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MIT 8 01T - Exam 3 Sample Problems

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8.01L Fall 2005 Exam 3 Sample Problems Note that some of these problems were from an exam allowing calculators. Your exam will not allow calculators. Problem 1 a) Two large spherical stars with masses of M and 2M are positioned a distance D apart (measured from the center of one star to the center of the other star) as shown. A small spherical asteroid with mass m is located with its center exactly halfway between the two large stars. Find the magnitude and direction of the total gravitational force acting on the asteroid. b) A small probe of mass m is released from rest at a distance of 2R above the surface of a spherical planet of mass M and radius R. Find the speed that the probe will have when it hits the surface of the planet. Assume that the planet does not have an atmosphere and no forces other than gravity act on the probe. c) Assume that the probe is released at the same distance from the planet but now with a velocity of vo=GM4R. The probe has a rocket which fires to put it into a circular orbit at that height. Find the speed the probe would have once it is in the circular orbit and, using that, find the work done by the rocket. Assume that, at all times, the probe is at the same distance from the planet and that there are no forces acting on the probe other than the planet’s gravity and the rocket. Extra Credit Question This question may require a lot of thought for the points available. Try it only if you have finished with all other problems d) Redo part (b) but assuming that the planet is a hollow shell and that the probe falls through a hole in the shell. Find the speed of the probe when it gets to the center of the hollow shell. Explain your answer. Problem 2 a) An object of mass, M=2 kg, is attached to a spring of spring constant k=50 N/m which is compressed a distance d=20 cm and then released at rest. Find the speed of the object when it has gone past the point where the spring is uncompressed and now the spring is stretched a distance of 10 cm. Assume that the mass is moving on a horizontal, frictionless surface. b) Write an equation for the position of the mass as a function of time with t=0 being the instant that the mass was first released from rest. Use this equation to find out how long it takes the mass to get from the initial point with the spring compressed by 20 cm to the point where the spring is stretched by 10 cm. (Hint: Think carefully about the units of ω when doing the trig functions on your calculator.) c) Write an equation for the velocity as a function of time, using the same definition of t=0 as in part (b). Use this equation to find the velocity (magnitude and direction) when the time is t =3T4 where T is the period of motion of the mass on the spring. M2MmDRm2RMRm2RMvokspringuncompressedMdProblem 3 Two hockey pucks collide on a horizontal, frictionless surface. The velocities of both pucks before the collision are shown in the drawing. After they collide, puck A is moving as shown. Assume that the mass of puck A is twice as large as the mass of puck B: MA = 2*MB. a) Find the magnitude and direction of the velocity of puck B after the collision. Clearly indicate on your drawing the angle you are using to specify the direction of puck B. b) Is kinetic energy conserved in this collision? Justify your answer. If not, clearly indicate whether KE is gained or lost. Problem 4 a) A baseball-player-turned-astronaut stands on Mimas, a small moon of Saturn, and throws a baseball with initial speed 37m/s. The mass of the baseball is 0.15kg; the mass of Mimas is 3.8x1019kg and its radius is 200km. If the astronaut throws the ball horizontally, will it go into a circular orbit around Mimas? Explain your answer. b) For the same situation given in part b), if the astronaut throws the ball vertically, how high above the surface of Mimas will it rise before stopping? Problem 5 Short Answer Questions: You must show your work or write an explanation of your answer to get credit for these problems. i) A ball hits a wall and bounces off as shown. Assume that the collision is elastic. Which vector best represents the direction of the change in momentum of the ball? a) Circle your choice and explain it b) Not enough information to answer. A BCDE FG H ii) A small car is moving along a straight level highway at a speed of 3v. The car hits from behind a large truck moving in the same direction at speed v. After the collision, the car is stuck onto the truck. During the collision, which vehicle experiences the greater average force? Explain your answer. a) The car b) The truck c) The forces are equal d) Impossible to determine without information about the masses AB25o25 m/s40oABBEFORE AFTER15 m/sv=?30 m/siii) Two balls of masses 2 kg and 3 kg slide along a frictionless horizontal surface with speeds of 4 m/s and 2 m/s, respectively. After an inelastic collision, the balls stick together and move at a speed of 2 m/s. What direction did the two balls move before the collision? a) In the same direction b) In opposite directions c) At an angle not equal to 0° or 180° d) The situation described is impossible e) Not enough information given to select an answer Problem 6 Two steel spheres are shot at each other and then collide head-on as shown. Sphere A has a mass of 10 kg and a velocity just before the collision of 3 m/s to the right. Sphere B has mass of 4 kg and velocity just before the collision of 4 m/s to the left. Immediately after the collision the velocity of sphere B is observed to be 6 m/s to the right. a) What is the magnitude and direction of the velocity of sphere A immediately after the collision? Clearly indicate your coordinate system and what direction is positive. b) Is this collision elastic or inelastic? Explain your answer. c) Assume that the collision lasts 10!3 seconds. Calculate the magnitude and direction of the average force that sphere A exerts on sphere B during the collision. Clearly indicate your coordinate system and what direction is positive. Problem 7 A spring-loaded toy gun is used to shoot a ball of mass M straight up in the air. The ball is not attached to the spring. The ball is pushed down onto the spring so that the spring is compressed a distance S below its unstretched point.


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MIT 8 01T - Exam 3 Sample Problems

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