m below the surface as shown in the figure. The cargo container is being loaded with small 40-kg boxes. What is the maximum number of boxes the cargo container can hold without sinking? (1) 400 (2) 320 (3) 500 (4) 240 4m 2m (5) 300 48. A large cargo container has a square base with an area of 4 m2 and height H = 6 m. When empty, it floats on the water (ρwater = 1, 000 kg/m3 ) with 4 meters above the surface of the water and 2 m below the surface as shown in the figure. The cargo container is being loaded with small 32-kg boxes. What is the maximum number of boxes the cargo container can hold without sinking? (1) 500 (2) 320 (3) 400 (4) 240 4m 2m (5) 300 77777 77777 49. What is the maximum total mass (including the mass of the empty balloon) that a spherical helium balloon with a radius of 1.5 m can lift off the ground? The density of helium and the air are ρHe = 0.18 kg/m3 and ρair = 1.2 kg/m3 , respectively. (1) 14.42 kg (2) 34.18 kg (3) 66.76 kg (4) 10.45 kg (5) 72.25 kg 50. What is the maximum total mass (including the mass of the empty balloon) that a spherical helium balloon with a radius of 2.0 m can lift off the ground? The density of helium and the air are ρHe = 0.18 kg/m3 and ρair = 1.2 kg/m3 , respectively. (1) 34.18 kg (2) 14.42 kg (3) 66.76 kg (4) 10.45 kg (5) 72.25 kg 51. What is the maximum total mass (including the mass of the empty balloon) that a spherical helium balloon with a radius of 2.5 m can lift off the ground? The density of helium and the air are ρHe = 0.18 kg/m3 and ρair = 1.2 kg/m3 , respectively. (1) 66.76 kg (2) 14.42 kg (3) 34.18 kg (4) 10.45 kg 52. A stationary motion detector on the x-axis sends sound waves of frequency of 500 Hz, as shown in the figure. The waves sent out by the detector are reflected off a truck traveling along the x-axis and then are received back at the detector. If the frequency of the waves received back at the detector is 750 Hz, what is the x-component of the velocity of the truck (in m/s)? (Take the speed of sound to be 343 m/s.) (1) −68.6 (2) 38.1 (3) 60.5 (2) −68.6 (3) 60.5 x-axis (4) 90.3 (5) 68.6 (2) −68.6 (3) 38.1 55. The figure shows two isotropic point sources of sound on the x-axis, source S1 at x = 0 and source S2 at x = d. The sources emit sound at the same wavelength λ and the same amplitude A, and they emit in phase. A point P is shown on the x-axis with 0 < x < d. Assume that as the sound waves travel to the point P, the decrease in their amplitude is negligible. If λ = d, at what points P along the x-axis does maximally destructive interference occur? x-axis (4) 90.3 (5) −38.1 (1) (2) (3) (4) (5) x = 0.25d and x = 0.75d x = 0.20d and x = 0.80d x = 0.15d and x = 0.85d x = 0.10d and x = 0.90d Only at x = 0.50d Truck Motion Detector x-axis (4) 90.3 (5) −60.5 S1 x=0 Truck Motion Detector 54. A stationary motion detector on the x-axis sends sound waves of frequency of 500 Hz, as shown in the figure. The waves sent out by the detector are reflected off a truck traveling along the x-axis and then are received back at the detector. If the frequency of the waves received back at the detector is 350 Hz, what is the x-component of the velocity of the truck (in m/s)? (Take the speed of sound to be 343 m/s.) (1) 60.5 Truck Motion Detector 53. A stationary motion detector on the x-axis sends sound waves of frequency of 500 Hz, as shown in the figure. The waves sent out by the detector are reflected off a truck traveling along the x-axis and then are received back at the detector. If the frequency of the waves received back at the detector is 400 Hz, what is the x-component of the velocity of the truck (in m/s)? (Take the speed of sound to be 343 m/s.) (1) 38.1 (5) 72.25 kg P x S2 x=d x-axis 77777 77777 56. The figure shows two isotropic point sources of sound on the x-axis, source S1 at x = 0 and source S2 at x = d. The sources emit sound at the same wavelength λ and the same amplitude A, and they emit in phase. A point P is shown on the x-axis with 0 < x < d. Assume that as the sound waves travel to the point P, the decrease in their amplitude is negligible. If λ = 1.2d, at what points P along the x-axis does maximally destructive interference occur? (1) (2) (3) (4) (5) x=0 S2 P x=d x x-axis x = 0.20d and x = 0.80d x = 0.25d and x = 0.75d x = 0.15d and x = 0.85d x = 0.10d and x = 0.90d Only at x = 0.50d 57. The figure shows two isotropic point sources of sound on the x-axis, source S1 at x = 0 and source S2 at x = d. The sources emit sound at the same wavelength λ and the same amplitude A, and they emit in phase. A point P is shown on the x-axis with 0 < x < d. Assume that as the sound waves travel to the point P, the decrease in their amplitude is negligible. If λ = 1.4d, at what points P along the x-axis does maximally destructive interference occur? (1) (2) (3) (4) (5) S1 S1 x=0 S2 P x=d x x-axis x = 0.15d and x = 0.85d x = 0.25d and x = 0.75d x = 0.20d and x = 0.80d x = 0.10d and x = 0.90d Only at x = 0.50d 58. A travelling wave on a string is described with the equation y(x, t) = 0.5 cos(5πt − 3πx + 0.5π), where t is in seconds, and x and y are in meters. How long does it take (in s) for the wave to travel a distance of 10 m along the string? (1) 6 (2) 9 (3) 12 (4) 3 (5) 15 59. A travelling wave on a string is described with the equation y(x, t) = 0.5 cos(5πt − 3πx + 0.5π), where t is in seconds, and x and y are in meters. How long does it take (in s) for the wave to travel a distance of 15 m along the string? (1) 9 (2) 6 (3) 12 (4) 3 (5) 15 60. A travelling wave on a string is described with the equation y(x, t) = 0.5 cos(5πt − 3πx + 0.5π), where t is in seconds, and x and y are in meters. How long does it take (in s) for the wave to travel a distance of 20 m along the string? (1) 12 (2) 6 (3) 9 (4) 3 (5) 15 FOLLOWING GROUPS OF QUESTIONS WILL BE SELECTED AS ONE GROUP FROM EACH TYPE TYPE 1 Q# S 1 Q# S 2 Q# S 3 TYPE 2 Q# S 4 Q# S 5 Q# S 6 TYPE 3 Q# S 7 Q# S 8 Q# S 9 TYPE 4 Q# S 10 Q# S 11 Q# S 12 77777 TYPE 5 Q# S 13 Q# S 14 Q# S 15 TYPE 6 Q# S 16 Q# S 17 Q# S 18 TYPE 7 Q# S 19 Q# S 20 Q# S 21 TYPE 8 Q# S 22 Q# S 23 Q# S 24 TYPE 9 Q# S 25 Q# S 26 Q# S 27 TYPE 10 Q# S 28 Q# S 29 Q# S 30 TYPE 11 Q# S 31 Q# S 32 Q# S 33 TYPE 12 Q# S 34 Q# S 35 Q# S 36 TYPE 13 Q# S 37 Q# S 38 Q# S 39 TYPE 14 Q# S 40 Q# S 41 Q# S 42 TYPE 15 Q# S 43 Q# S 44 Q# S 45 TYPE 16 Q# S 46 Q# S 47 Q# S 48 TYPE 17 Q# S 49 Q# S 50 Q# S 51 TYPE 18 Q# S 52 Q# S 53 Q# S 54 TYPE 19 Q# S 55 Q# S 56 Q# S 57 TYPE 20 Q# S 58 Q# S 59 Q# S 60 77777

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