Physics 2210 (Fall 2010) Midterm IV - Practice Problems Mike McLaughlin Quick Note: At the time of publication of this document, I have not yet seen the Midterm Exam. The practice problems in this document are what I would consider useful problems for you to study in preparation for the exam. Your studying approach should not be limited to working through this set of problems, nor should you interpret these problems as an official study guide for your exam. Email me (michaelm@physics.utah.edu) if you have any questions about the problems. I consider the Parallel Axis Theorem to be an important topic for you to master. 1 While Al Gore Sleeps The ice caps of the Earth completely melt and increase the ocean s depth by 30 meters. Estimate the new length of a day on Earth. 2 Ask Me About My Tie Joke Two spheres are rolled down an incline of height h. One is hollow and the other is solid. Each sphere has a mass of M. Determine the relationship between their radii (radiuses for you non-latin speaking losers) if the two are to finish in a tie. 3 Sew What A ball of yarn with radius of 24 centimeters is rolled across the floor with a center of mass velocity equal to 3.0 m/s. It comes to rest some time later and you notice that it left a trail of 4.2 meters worth of yarn in its wake. Determine α. (assume that the loss of yarn has a negligible effect on the ball of yarn s moment of inertia). 4 A-Rod A uniform rod of length L and mass M is rotated about an axis perpendicular to the length of the rod centered at a point 1/4 of the way down the rod s length. Determine its moment of inertia about this axis. 5 Double Pulley Two pulleys are set up on opposite sides of a table. Each has moment of inertia of I. Two masses (m 1 and m 2 ) are connected to each other by a string which crosses over both pulleys (these masses are hanging off opposites sides of the table). Find the acceleration of the masses as well as the tension in each section of the cord (the two hanging sections and the section in between the pulleys).
6 Single Pulley Consider two masses connected by a string passing over a pulley which has a moment of inertia of I and a radius of R. The system is initially at rest. Find the linear velocities of the masses after m 2 descends through a distance of h. Also, determine the angular velocity of the pulley at this time. 7 MinimIze Two masses, M and m, are attached by a rigid rod of length L and negligible mass. For an axis perpendicular to the rod, show that the moment of inertia is a minimum for the axis positioned at the center of mass of the system. What is this moment of inertia? 8 Barney and Betty Ok, you have this wheel that is standing up vertically. A force of 10 Newtons is directed at the top of the wheel to the East. Another force is directed at the side of the wheel to the South (this one has a strength of 9 Newtons). These two forces occur on the edge of the wheel (b= 25 cm). A third force is directed at the edge of an inner radius (a= 10 cm). This forces is 12 Newtons and is directed 30 South of West. Find the net torque exerted on the wheel. 9 Analyzing the Earth Compute the ratio of the Earth s rotational kinetic energy about the Sun to its rotational kinetic energy about its own axis. 10 Can You Detect Crabs With a Radio Antenna? A pulsar is a rapidly rotating neutron star that emits radio pulses with each rotation of the star. A pulsar is known to exist in the Crab Nebula. Its period is T=.033 seconds and this has been observed to increase at the rate of 1.26 X 10 5 seconds per year. Find the current value of ω. What is the value of α if you assume constant acceleration? In how many years will the pulsar stop rotating? The pulsar originated from a supernova explosion in the year 1054. What was the period of rotation back then? 11 Determining the Speed of Light An early method for measuring the speed of light had a beam of light passing through a slot on the outside edge of a rotating (slotted) wheel. The light then continues on and reflects off of a distant mirror and returns to the wheel just in time to pass through the next slot in the wheel. Consider a wheel of radius 5 centimeters with 500 slots along its edge (or circumference). The distance from wheel to mirror was 500 meters and the speed of light was determined to be the speed of light as you know it. What was the angular velocity of the wheel? What was the linear speed on the edge of the wheel?
