PHYS 211 вЂ“ MT3 Fall 2012 Sample 3 Solutions 1. The following true-false questions are all based on the following set-up and each is worth 2 points (so these 5 questions together are worth 2 of the multiple choice questions, 10 points) A particle moving in one dimension is subject to the potential energy curve shown in the plot. A. T or F: At point xa, the force on the particle is in the negative x direction. Mathematically: The force is the negative slope (slope is negative so force is to the right). Conceptually: Objects like to roll down hill (to lower potential energy) B. T or F: If the particle has total energy of 3 Joules at point xc and its velocity is in the positive x direction, it will eventually reach point xa. It wonвЂ™t be able to make it over the hump between xb and xc C. T or F: If the particle has kinetic energy of 3 Joules at point xc it can reach point xb. The total energy, 4 J, is enough to make it over the hump D. T or F: If the total energy of the particle at point xb is 3 Joules it can reach point xc. For the same reason as case B E. T or F: If the particle is released from rest at xc, it will travel in the positive x direction. The force will be to the left (downhill) Page 1 of 10 PHYS 211 вЂ“ MT3 Fall 2012 Sample 3 Solutions 2. A boy walks on level ground at a constant speed of 2.0 m/s while holding a 40 N box horizontally at arm's length. His arm is 1.5m above the ground. What is the work done by the force of the boy on the box while he walks with it? A. B. C. D. E. 0J 27 J 40 J 60 J 90 J No change in energy, no work done (F perpendicular to distance) 3. You and your friend are both standing on a horizontal frictionless surface. To get your friend's attention, you throw a ball due west at your friend. The ball bounces elastically off your friend's back. Which one of the following statements is true about this situation? A. Your friend will remain stationary because the ball bounces elastically and does not impart its momentum to her. B. Your friend will remain stationary because the kinetic energy of the ball is conserved in an elastic collision. C. Your friend will remain stationary due to conservation of both linear momentum and kinetic energy. D. You will remain stationary after you throw the ball due to conservation of both linear momentum and kinetic energy. E. You will move east after you throw the ball due to conservation of linear momentum. Ball goes W, you go E. Your friend will go W (in all conditions) F. More than one of the above 4. A large car and a hard-headed bird collide, with the bird bouncing off the windshield. Which of the following statements is true about the vector impulse on the bird ( J bird ) and the vector impulse on the car ( J car ). A. J bird пЂѕ J car B. J bird пЂј J car C. J bird пЂЅ пЂJ car Equal and opposite forces, same time D. J bird пЂЅ J car E. None of the above Page 2 of 10 PHYS 211 вЂ“ MT3 Fall 2012 Sample 3 Solutions 5. Two bodies, A and B, have equal kinetic energies. The mass of A is nine times that of B. The ratio of the momentum of A to that of B is: A. B. C. D. E. 1:3 1:1 1:9 9:1 3:1 1пЂЅ K A p A2 2mA p A2 mB p A2 mB p2 p пЂЅ 2 пЂЅ 2 пЂЅ 2 пѓћ A2 пЂЅ 9 пѓћ A пЂЅ 3 K B pB 2mB pB mA pB 9mB pB pB 6. Each of the figures shown below indicates the direction