POTENTIAL ENERGY AND ENERGY CONSERVATION 1. Sky Jump: You have landed a summer job with a company that has been given the contract to design the ski jump for the next Winter Olympics. The track is coated with snow and has an angle of 25 o from the horizontal. A skier zips down the ski jump ramp so that he leaves it at high speed. The winner is the person who jumps the farthest after leaving the end of the ramp. Your task is to determine the height of the starting gate above the end of the ramp, which will determine the mechanical structure of the ski jump facility. You have been told that the typical ski-jumper pushes off from the starting gate at a speed of 2.0 m/s. For safety reasons, your design should be such that for a perfect run down the ramp, the skier's speed before leaving the end of the ramp and sailing through the air should be no more than 80 km/hr. You run some experiments on various skies used by the jumpers and determine that the coefficient of static friction between the snow and the skis is 0.10 and its coefficient of kinetic friction is 0.02. Since the ski-jumpers bend over and wear very aerodynamic suits, you decide to neglect the air resistance to make your design. [LO 21, LO 23, LO 26, LO 28, LO 34, LO 36, LO 38, LO 39] 2. Aircraft Launcher: The Navy wants a new airplane launcher for their aircraft carriers that is basically a large spring and your job is to determine the necessary spring constant. The launcher pushes the plane for a short distance along a much longer runway. During that same time, the plane's jet engines supply a constant thrust force for the entire length of the runway. The planes need to have a minimum velocity by the time they reach the end of the runway in order to take off successfully. [LO 34, LO 36, LO 37, LO 38, LO 39] 3. Human Cannonball: In a weak moment you have volunteered to be a human cannonball at an amateur charity circus. The "cannon" is actually a 3 foot diameter tube with a big stiff spring inside which is attached to the bottom of the tube. A small seat is attached to the free end of the spring. The ringmaster, one of your soon to be exfriends, gives you your instructions. He tells you that just before you enter the mouth of the cannon, a motor will compress the spring to 1/10 its normal length and hold it in that position. You are to gracefully crawl in the tube and sit calmly in the seat without holding on to anything. The cannon will then be raised to an angle such that your speed through the air at your highest point is 10 ft/sec. When the spring is released, neither the spring nor the chair will touch the sides of the 12 foot long tube. After the drum roll, the spring is released and you will fly through the air with the appropriate sound effects and smoke. With the perfect aim of your gun crew, you will fly through the air over a 15 foot wall and land safely in the net. You are just a bit worried and decide to calculate how high above your starting position you will be at your highest point. Before the rehearsal, the cannon is taken apart for maintenance. You see the spring, which is now removed from the cannon, is hanging straight down with one end attached to the ceiling. You determine that it is 10 feet long. When you hang on its free end without touching the ground, it stretches by 2.0 ft. Is it possible for you to make it over the wall? [LO 21, LO 23, LO 25, LO 34, LO 37, LO 38, LO 39] 1
4. Construction Site: At a construction site, a 65.0 kg bucket of concrete hangs from a light (but strong) cable that passes over a light, friction-free pulley and is connected to an 80.0 kg box on a horizontal roof (see figure). The cable pulls horizontally on the box, and a 50.0 kg bag of gravel rests on top of the box. The coefficients of friction between the box and roof are shown. (a) Find the friction force on the bag of gravel and on the box. (b) Suddenly a worker picks up the bag of gravel. Use energy conservation to find the speed of the bucket after it has descended 2.00 m from rest. (Use Newton s laws to check your answer.) [LO 27, LO 28 or 29, LO 32, LO 34, LO 36, LO 37, LO 38, LO 39] 5. Bungee Jump: A bungee cord is 30.0 m long and, when stretched a distance x, it exerts a restoring force of magnitude kx. Your father-in-law (mass 95.0 kg) stands on a platform 45.0 m above the ground, and one end of the cord is tied securely to his ankle and the other end to the platform. You have promised him that when he steps off the platform he will fall a maximum distance of only 41.0 m before the cord stops him. You had several bungee cords to select from, and you tested them by stretching them out, tying one end to a tree, and pulling on the other end with a force of 380.0 N. When you do this, what distance will the bungee cord that you should select have stretched? [LO 25, LO 32, LO 34, LO 36, LO 37, LO 38, LO 39] 6. Delivery Ramp: You are designing a delivery ramp for crates containing exercise equipment. The 1470 N crates will move at 1.8 m/s at the top of a ramp that slopes downward at 22.0 o. The ramp exerts a 515 N kinetic friction force on each crate, and the maximum static friction force also has this value. Each crate will compress a spring at the bottom of the ramp and will come to rest after traveling a total distance of 5.0 m along the ramp. Once stopped, a crate must not rebound back up the ramp. Calculate the largest force constant of the spring that will be needed to meet the design criteria. [LO 28, LO 32, LO 34, LO 36, LO 37, LO 38, LO 39] 7. The Great Sandini: The Great Sandini is a 60 kg circus performer who is shot from a cannon (actually a spring gun). You don t find many men of his caliber, so you help him design a new gun. This new gun has a very large spring with a very small mass and a force constant of 1100 N/m that he will compress with a force of 4400 N. The inside of the gun barrel is coated with Teflon, so the average friction force will be only 40 N during the 4.0 m he moves in the barrel. At what speed will he emerge from the end of the barrel, 2.5 m above his initial rests position? [LO 25, LO 28, LO 32, LO 34, LO 36, LO 37, LO 38, LO 39] 2
8. Pendulum: A pendulum is suspended from the ceiling and attached to a spring fixed to the floor directly below the pendulum support (see figure). The mass of the pendulum bob is m, the length of the pendulum is L, and the force constant is k. The unstressed length of the spring is L/2 and the distance between the floor and ceiling is 1.5 L. The pendulum is pulled aside so that it makes an angle with the vertical and is then released from rest. Obtain an expression for the speed of the pendulum bob as the bob passes through a point directly below the pendulum support. [LO 34, LO 37, LO 38, LO 39] 9. Mass on a String: A small object of mass m moves in a horizontal circle of radius r on a rough table. It is attached to a horizontal string fixed at the center of the circle. The speed of the object is initially v o. After completing one full trip around the circle, the speed of the object is 0.5 v o. (a) Find the energy dissipated by friction during that one revolution in terms of m, v o, and r. (b) What is the coefficient of kinetic friction? (c) How many more revolutions will the object make before coming to rest? [LO 32, LO 34, LO 36, LO 37, LO 38, LO 39] 10. Jet Car: In 1964, after the 1250 kg jet-powered car Spirit of America lost its parachute and went out of control during a run at Bonneville Salt Flats, Utah, it left skid marks about 8.00 km long. (This earned a place in the Guiness Book of World Records for longest skid marks.) (a) If the car was moving initially at a speed of about 800 km/h, and was still going at about 300 km/h when it crashed into a brine pond, estimate the coefficient of kinetic friction k. (b) What was the kinetic energy of the car 60 s alter the skid began? [LO 23, LO 32, LO 34, LO 36, LO 37, LO 38, LO 39] 3
11. Roller-Coaster: Your firm is designing a new roller-coaster ride. The permit process requires the calculation of forces and acceleration at various important locations on the ride. Each roller-coaster car will have a total mass (including passenger) of 500 kg and will travel freely along the winding frictionless track shown in the figure. Points A, E, and G are on horizontal straight sections, all at the same height of 10 m above the ground. Point C is at a height of 10 m above the ground on an inclined section of slope angle 30. Point B is at the crest of a hill, while point D is at ground level at the bottom of a valley; the radius of curvature at both of these points is 20 m. Point F is at the middle of a banked horizontal curve with a radius of curvature of 30 m, and at the same height as points A, E, and G. At point A, the speed of the car is 12 m/s. (a) If the car is to just barely make it over the hill at point B, what must be the height of point B above the ground? (b) If the car is to just barely make it over the hill at point B, what should be the magnitude of the force exerted by the track on the car at that point? (c) What will be the acceleration of the car at point C? (d) What will be the magnitude and direction of the force exerted by the track on the car at point D? (e) What will be the magnitude and direction of the force exerted by the track on the car at point F? (f) At point G, a constant braking force is to be applied to the car, bringing it to a halt in a distance at 25 m. What is the magnitude of this required braking force? [LO 19, LO 21, LO 23, LO 26, LO 32, LO 34, LO 36, LO 37, LO 38, LO 39] 12. Energy Concept Map: A concept map presents the relationships among a set of connected concepts and ideas. Concept mapping is a method for organizing and representing knowledge. It can help you visualize relationships between various concepts, test your understanding of complex subjects, and increase your understanding of the connection between new and old knowledge. In a concept map, the concepts, usually represented by one or two words enclosed in a rectangular box, are connected to other concepts by arrows. A word, or brief phrase, written by the arrow, defines the relationship between the connected concepts. Major concept boxes will have lines to and from several other concept boxes generating a network. An example of a concept map of concept maps (!) is shown below, followed by a concept map for an atom. 4
Write down the key concepts related to energy that were covered in Chapters 6 and 7 in the text. Order these concepts by placing the most general concepts toward the top of your list and more specific concepts toward the bottom. Starting with your broadest concepts, start connecting them to other concepts by lines and label the lines with action, or linking, words. These links help to illustrate how the concepts are related to one another. 5