ps06sol - MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department...

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1 MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Physics Physics 8.01 Fall Term 2010 Problem Set 6: Conservation of Energy and Simple Harmonic Motion Solutions Problem 1: Escape Velocity Toro The asteroid Toro, discovered in 1964, has a radius of about R = 5.0km and a mass of about m Toro = 2.0 ! 10 15 kg . Let’s assume that Toro is a perfectly uniform sphere. What is the escape velocity for an object of mass m on the surface of Toro? Could a person reach this speed (on earth) by running? Solution: The only potential energy in this problem is the gravitational potential energy. We choose the zero point for the potential energy to be when the object and Toro are an infinite distance apart, U gravity (r 0 = ! ) " 0 . With this choice, the potential energy when the object and Toro are a finite distance r apart is given by Toro gravity ( ) = Gm m U r r ! (1.1.1) with U gravity (r 0 = ! ) " 0 . The expression escape velocity refers to the minimum speed necessary for an object to escape the gravitational interaction of the asteroid and move off to an infinite distance away. If the object has a speed less than the escape velocity, it will be unable to escape the gravitational force and must return to Toro. If the object has a speed greater than the escape velocity, it will have a non-zero kinetic energy at infinity. The condition for the escape velocity is that the object will have exactly zero kinetic energy at infinity. We choose our initial state, at time t 0 , when the object is at the surface of the asteroid with speed equal to the escape velocity. We choose our final state, at time t f , to occur when the separation distance between the asteroid and the object is infinite. Initial Energy: The initial kinetic energy is K 0 = 1 2 mv esc 2 . The initial potential energy is U 0 = ! G m Toro m R , and so the initial mechanical energy is
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2 E 0 = K 0 + U 0 = 1 2 mv esc 2 ! G m Toro m R . (1.1.2) Final Energy: The final kinetic energy is K f = 0 , since this is the condition that defines the escape velocity. The final potential energy is zero, U f = 0 since we chose the zero point for potential energy at infinity . T he final mechanical energy is then E f = K f + U f = 0 . (1.1.3) There is no non-conservative work, so the change in mechanical energy nc mech 0 W E = = ! , (1.1.4) is then 0 = 1 2 esc 2 ! G m Toro m R . (1.1.5) This can be solved for the escape velocity, v esc = 2 Gm Toro R = 2(6.67 ! 10 " 11 N # m 2 # kg " 2 )(2.0 ! 10 15 kg) (5.0 ! 10 3 m) = 7.3m # s " 1 . (1.1.6) Considering that Olympic sprinters typically reach velocities of 12m ! s " 1 , this is an easy speed to attain by running on earth. It may be harder on Toro to generate the acceleration necessary to reach this speed by pushing off the ground, since any slight upward force will raise the runner’s center of mass and it will take substantially more time than on earth to come back down for another push off the ground.
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3 Problem 2: A particle is released from rest at the position x = x 0 in the potential described below.
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This note was uploaded on 04/13/2011 for the course PHYSICS 8.01 taught by Professor Guth during the Fall '09 term at MIT.

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ps06sol - MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department...

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