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Physics 325 Homework 2

# Physics 325 Homework 2 - a Air resistance is neglected b...

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Physics 325, Fall 2010 Homework Assignment #2 Due Thursday 9 September at 11:15 am 1) (15 points) A particle of mass m moves in one dimension with this potential energy function: U ( x ) = αxe - βx 2 where α and β are positive, real numbers. a) Compute the force as a function of x . b) Sketch the potential energy and discuss the motion that may occur. (For example, you could sketch the potential for the case when α = β = 1.) c) Locate all equilibrium points and classify them as either stable or unstable. d) Determine the frequency of small oscillations about all stable equilibrium points. e) A different system has a particle of mass m acting under the force - F ( x ), where F ( x ) is defined above. Find the frequency (or frequencies) of small oscillations about the stable equilibrium point(s) of this system. 2) (15 points) A ball of mass m is thrown vertically upward with an initial velocity v 0 . Find the maximum height and the time it takes to reach the maximum height in each of the following three cases:

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Unformatted text preview: a) Air resistance is neglected. b) The force of air resistance is F air = αv where v is the instantaneous speed. The direction of the force of air resistance is to oppose the motion of the ball through the air. c) The force of air resistance is F air = βv 2 . d) Graph the velocity versus time and distance versus time in all three cases. Compare and discuss the results. 3) (10 points) Find the velocity ˙ x and the position x as a function of time t for a particle of mass m which starts at rest at x = 0 at t = 0, subject to the following force functions ( F and c are constant real numbers): a) F x = F + ct b) F x = F e ct For the following two cases, ﬁnd the velocity ˙ x as a function of the position x for a particle of mass m which starts from rest at x = 0 and t = 0, subject to the following force functions. c) F x = F + cx d) F x = F e-cx 40 Points Total...
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