12 Roger Went Loopy on Us There is a rotation axis at point P. A thin rod of length L and mass M is fastened to the rotation axis. Attached to the end of the rod (the end away from the axis) is a particle of mass m. Attached to that particle is another rod of length L and mass M. And at the end of the 2nd rod is another particle of mass m. (the system forms a straight line). The system rotates about the rotation axis with angular speed ω. Determine the rotational moment of inertia of this system. What is the kinetic energy of rotation about the point P? 13 See-Saw A block of mass m is suspended from each end of a massless, rigid rod. The rod is 1 meter long. A fulcrum is placed at x=20 centimeters, but you hold the rod motionless and exactly horizontal. Then, the systems is released. Find the accelerations of the two blocks as they begin to move. 14 The Acceleration of Two Balls Two uniform solid spheres have the same mass (1.65 kilograms). One has radius.226 meters while the other has radius.854 meters. Find the torque required to bring each sphere from rest to a speed of 317 rad/sec in 15.5 seconds. Each sphere rotates about its center. For each sphere, what force applied tangentially at the equator would provide the needed torque? 15 Jaime Escalante Use the calculus to determine the moment of inertia for a uniform, solid cylinder. 16 Circular Track with Friction A particle of mass m slides down a frictionless surface of height h and collides with a uniform vertical rod (sticking to it). The rod is free to pivot about a point P. The rod has length L and mass M. The rod does pivot and displaces through an angle θ as measured from the vertical before momentarily coming to rest. Find θ in terms of the other parameters in the problem. 17 http://www.physics.utah.edu/ michaelm/starman.flv An astronaut weighs 140 Newtons on the surface of the Moon. What gravitational force does the Moon exert on the astronaut when the astronaut is located exactly one Moon radius above the surface of the Moon? NOTE: You are not allowed to look up any data for the Moon while working this problem. If you do, I ll burn your house down. 18 Like a G 0.5 How high above the Earth s surface must you be in order to feel an acceleration due to gravity equal to 1/2 the value at the Earth s surface. At this distance, what speed is required to maintain a circular orbit?
19 Fear Phobos is a satellite (or moon) in our Solar System. It has an orbital radius of 9.4 million meters. A complete trip around its host planet requires 7 hours and 39 minutes. Use physics to determine the mass of this planet. Then, look in Appendix E of your textbook to determine the planet s name and to compare the mass. 20 http://www.youtube.com/watch?v= SSrH o qnrl0 Read in your text and convince yourself you know what a geosynchronous orbit is. Now, suppose you put a weather satellite in a synchronous orbit above the Great Red Spot of Jupiter. What altitude is required of your satellite? You might like to know that a day on Jupiter only lasts 9.9 hours. You may wish to look up the mass of Jupiter. 21 Until it Goes Click A vertical spring is attached to the surface of the Moon. A mass of.25 kilograms is placed atop the spring and then the spring is pushed down by your hand a total distance of 10 centimeters (note: this includes the compression by the mass under the influence of gravity). The spring has a spring constant of 5000 N/m. To what height, above the spring s compressed length, will the mass rise? How high does it rise on Earth? 22 Extinction of the Dinosaurs A certain asteroid whose mass is.0002 times that of the Earth orbits the Sun with a radius twice as large as that of the Earth s orbit. Find the asteroid s orbital period in years. Determine the ratio of KE for the asteroid as compared to the Earth. 23 A Nicer Rock Than Your Husband Will Ever Buy After the Sun runs out of nuclear fuel, it will collapse to the white dwarf state. Its mass will be roughly the same as it is now (for our calculation), but its radius will equal that of the Earth (no joke). All of this occurs after it becomes a Red Giant (the stage we discussed in section where the Sun engulfs the Earth and a few other unlucky planets). Determine the density of the Sun when in the white dwarf stage. Then determine the surface gravity of the white dwarf. While you re at it, calculate the gravitational potential energy of a 1 kilogram object on the surface of this dead star (assume the potential is equal to zero at infinity). 24 Stellar Death Another death stage possible for stars is called a neutron star (the leftover remnant of a massive star that has exploded as a Supernova). This death stage is really only available to stars a bit more massive than our Sun (typically stars with masses about 2-3 times larger than the Sun). These stars are compacted so tightly by gravity that all of the protons and electrons combine to yield a spherical ball of only neutrons that has a radius of 10 kilometers (if you re in shape, you could run that distance in about 45 minutes). Find the density of a neutron star. Then, find its surface gravity. If you happen to drop your keys from a distance of 1 meter above the surface of a neutron star, how fast are they traveling
when they reach the ground? Go ahead and assume a mass that is three times as large as our Sun for this problem. 25 John Left Us on a Jet Plane Determine g for Denver, the mile high city. 26 She Packed My Bags Last Night Pre-Flight Consider a rocket at rest deep in space. What must be its mass ratio (ratio of initial mass to final mass) in order that, after firing its engine, the rocket s speed is twice that of the exhaust. 27 And I m Gonna Be High as a Kite by Then A rocket whose initial mass is M i =850 kg consumes fuel at a rate of 2.3 kg/s. The speed of the exhaust relative to the rocket engine is 2800 m/s. What thrust does the rocket engine provide? What is the initial acceleration of the rocket? Suppose now, instead, that the rocket is launched from a spacecraft already in deep space, where we can neglect any gravitational force acting on it. The mass M f of the rocket when its fuel is exhausted is 180 kg. What is its speed relative to the spacecraft at that time? 28 A Proof There exists a maximum rate of rotation for planets such that the gravitational force on material at the equator just barely provides the centripetal force required for the rotation. Why? Determine the shortest period possible for a planet assuming that the planet has a density of ρ. 29 Distorting Your Idea of a Pendulum A rod of length L is being used as pendulum for time-keeping purposes (its period is critical to you). Afterwards, it is melted down so that all of the mass is concentrated into a small sphere (which you can consider as a point mass). If you are to now reconstruct your timekeeping device, with this point mass and a piece of massless string, how much string should you use? 30 Changes in the Period - Part I A mass m is attached to a vertical spring (attached to the ceiling). When the mass is 810 grams, the period is 0.91 seconds. An unknown mass is observed to have a period of 1.16 seconds. Determine the spring constant and the unknown mass. 31 Changes in the Period - Part II By what fraction does the period of a simple pendulum change if taken from the surface of the Earth to the surface of the Moon?