and magnitude of a force F acting on a system which is moving with a velocity v. Which figure represents the most negative (or least positive if none are negative) power being done by the force F at the instant pictured ? Need F & v to point in opposite directions to be negative 7. A student standing on a third floor balcony throws a water balloon onto the ground below. The work done on the balloon by gravity (during the fall) and the ground (as the balloon smashes into it), respectively, are: A. B. C. D. E. Negative, Negative Positive, Negative Negative, Positive Positive, Positive None of the above The ground doesnвЂ™t move so it does no work. Gravity does positive work (force and motion both down) Page 3 of 10 PHYS 211 вЂ“ MT3 Fall 2012 Sample 3 Solutions 8. A 2500 kg car traveling down I-80 at 100 km/hr smacks into a 0.001 kg fly traveling in the opposite direction at 1 km/hr. Which of the following statements is true? A. B. C. D. The car exerts a much bigger force on the fly than does the fly on the car. The fly exerts no force on the car whatsoever. If the fly sticks to the car, it exerts a greater force on it than if it bounces off it. The magnitude of the change in momentum of the fly is the same as the magnitude of the change in momentum of the car, and points in the same direction. E. The magnitude of the change in momentum of the fly is the same as the magnitude of the change in momentum of the car, and points in the opposite direction. Equal & opposite force, impulse and hence change in momentum пЂЁ пЂ© пЂЁ пЂ© 9. A particle of mass 5 kg is acted on by two forces, F1 пЂЅ 3Л†i пЂ« 4Л†j N and F2 пЂЅ пЂ2Л†i пЂ« Л†j N пЂЁ пЂ© пЂЁ пЂ© and is moved from ri пЂЅ Л†i пЂ« Л†j m to rf пЂЅ 5Л†i пЂ 4Л†j m . What is the net work done on the particle? A. B. C. D. E. -21 J -11 J -9 J -1 J 7J пЂЁ пЂ© пЂЁ пЂ© пЂЁ пЂ© пЃ„r пЂЅ r пЂ r пЂЅ пЂЁ 5Л†i пЂ 4Л†jпЂ© m пЂ пЂЁ Л†i пЂ« Л†jпЂ© m пЂЅ пЂЁ 4Л†i пЂ 5Л†jпЂ© m W =F пѓ— пЃ„r пЂЅ пЂЁ Л†i пЂ« 5Л†jпЂ© N пѓ— пЂЁ 4Л†i пЂ 5Л†jпЂ© m пЂЅ 4 пЂ 25J пЂЅ пЂ21J Fnet пЂЅ F1 пЂ« F2 пЂЅ 3Л†i пЂ« 4Л†j N пЂ« пЂ2Л†i пЂ« Л†j N= Л†i пЂ« 5Л†j N f net i net 10. A paratrooper whose chute fails to open land in snow; he is hurt slightly. Had he landed on bare ground, the stopping time would have been 10 times shorter. Does the presence of the snow increase, decrease, or leave unchanged the values of the impulse stopping the paratrooper and the force stopping the paratrooper? A. B. C. D. E. impulse unchanged, force unchanged impulse unchanged, increased force impulse unchanged, decreased force decreased impulse, decreased force decreased impulse, force unchanged Same momentum to stop, so same impulse. But longer time means smaller force. Page 4 of 10 PHYS 211 вЂ“ MT3 Fall 2012 Sample 3 Solutions 11. A firecracker at rest on the ground explodes into just two pieces (A and B): Piece A: mass M Piece B: mass 3M What can we say about the magnitude of momentum (p) and the kinetic energy (K) of these two pieces? A. Both pieces have the same magnitude of momentum, but A has 3 times the kinetic energy of B. p conserved so same, small m пѓћ big K= p 2 2m B. Both pieces have the same magnitude of momentum, but B has 3 times the kinetic energy of A. C. A has three times the magnitude of momentum and three times the kinetic energy of B. D. B has three times the magnitude of momentum and three times the kinetic energy of A. E. Both pieces have the same kinetic energy, but B has three times the magnitude of momentum of A. F. None of the above 12. The following 5 вЂњmatchingвЂќ questions are all based on the following set-up and are each worth two points (so these 5 questions together are worth 2 of the other multiple choice questions, 10 points) A stock person at the local grocery store has a job consisting of the following five segments. During each segment determine whether she is doing: A. Positive work B. Negative work C. No work D. Either positive or negative work (not enough info to determine) 1. 2. 3. 4. 5. picking up boxes of tomatoes from the stockroom floor (A: F, пЃ„s up) accelerating to a comfortable speed (A: F, пЃ„s forward) carrying the boxes to the tomato display at constant speed (C: const. v пѓ no F) decelerating to a stop (B: F backward, пЃ„s forward) lowering the boxes slowly to the floor. (B: F up, пЃ„s down) Page 5 of 10 PHYS 211 вЂ“ MT3 Fall 2012 Sample 3 Solutions 13. A mass M moving in the x-direction with a velocity of 10 m/s Л†i collides with a mass m (with half the mass of M) that has a velocity -5 m/s Л†i . Immediately afterwards, M is observed to have a velocity of 0 m/s Л†i and m is observed to have a velocity of 15 m/s Л†i . What can we say about this collision? A. It is an elastic collision. B. It is an inelastic (but not completely inelastic) collision. C. It is a completely inelastic collision. D. It is possible only if some other energy was released in the collision. E. It is not possible. 1 2 ? M пЂЁ10 ms пЂ© пЂ« 12 пЂЁ 12 M пЂ© пЂЁ 5 ms пЂ© пЂЅ 12 M пЂЁ 0 ms пЂ© пЂ« 12 пЂЁ 12 M пЂ© пЂЁ15 ms пЂ© 2 2 2 2 Multiply by 4/M and the units, and we get ? ? 2 пЂЁ10 пЂ© пЂ« пЂЁ 5пЂ© пЂЅ пЂЁ15пЂ© пЂЅ 225 пЂЅ 2 пѓ— 100 пЂ« 25 YES! 2 2 2 14. A single force, shown below on the graph, acts on a 2 kg object. If the object's speed at x = 0 m is 2 m/s, what is its speed at 8 m? A. 2 m/s пЂЁ пЂ© W пЂЅ пѓІ Fdx пЂЅ 1 2 пЂЁ 4N пЂ©пЂЁ 4m пЂ© пЂ« пЂЁ 4N пЂ©пЂЁ 2m пЂ© пЂ« 12 пЂЁ 4N пЂ©пЂЁ 2m пЂ© пЂЅ 20J B. 2 1 пЂ« 5 m/s Start with Kinetic energy, add this, get new kinetic energy: C. 2 6 m/s KE f пЂЅ 12 mv 2f пЂЅ KE0 пЂ« W D. 20 m/s E. 24 m/s F. None of the above пѓћ v f пЂЅ v02 пЂ« m2 W пЂЅ пЂЁ 2 ms пЂ© 2 пЂ« 2 2kg 20J пЂЅ 24 m s пЂЅ 2 6 ms Page 6 of 10 PHYS 211 вЂ“ MT3 Fall 2012 Sample 3 Solutions 15. A large care package, attached securely to a bungee cord, is released from rest from a stationary helicopter. The helicopter is 30 meters above the ground. The bungee cord has an unstretched length of 20 meters and a spring constant of 100 Newtons per meter. The mass of the package is 50 kg. Unfortunately the helicopter is too low and the package hits the ground. With approximately what velocity does it hit the ground? As always, use g = 10 m/s2 and ignore air resistance A. 0 m/s B. 2 10 m/s C. 20 m/s D. 20 2 m/s E. 20 6 m/s F. None of the above Use conservation of energy, with the initial energy at the helicopter and the final at the ground. WeвЂ™ll call the ground zero height for gravitational potential. The bungee cord only pulls, so it has no energy in the helicopter (as opposed to a compression spring, which would) E0 пЂЅ U 0 пЂЅ mgy0 пЂЅ E f пЂЅ U g , f пЂ« U B , f пЂ« K f пЂЅ 0 пЂ« 12 k пЂЁ пЃ„s пЂ© пЂ« 12 mv 2 2 v пЂЅ 2 gy0 пЂ пЂЁ пЂ© k 100 N 2 2 пЂЁ пЃ„s пЂ© пЂЅ 2 10 sm2 пЂЁ 30 m пЂ© пЂ m пЂЁ 30m - 20m пЂ© пЂЅ 600 пЂ 200 m 50 kg m s пЂЅ 20 ms Start with Kinetic energy, add this, get new kinetic energy: KE f пЂЅ 12 mv 2f пЂЅ KE0 пЂ« W пѓћ v f пЂЅ v02 пЂ« m2 W пЂЅ пЂЁ 2 ms пЂ© 2 пЂ« 2 2kg 20J пЂЅ 24 m s пЂЅ 2 6 ms Page 7 of 10 PHYS 211 вЂ“ MT3 Fall 2012 Sample 3 Solutions 16. A massless spring connects a ball of mass M to a vertical post on a horizontal frictionless table. The spring has a spring constant k and an unstretched length of Lo. If the post is rotated at a constant rate, the ball travels at constant speed in a horizontal circle on the table (so gravity plays no part in this problem). The spring stretches when the ball is moving in such a circle. At what speed, v, should the ball be moving so that the spring is stretched to twice its unstretched length (i.e. so that it has a length of 2Lo)? FBD Acc aC A. L0 4k m This problem is a dynamics problem. Draw the FBD & acc diagrams (above). Then solve: B. L0 2k m C. L0 k m D. L0 k 2m Fspring пЂЅ k пЃ„s пЂЅ k пЂЁ 2 L0 пЂ L0 пЂ© пЂЅ mac пЂЅ m v пЂЅ 2 L0 v2 2 L0 k k L0 пЂЅ L0 2 m m E. L0 k 4m Page 8 of 10 PHYS 211 вЂ“ MT3 Fall 2012 Sample 3 Solutions 17. Two rods are connected to make an "L" shape. Each rod is uniform, and they have the same length l, but one is twice as massive as the other. If the rods are placed as shown below, with the more massive rod on the y-axis, what is the coordinate of the center of mass of the "L"? To find the center of mass of the system, we can work from the CM of the two rods. We have 2m @ (0,l/2) and m @ (l/2,0) 1 пѓ© 1 l rCM пЂЅ 2m 0, l пЂ« m l ,0 пѓ№ пЂЅ ,l пЂЅ l , l 2 2 2 6 3 пѓ« пѓ» m пЂ« 2m 3 пЂЁ A. B. C. D. E. пЂ© пЂЁ пЂ© пЂЁ пЂ© пЂЁ пЂ© (l/2, l/2) (l/6, l/3) (l/6, l/2) (l/2, l/3) (l, l) Page 9 of 10 PHYS 211 вЂ“ MT3 Fall 2012 Sample 3 Solutions 18. A 1 kg box is given a quick push (from rest) up a ramp angled at 30В° to the horizontal, using a spring launcher of spring constant k = 4000 N/m, initially compressed by 0.1 m. The box slides up the ramp with a coefficient of kinetic friction of 1 3 until it comes to a halt. What total distance D along the slide does the box travel before it stops? FBD Acc f A. 2 m B. 4 We use conservation of energy, remembering that the work done will add (negatively) to the initial energy. We start with spring potential energy (weвЂ™ll define the height to be zero at the spring) and end with gravitational potential energy (since it comes to rest). We have to draw the FBD and acc diagrams to get the friction force right: N пЂ mg cosпЃ± пЂЅ 0; f пЂЅ пЃk N пЂЅ пЃk mg cosпЃ± 3m C. 4 m D. 4 3 m E. 4 3 F. 4 пЃk пЂЅ 1 3 a 5 m пЂЁ1 пЂ« 3 пЂ© G. 4 3 пЂЁ H. 4 3 пЂЁ m Now we do energy/work: E0 пЂ« W пЂЅ 12 k пЂЁ пЃ„s пЂ© пЂ fD пЂЅ E f пЂЅ mgy f пЂЅ mgD sin пЃ± 2 пЂ© 3 пЂ«1 пЂ© 3 пЂ1 m m 1 2 k пЂЁ пЃ„s пЂ© пЂ пЃk mg cos пЃ± D пЂЅ mgD sin пЃ± 2 Dmg пЂЁ пЃk cos пЃ± пЂ« sin пЃ± пЂ© пЂЅ k пЂЁ пЃ„s пЂ© 1 2 2 k пЂЁ пЃ„s пЂ© пѓћDпЂЅ mg пЂЁ пЃk cos пЃ± пЂ« sin пЃ± пЂ© 1 2 2 Now we just plug in numbers: DпЂЅ 1 2 пЂЁ 4000 mN пЂ©пЂЁ 101 m пЂ© 2 пЂЁ1kg пЂ© пЂЁ10 sm пЂ© пЂЁ пЂЁ 13 пЂ©пЂЁ 23 пЂ© пЂ« пЂЁ 12 пЂ©пЂ© пЂЅ 2m 2 Page 10 of 10