SampleProbs1

SampleProbs1 - Physics 325, Fall 2006 REVIEW QUESTIONS #1...

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Physics 325, Fall 2006 REVIEW QUESTIONS #1 and SOLUTIONS 1) A particle of mass rn is moving in the potential i(r).: ar2 + bra Depending on the sign of the constants o and b, there are four cases shown in the following table b>0 b<0 a)0 A B a<0 C D For the four cases in the table: a) Sketch the potentialU(r). b) Find all the equilibrium points of U (r). Specify whether they are stable or un- stable. c) Find the frequency of small oscillations around points of stable equilibrium. I o: \ur s\ \ o- d) In the above figure, the acceleration of the particle is plotted as a-function of r f.or the four cases mentioned above. Match every acceleration graph to its corresponding potential. e) Find the work done when the particle moves from point p to point g as shown in the acceleration graph (III). lt I I I \-/ OI \ 0
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2) A cannon ball is frred uertically from ground level at time t : 0 with initial velocity u6. The ball encounters a force Fai. : -4v due to air resistance (v is the velocity vector for the ball and 4 is a constant). Derive the diffrential equation for the velocity of the ball as a function of time for all times before the ball first hits the ground on return. Solve the equation derived in (p). Determine the terminal velocity of the ball u7. Calculate the height of the ball as. a function of time. Use the results of (b) ald (c) to show that the condition that the ball hits the ground with u - u7 is that u6 D ur. 4) A rocket starts from rest at the center of a galaxy (r 0) at time t 0 and moves radially outward following a straight line path. Gravitational and other forces cause the rocket to move in the spherically symmetric potential u(r) -@ lro)2 where rn is the rocket mass (which is a function oi time) and 26, 76 &r€ some known characteristic distance and time. In addition, the rocket engine is fired at launch, ejecting fuel mass at the constant rate a > 0 with speed u relative to the rocket frame. All relativistic effects are ignored in this problem.
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SampleProbs1 - Physics 325, Fall 2006 REVIEW QUESTIONS #1...

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