32 I Make Incredible Lasagna You are preparing to enjoy a nice piece of lasagna for dinner cooked by your TA. You have a sensitive mouth and realize that the meal is too hot. In an effort to cool it down, you move the plate up and down (in the vertical plane) with a period of 1.2 seconds. What is the maximum amplitude for this motion for which the lasagna remains in contact with the plate. 33 It s Not Where You Think It Is At what position (how far from equilibrium) will a simple harmonic oscillator s velocity be equal to exactly 1/2 of its maximum velocity? You may assume the amplitude of this system is A. 34 Spring + Elevator A particle is attached to a spring and is capable of oscillating at a frequency of ω 0. This spring is attached to the ceiling of an elevator which descends with a constant speed of 1.5 m/s. The elevator suddenly comes to a stop. What is the amplitude of oscillations for the mass-spring system? 35 Vector Medley Given the following vectors: (3, 4, -2) and (-1, 2, -3), compute the dot product, the cross product, the angle between these vectors, and the length of the vector which results upon addition of these two vectors. 36 Celebrity Deaths Do Not Happen in Threes A particle with a mass of 3 kg is traveling at 3 m/s along a straight line path in the positive x-direction. If the particle intersects the y-axis at y = 3m, what is the particle s angular momentum? 37 The Worst Sketch Ever You ve probably realized that its not always convenient, nor reasonable, to use the Calculus to determine the moment of inertia for an object that has an odd shape. So, in practice, how can a physicist or engineer determine the moment of inertia for any object? Imagine the following experiment (see Figure below). A mass m is suspended by a thin cord would around the inner shaft (radius r) of a turntable which supports the object for which you are trying to determine the moment of inertia. The mass is released from rest and acquires a speed of v after falling a height of h. Show that the moment of inertia for the system (object + turntable) is: I = mr 2 2gh v 2 1 (1)
38 Circular Motion or Oscillations? A particle rotates counterclockwise in a circle of radius 3.0 meters with a constant angular speed of 8 rad/sec. At t=0, the particle has an x-coordinate of 2.0 meters. Find the x-coordinate as a function of time. Find the x-components of the particle s velocity and acceleration at any time t. What is v max? What is a max? What physical quantities from circular motion can you relate these last concepts to? 39 500 A light rope passes over a light, frictionless pulley. One end is attached to a bunch of bananas of mass M. The other end is attached to a monkey of mass M. Lets assume the monkey starts at a lower height than the bananas and begins climbing the rope in an attempt to reach the bananas. What is the net torque about the pulley axis? What is the total angular momentum about the pulley axis and describe the motion of the system. Will the monkey reach the bananas? If not, what can the monkey do to get the bananas? 40 The Physics of Rubberbands Take two rubberbands and break them so that they are rubber strands and not circles. Suppose you are standing in a doorway. Glue one end of a rubberband to the left hand side of the doorway and glue one end of the other rubberband to the right hand side of the doorway (at exactly the same height). Next, take the loose ends of each rubberband and glue those to either side of a marble. Each rubberband has a length of L. The marble has a mass of m. Assume the tension in the rubberbands is T and remains effectively constant in this problem. Suppose you displace the the marble a very small distance y in the vertical direction. Determine the restoring force for this system (the force that behaves like the -kx term in a spring). Then, determine the angular frequency of oscillations ω for this system. 41 Two Springs - Asymmetry A mass m is attached to a spring of constant k 2 which is attached to a spring of constant k 1, which is attached to a rigid wall. The mass is allowed to oscillate on a frictionless, horizontal surface. Determine its period of oscillation. What happens if k 1 = k 2?
42 Two Springs - Symmetry Now, that same mass has spring with k 2 attached to one side and the spring with k 1 attached to the other side. Both springs are attached to rigid walls and the mass is free to oscillate on a frictionless, horizontal surface. Determine its period of oscillation. What happens if k 1 = k 2?