Physics 8, Fall 2017, Homework #6. Due at start of class on Friday, October 20, 2017 Problems marked with (*) must include your own drawing or graph representing the problem and at least one complete sentence describing your reasoning. Remember online response at positron.hep.upenn.edu/wja/jitt/?date=2017-10-20 (Chapter 10 problems.) 1. Three forces are exerted on a 2.00 kg block initially at rest on a slippery surface: a 100 N force directed along the x-axis, a 50.0 N force making an angle of 30.0 (counterclockwise) from the x-axis, and a 144 N force making an angle of 190 (counterclockwise) from the x-axis. (These forces are all in the horizontal plane, so gravity is irrelevant.) (a) Draw a diagram of the three forces, indicating their directions. (b) What is the vector sum of the forces acting on the block? (c) What is the work done on the block (by the vector sum of these forces) in 10.0 s? (Part c is tricky, because time is given, not displacement.) 2*. A janitor is pushing an 12 kg trashcan across a level floor at constant speed. The coefficient of friction between can and floor is 0.12. (a) If he is pushing horizontally, what is the magnitude of the force he is exerting against the can? (b) If he pushes not horizontally but rather at an angle of 35 down from the horizontal, what must be the magnitude of his pushing force to keep the can moving at constant speed? (Chapter 11 problems.) 3*. You have a weekend job selecting speed-limit signs to put at road curves. The speed limit is determined by the radius of the curve and the bank angle of the road w.r.t. horizontal. Your first assignment today is a turn of radius 250 m at a bank angle of 4.8. (a) What speed limit sign should you choose for that curve such that a car traveling at the speed limit negotiates the turn successfully even when the road is wet and slick? (So at this speed, it stays on the road even when friction is negligible.) (b) Draw a free-body diagram showing all of the forces acting on the car when it is moving at this maximum speed. (Your diagram should also indicate the direction of the car s acceleration vector.) phys008/hw06.tex page 1 of 4 2017-10-12 15:43
4. An automobile accelerates from rest starting at t = 0 such that its tires undergo a constant rotational acceleration α = 5.7 s 2. The radius of each tire is 0.27 m. At t = 12 s after the acceleration begins, find (a) the instantaneous rotational speed ω of the tires, (b) the total rotational displacement ϑ of each tire, (c) the linear speed v of the automobile (assuming the tires stay perfectly round) and (d) the total distance the car travels in the 12 s. 5. A long, thin rod is pivoted from a hinge such that it hangs vertically from one end. (The hinge is at the top.) The length of the rod is 1.23 m. You want to hit the lower end of the rod just hard enough to get the rod to swing all the way up and over the pivot (i.e. to swing more than 180 ). How fast do you have to make the end go? (Hint: consider gravitational potential energy and rotational kinetic energy. For rotational inertia, remember that the rod rotates about its end here, not about its center, so you ll need to use the parallel-axis theorem to get I about the end.) 6*. You accidentally knock a full bucket of water off the side of the well shown in the figure at right. The bucket plunges 17 m to the bottom of the well. Attached to the bucket is a light rope that is wrapped around the crank cylinder. How fast is the handle turning (rotational speed) when the bucket hits bottom? The inertia of the bucket plus water is 12 kg. The crank cylinder is a solid cylinder of radius 0.63 m and inertia 5.2 kg. (Assume the small handle s inertia is negligible in comparison with the crank cylinder.) 7*. You have a pail of water with a rope tied to the handle. If you whirl it fast enough in a vertical circle (i.e. a circle whose central axis is horizontal), none of the water spills out of the bucket, even when the bucket is upside down. (a) Explain how this works. (b) If the bucket rotates at constant speed v on the end of a rope of length L, what minimum speed is required to keep the water from falling out of the pail? (c) If you plug in L = 0.9 m, what number of revolutions per second does this speed correspond to? (You may remember that I spun the bucket about two or three times faster than this in the in-class demo.) 8*. A ball is attached to a vertical rod by two strings of equal strength and equal length. (See figure, below left.) The strings are very light and do not stretch. The rod begins to rotate under the influence of a constant rotational phys008/hw06.tex page 2 of 4 2017-10-12 15:43
