Practice Problems from Chapters 11-13, for Midterm 2. Physics 11a Fall 2010 Chapter 11 1. The Ferris wheel shown below is turning at constant speed. Draw and label free-body diagrams showing the forces on a passenger at each of the four numbered positions. Make sure you get the relative lengths of the vector arrows about right. 2. An ice cube and a rubber ball are both placed at one end of a warm cookie sheet, and the sheet is then tipped up. The ice cube slides down with virtually no friction, and the ball rolls down without slipping. If the ball and the ice cube have the same inertia, which one makes it to the bottom first? 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.. Your first assignment today is a turn of radius 400 m at a bank angle of 7. What speed limit sign should you choose for that curve such that a car traveling at that speed negotiates the turn successfully even when the road is wet and slick? 4. A pendulum is made of a bob dangling from a lightweight string of length L. The bob is pulled sideways so that the string makes an angle θ i with respect to the vertical, then released. As it swings down, what is the rotational speed of the bob as a function of the changing angle θ? 5. A roller-coaster car begins its run on a long, steeply sloping track and then goes into a circular loop-the-loop of radius R whose bottom is a distance D above the ground (see Figure). (a) What is the car s speed at the bottom of the loop? (b) What is the magnitude of the normal force acting on the car at that instant? (b) What is its speed one-quarter of the way around the loop? (c) What is the magnitude of the normal force acting on it at the one-quarter point? (c) What is the car s total acceleration at the one-quarter point? 1
6. Two skaters skate toward each other, each moving at 3.3 m/s. Their lines of motion are separated by a perpendicular distance of 2.0 m. Just as they pass each other (still 2.0 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, one with an inertia of 75 kg and the other with an inertia of 48 kg. 7. A 20-kg child running at 1.4 m/s jumps onto a playground merry-go-round that has inertia 180 kg and radius 1.6 m. She is moving tangent to the platform when she jumps, and she lands right on the edge. What is the rotational speed of the merry-go-round and child if the merry-go-round started from rest? Ignore any friction in the axle about which the platform rotates 8. A pair of blocks is connected by a string that is draped over a pulley. One block has inertia 0.20 kg, the other has inertia 0.25 kg, and the pulley is a solid disk of inertia 0.50 kg and radius 0.10 m. When released, the 0.25-kg block is 0.30 m off the ground. What speed does this block have when it hits the ground? 9. A thick-walled cylinder of inertia M has an outer radius R, an inside radius r = R/2, and a length! (see Figure). It rotates around an axis that is attached to its outside along its length. What is its rotational inertia about this axis? 2
Chapter 12 10. A father and son are carrying a sofa down the stairs, with the son leading the father. The center of mass of the sofa is more or less at its geometric center. (a) Draw the extended free-body diagram for the sofa. (b) Who has the greater share of the load? (c) Would your answer to part b be the same if instead of the sofa the men were carrying a thin, flat sheet of plywood? 11. The top end of a ladder of inertia m rests against a smooth wall, and the bottom end rests on the ground. The coefficient of static friction between the ground and the ladder is µ s. What is the minimum angle between the ground and the ladder such that the ladder does not slip? 12. Standing on the edge of a rotating playground carousel, you take sight on the center of the carousel and walk directly toward that point. (a) What happens to the rotational speed of the carousel? (b) In the system consisting of only the carousel (a nonisolated system), any change in angular momentum is due to a torque. Because the hub of the carousel is well-lubricated, the only candidate for what causes this torque is the force of friction exerted by your feet on the carousel. But, you say, I walked radially inward. How could I possibly cause a torque with my feet? What s going on? 13. You want to hang a 10-kg sign that advertises your new business (Figure P12.39). To do this, you attach a 5.0-kg beam of length L to a wall at its base by a pivot P. You then attach a thin cable to the beam and to the wall in such a way that the cable and beam are perpendicular to each other. The beam makes an angle of 37 with the vertical. You hang the sign from the end of the beam to which the cable is attached. (a) What must be the minimum tensile strength of the cable (the minimum amount of tension it can sustain) if it is not to snap? (b) Determine the horizontal and vertical components of the force the pivot exerts on the beam. 3
14. A 3.0-kg ball rolls without slipping down a ramp inclined to the horizontal by an angle of 30. (a) What is the linear acceleration of the ball's center of mass? (b) What is magnitude of the frictional force exerted on the ball? 15. A block of inertia m is placed on a smooth surface. A light string is attached to it and placed over a solid pulley of inertia 3m and radius R, as in Figure P12.71, and a ball of inertia m is attached to the free end of the string. (a) If there is no slippage between the string and pulley, determine the magnitude of the linear acceleration of the block. (b) If you detach the ball and pull down on the hanging string with a force of magnitude mg, what is the acceleration of the block now? Chapter 13 16. A commercial airliner hits a pocket of turbulence and experiences a downward acceleration of about 2g. A person not fastened into a seat by a safety belt falls to the ceiling and suffers a broken neck. Explain how this injury can be so severe. 17. A neutron star has about two times the gravitational mass of our Sun but has collapsed to a diameter of 10 km. What is the acceleration due to gravity on the surface of this star in terms of the free-fall acceleration at Earth's surface? 18. (a) On a trip from Earth to the Moon, at what point does the force of gravity exerted by Earth on a coasting spacecraft cancel the force of gravity exerted by the Moon on the spacecraft? (b) What do the passengers in the spacecraft notice, if anything, before and after this point? 19. (a) How much gravitational potential energy does a 100-kg object-earth system have if the object is one Earth radius above the ground? (b) How fast would a 100-kg object have to be moving at this height to have a total energy of zero? 20. Find expressions for the orbital speed and total energy for a satellite of inertia m s traveling in a circular orbit of radius a around a planet of inertia m p >> m s. 4
21. What is the speed of a space probe when it is very far from Earth if it was launched from the surface of Earth at twice its escape speed? 5