PHYSICS 218 Final Exam FALL, 2009 Do not fill out the information below until instructed to do so! Name:__________________________________ Signature:_______________________________ Student ID:______________________________ E-mail:__________________________________ Section Number: _________________________ • You have the full class period to complete the exam. • Formulae are provided. You may NOT use any other formula sheet. • When calculating numerical values, be sure to keep track of units. • You may use blank pages provided with this packet as scratch paper or come up front to get more. • Be sure to put a box around your final answers and clearly indicate your work to your grader. • All work must be shown to get credit for the answer marked. If the answer marked does not obviously follow from the shown work, even if the answer is correct, you will not get credit for the answer. • Clearly erase any unwanted marks. No credit will be given if we can’t figure out which answer you are choosing, or which answer you want us to consider. • Partial credit can be given only if your work is clearly explained and labeled. Put your initials here after reading the above instructions: ________SCORE Part 1 (20) ___________ Part 2 (20) ___________ Part 3 (20) ___________ Part 4 (20) ___________ Part 5 (30) ___________ Part 6 (30) ___________ Part 7 (30) ___________ Part 8 (30) ___________ Total (200)___________Problem 1 (20 points): In the first stage of a two-stage rocket, the rocket is fired from the launch pad starting from rest but with a constant acceleration of a upward. At t=t0 after launch, the rocket fires the second stage, which suddenly boosts its speed to v upward. This firing uses up all the fuel, however, so then the only force acting on the rocket its gravity. Air resistance is negligible. Find the maximum height that the stage-two rocket reaches above the launch pad In terms of g, v, a, t0 How much time after the stage-two firing will it take for the rocket to fall back to the launch pad? How fast will the stage-two rocket be moving just as it reaches the launch pad?Problem 2 (20 points): A uniform solid cylinder of mass M is supported on a ramp that rises at an angle above the horizontal by a wire that is wrapped around its rim and pulls on it tangentially parallel to the ramp. Show that there must be friction on the surface for the cylinder to balance this way. Show that the tension in the wire must be equal to the friction force. Find this tension.Problem 3 (20 points): On a horizontal surface, a crate with mass m is placed against a compressed spring that stores energy E. The spring is released and the crate slides distance d before coming to rest counting from its initial position. The initial compression y of the spring is very small compared to d. What is the speed of the crate when it is a distance x (y<x<d) from its initial position when the spring was released?Problem 4 (20 points): Two identical masses are released from rest in a smooth hemispherical bowl of radius R from the positions shown in the figure. You can ignore friction between the masses and the surface of the bowl. If they stick together when they collide, how high above the bottom of the bowl will the masses go after colliding?Problem 5 (30 points): A small block with mass m is placed inside an inverted cone that is rotating about a vertical axis such that the time for one revolution of the cone is T. The walls of the cone make an angle with the vertical. The coefficient of static friction between the block and the cone is . If the block is to remain at a constant height above the apex of the cone, what is the maximum value of T? If the block is to remain at a constant height above the apex of the cone, what is the maximum value of T?Problem 6 (30 points): The mechanism shown in the figure is used to raise a crate of supplies from a ship's hold. The crate has total mass m. A rope is wrapped around a wooden cylinder that turns on a metal axle. The cylinder has radius R and a moment of inertia I about the axle. The crate is suspended from the free end of the rope. One end of the axle pivots on frictionless bearings; a crank handle is attached to the other end. When the crank is turned, the end of the handle rotates about the axle in a vertical circle of radius r, the cylinder turns, and the crate is raised. What magnitude of the force applied tangentially to the rotating crank is required to raise the crate with acceleration a? (You can ignore the mass of the rope as well as the moments of inertia of the axle and the crank.) rProblem 7 (30 points): An unmanned spacecraft is in a circular orbit around the moon, observing the lunar surface from an altitude of h. To the dismay of scientists on earth, an electrical fault causes an on-board thruster to fire, decreasing the speed of the spacecraft by v (v is a positive number defined as vnew=v0 -v). If nothing is done to correct its orbit, with what speed will the spacecraft crash into the lunar surface? (Take the following to be given: mass of the moon M, its radius R, mass of the spacecraft m, gravitational constant is G)Problem 8 (30 points): Two solid cylinders connected along their common axis by a short, light rod have radius R and total mass M and rest on a horizontal tabletop. A spring with force constant has one end attached to a clamp and the other end attached to a frictionless ring at the center of mass of the cylinders. The cylinders are pulled to the left a distance x, which stretches the spring, and released. There is sufficient friction between the tabletop and the cylinders for the cylinders to roll without slipping as they move back and forth on the end of the spring. Show that the motion of the center of mass of the cylinders is simple harmonic. Calculate its period in terms of M and k. [Hint: The motion is simple harmonic if a and x are related by a= - w2x, and the period then is T=2/w. Apply =Icm and F=Ma to the cylinders in order to relate and the displacement of the cylinders from their equilibrium
View Full Document
Unlocking...