acceleration. (a) Which string breaks first? (b) Draw a free-body diagram for the ball, indicating all forces and their relative magnitudes, to justify your answer to (a). 9*. An open door of inertia M and width L is at rest when it is struck by a thrown putty ball of inertia m that is moving at linear speed v at the instant it strikes the door. (See figure, above right.) The impact point is a distance D = 2 L from the rotation axis through the hinges. The putty ball strikes at 3 a right angle to the door face and sticks after it hits. What is the rotational speed of the door and putty? Do not ignore the inertia m. (Hint: angular momentum.) 10*. Two skaters skate toward each other, each moving at 3.1 m/s. Their lines of motion are separated by a perpendicular distance of 1.8 m. Just as they pass each other (still 1.8 m apart), they link hands and spin about their common center of mass. What is the rotational speed of the couple about the center of mass? Treat each skater as a point particle, each with an inertia of 52 kg. Remember online response at positron.hep.upenn.edu/wja/jitt/?date=2017-10-20 XC1*. Optional / extra-credit. The spacecraft in the movie 2001: A Space Odyssey has a rotating cylinder to create the illusion of gravity, inside of which the crew walks and exercises. (a) If the radius of the cylinder is 7.0 meters, what should the rate of revolution of the cylinder be in order to replicate Earth s gravity at this radius? (b) For a person, of height 1.7 meters, standing in this cylinder, how does the gravitational acceleration at the top of her head compare with the gravitational acceleration at her feet? (Might this be uncomfortable?) phys008/hw06.tex page 3 of 4 2017-10-12 15:43
XC2*. Optional/extra-credit. Your aunt owns an amusement park, and she wants you to add a circular loop to an exiting roller coaster ride to make it more fun. The first hill for the existing roller coaster is 43 m tall, and your aunt wants you to build right after this hill the tallest loop possible without having the cars fall out of the track or the passengers fall out of the cars. You think for a minute and realize what the minimum speed at the top of the loop has to be, and this gives you what you need to design the loop. XC3*. Optional/extra-credit. (a) Calculate the rotational inertia of a flat sheet of plywood, of width w and height h, about an axis that passes through the center of the sheet and is perpendicular to the plane of the sheet. Since it s easy to look this answer up, you need to compute it using calculus and to show your work. (b) Calculate the rotational inertia of this same sheet about an axis that passes through the center of the sheet and is parallel to the long edge of the sheet. (c) Calculate the rotational inertia of this same sheet about an axis that passes through the center of the sheet and is parallel to the short edge of the sheet. (d) Verify that your answer for part (a) is the sum of your answers for parts (b) and (c). XC4*. Optional/extra-credit. (a) Repeat problem 10, but let the two skaters have different masses: the first skater has inertia m 1 = 45.0 kg and the second skater has inertia m 2 = 90.0 kg. (b) What is the center-of-mass velocity of the two-skater system after they link hands? (Assume, somewhat unrealistically, zero friction between the skaters and the ice. This should remind you of the two-male-moose problem from an earlier homework.) phys008/hw06.tex page 4 of 4 2017-10-12 15:43
Physics 008 2017 homework #6 score sheet Problem 1: /4 Problem 8: /4 Problem 2: /4 Problem 9: /4 Problem 3: /4 Problem 10: /4 Problem 4: /4 Problem 11: /4 Problem 5: /4 Problem 12: /4 Problem 6: /4 Problem 13: /4 Problem 7: /4 The following 4 points reflect your maintaining good habits for all of the above problems: Reasonable number of significant digits reported in answers: /2 Sufficient level of explanation for reasoning behind answers: /2 (You can earn 3 out of 2 for consistently clear presentation/explanation of your answers.) Total (out of 44): + Extra credit (if any): scoring guideline (for each problem): 4 = complete and correct solution 3 = good but minor error or minor omission 2 = serious error or omission but shows decent effort 1 = we can't make sense of what is written down 0 = no serious attempt to solve problem Please write your name only on the bottom edge of this sheet (not on your own solutions), and staple the sheet to your own solutions, so that we don t know whose paper we are grading until after we have finished grading it. Name: