S08 AM261 MT Q

# S08 AM261 MT Q - ¨ x 10 ˙ x 16 x = 0(a Find the damping...

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MIDTERM Phys 263 / Amath 261 June 16, 2008: 4:30 to 6:30pm 1. For two particles with mass m 1 and m 2 and positions r 1 = (2 t 2 , 3 t, 4) r 2 = (1 + t 2 , 0 , 4 t 2 ) (a) Find the center of mass coordinates. (b) Find the velocity and acceleration for the center of mass. (c) If m 1 = 1 kg and m 2 = 2 kg, ﬁnd the linear momentum of the center of mass at t = 1 s. 2. A block of mass m is initially at rest on a frictionless surface at the origin. At time t = 0, a decreasing force given by F = exp( - λt ), where 0 < λ < 1. (a) Calculate v ( t ). (b) Calculate x ( t ). (c) If λ = 0 . 5 and the block has mass m = 2, what is its asymptotic velocity v ( t )? (d) How long will it take for the velocity of the block to be e - 1 less then the asymptotic velocity? 3. A particle of mass m is subjected to a force F = - kx + kx 3 /a 2 , where k and a are positive constants. (a) Find the potential U ( x ). (b) Find the equilibrium positions. (c) Sketch U ( x ) and label each equilibrium position as stable or unstable. (d) What is the maximum energy for which the motion can be bounded? 1

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4. A particular harmonic oscillator satisