Prof. Anchordoqui Problems set # 6 Physics 168 October 14, 2014 1. During your winter break you enter a “dogsled” race across a frozen lake. This is a race where each sled is pulled by a person not dogs. To get started you pull the sled (total mass 80 kg) with a force of a 180 N at 40 degrees above the horizontal. Find (a) the work you do, and (b) the final speed of the sled after it moves ∆x = 5.0 m, assuming that it starts from rest and there is no friction. 2. A force F~ = Fxˆi varies with x as shown in Fig. 1. Find the work done by the force on a particle as the particle moves from x = 0 m to x = 6 m. 3. You and your friend are at a ski resort with two ski runs, a beginner’s run and an expert’s run. Both runs begin at the top of the ski lift and end at finish line at the bottom of the same lift. Let h be the vertical descent for both runs. The beginner’s run is longer and less steep than the expert’s run. You and your friend, who is a much better skier than you, are testing some experimental frictionless skis. To make things interesting, you offer a wager that if she takes the expert’s run and you take the beginner’s run, her speed at the finish line will not be greater than your speed at the finish line. Forgetting that you study physics, she accepts the bet. The conditions are that you both start from rest at the top of the lift and both of you coast for the entire trip. Who wins the bet? (Assume air drag is negligible). 4. (a) Determine the work a hiker must do on a backpack (with a mass of 15 kg) to carry it up a hill of height h = 10 m. Determine also (b) the work done by gravity on the backpack, and (c) the net work done on the backpack. For simplicity assume the motion is smooth and at constant velocity (i.e, acceleration is negligible). 5. Jane looking for Tarzan, is running at top speed 5.3 m/s and grabs a vine hanging vertically from a tall tree in the jungle. How high can she swing upward? Does the length of the vine affect your answer? 6. A 62-kg bungee jumper jumps from a bridge. He is tied to a bungee cord whose unstretched length is L1 = 12 m, and falls a total of L1 + L2 = 31 m. (a) Calculate the spring stiffness constant k of the bungee cord, assuming Hooke’s law applies. (b) Calculate the maximum acceleration he experiences. (c) Calculate the velocity just before the cord is starting to stretch. 7. Imagine you have time-traveled back to late 1800 s and you are watching your great-greatgreat-grandparents on their honeymoon taking a ride on the Flip Flap Railway, a Coney Island roller coaster with a circular loop-the-loop. The car they are in is about to enter the loop-the-loop when a 100 lb sack of sands falls from a construction-site platform and lands in the back seat of the car. No one is hurt, but the impact causes the car to lose 25% of its speed. The car started from rest at a point 2 times as high as the top of the circular loop. Neglect losses due to friction on air drag. Will their car make it over the top of the loop-the-loop without falling off? 8. A pendulum consists of a bob of mass m attached to a string of length L. The bob is pulled aside so that the string makes an angle θ0 with the vertical, and is released from rest. As it passes through the lowest point of the arc, find expressions for (a) the speed of the bob and (b) the tension in the string. Effects due to air resistance are negligible. 9. A spring with stiffness constant k is cut in half. What is the spring stiffness constant for each of the two resulting springs? 10. A water molecule consists of an oxygen atom and two hydrogen atoms. An oxygen atom has a mass of 16 u (unified mass units) and each hydrogen atom has a mass of 1 u. The hydrogen atoms are each at an average distance of 96 pm from the oxygen atom and are separated from one another by an angle of 104.5 degrees. Find the center of mass of a water molecule.(See Fig. 2.) 11. Determine the center of mass of a uniform semicircular hoop of radius R. 12. Pete (mass 80 kg) and Dave (mass 120 kg) are in a rowboat (mass 60 kg) on a calm lake. Dave is near the bow of the boat, rowing, and Pete is at the stern, 2 m from Dave. Dave gets tired and stops rowing. Pete offers to row, so after the boat comes to rest they change places. How far does the boat moves as Pete and Dave change places. (Neglect any horizontal force exerted by the water). 13. Two pucks, each of mass m, are initially at rest in the configuration shown in Fig. 3. A constant force of magnitude F accelerates the system towards the right. After the point of application P of the force has moved a distance d, the pucks collide and stick together. What is the speed of the pucks immediately after the collision? 14. A system consists of a 110-kg basketball player, the rim of a basketball hoop, and Earth. Assume that the potential energy of the system is zero when the player is standing on the floor and the rim is horizontal. Find the total potential energy of this system when the player is hanging on the front of the rim. Also assume that the center-of-mass of the player is 0.8 m above the floor when he is standing and 1.30 m above the floor when he is hanging. The force constant of the rim is 7.2 kN/m and the front of the rim is displaced downward a distance of 15 cm. 15. (a) If a volcano spews a 500-kg rock vertically upward a distance of 500 m, what is its velocity when it left the volcano. (b) If the volcano spews the equivalent of 1000 rocks of this size every minute, what is the power output? Enrichment 16. (a) If the human body could convert a candy bar directly into work, how high could an 82-kg man climb a ladder if he were fueled by one bar (= 1100 kJ)? (b) If the man jumped off the ladder, what will be its speed when he reaches the bottom? 17. A block of mass m is attached to the end of a spring (spring stifness constant k). The block is given an initial displacement x0 , after which it oscillates back and forth. Write a formula for the total mechanical energy (ignore friction and the mass of the spring) in terms of x0 , position x, and speed v. (See Fig. 4.) 18. Find the center of mass of the uniform sheet of plywood shown in Fig. 5. 19. A ball is attached to a horizontal cord of length L whose other end is fixed. (a) If the ball is released, what will be its speed at the lowest point of its path? (b) A peg is located a distance h directly below the point of attachment of the cord. If h = 0.8L, what will be the speed of the ball when it reaches the top of its circular path above the peg? (See Fig. 6) 20. Three people of roughly equal masses m on a lightweight (air-filled) banana boat sit along the x axis at positions xA = 1 m, xB = 5 m, and xC = 6 m measured from the left-hand end as shown in Fig. 7. Find the position of the center-of-mass. Ignore the mass of the boat. 21. During a workout, the football players at Harvard University ran up the stadium stairs in 66 s. The stairs are 140 m long and inclined at an angle of 32◦ . If a typical player has a mass of 95 kg, estimate the average power output on the way up. Ignore friction. H 22. For F~ = Ax ˆi calculate C F~ . d~` for the path C shown in Fig. 8.