8.012 Physics I: Classical Mechanics Fall 2008

Transcription

1 MIT OpenCourseWare Physics I: Classical Mechanics Fall 2008 For inforation about citing these aterials or our Ters of Use, isit:

2 MASSACHUSETTS INSTITUTE OF TECHNOLOGY Departent of Physics Physics Fall 2008 Exa 2 NAME: Instructions: 1. Do all FIVE (5) probles. You hae 90 inutes. 2. SHOW ALL WORK. Be sure to circle your final answer. 3. Read the questions carefully. 4. All work ust be done in this booklet in workspace proided. Extra blank pages are proided at the end if needed. 5. NO books, notes, calculators or coputers are peritted. A sheet of useful equations is proided on the last page. Proble Maxiu Score Grader Total 100

3 Proble 1: Quick Multiple Choice Questions [10 pts] For each of the following questions circle the correct answer. You do not need to show any work. (a) Which of the following is the result of an elastic collision between the two isolated objects at right? /2 /3 2 2/3 2 (b) Which experiences the largest ipulse? Fully elastic collision of 10 kg ball dropped fro 10 Fully inelastic collision of 20 kg ball dropped fro 10 Fully elastic collision of 20 kg ball dropped fro 5 These all hae the sae ipulse. Page 2 of 7

4 (c) A cart traeling at speed on a frictionless track starts to leak sand. What is the cart s speed at a later tie? Greater than Less than Equal to! (d) Consider the coposite disk at right, with a ain section of ass density ρ and radius d, and a subsection of ass density 2ρ and radius d/2. If O arks the origin of our coordinate syste, at the center of the coposite disk, where is the disk s center of ass located? O 2! at O d/2 to right of O d/6 to right of O d/10 to right of O d (e) A block connected to a fixed wall by a spring oscillates back and forth on a frictionless surface. When it reaches its axiu extent, a piece of clay is dropped onto it and instantaneously sticks. Which of the following is consered in this collision? X MAX Horizontal oentu Total echanical energy Total echanical energy and horizontal oentu None of these Page 3 of 7

5 Proble 2: Collision [20 pts] U! 2 2 Two balls, one of ass and one of ass 2, approach fro orthogonal directions with identical speeds and collide. After the collision, the ore assie ball oes with the sae speed but downward (orthogonal to its original direction) and the less assie ball oes with speed U at an angle θ with respect to horizontal. Assue that no external forces act during the collision. (a) [10 pts] Calculate the final speed U of the less assie ball and the angle θ. (b) [10 pts] Deterine how uch kinetic energy is lost or gained by the two balls during the collision. Is this collision elastic, inelastic or superelastic? Page 4 of 7

6 Proble 3: Cube on a Sphere [20 pts] R A sall block starts fro rest and slides down fro the top of a fixed sphere of radius R, where R >> size of the block. The surface of the sphere is frictionless and constant graitational acceleration g acts downward. (a) [10 pts] Deterine the speed of the block as a function of angle fro the top while it reains in contact with the sphere. (b) [10 pts] At what angle does the block lose contact with the sphere? Page 5 of 7

7 Proble 4: Tabletop Rocket [25 pts] ex M 0 /2 M 0 /2 µ A rocket of total ass M 0, half of which is fuel, starts at rest on a long horizontal table. The coefficient of friction between the rocket and table surfaces is µ. At tie t = 0, the rocket is ignited, ejecting fuel out at a constant rate γ = dm/dt with elocity ex relatie to the rocket. Constant graitational acceleration g acts downward. (a) [10 pts] What condition ust be et for the rocket to start oing at t = 0? (b) [10 pts] Assuing that the rocket satisfies this requireent, what is the axiu speed V MAX achieed by the rocket? (c) [5 pts] How far does the rocket go after it runs out of fuel? You can express your answer in ters of V MAX. (d) [BONUS 5 pts] How far does the rocket trael in total? For this you will need to ake use of the following integral: Page 6 of 7

8 Proble 5: Planet Orbit [25 pts] R M r A sall planet of ass is in a circular orbit of radius r around a star of ass M and radius R in otherwise epty space (assue M >> so the star is stationary). (a) [10 pts] Deterine the potential energy U(r), the kinetic energy K(r) and the total echanical energy E(r) of the planet in ters of G, M and r assuing U 0 as r. (b) [5 pts] Deterine the iniu aount of echanical energy that ust be added to the planet to cause it to escape fro the star (i.e., r ). By what factor ust the speed of the planet be increased to cause it to escape? (c) [5 pts] Now assue that the planet in subject to a iscous force of the for where A is a constant and is the direction of otion. Copute the loss of echanical energy in one orbital period in ters of G, M, r and A. Assue that this loss is sall enough that neither the orbital radius nor speed of the planet changes appreciably in one orbit. (d) [5 pts] Building fro (c), copute the change in radius of the planet in one orbital period due to the iscous force and the corresponding radial elocity based on the assuptions aboe, in ters of G, M, r and A. Does the planet fall into the star or away fro it? Page 7 of 7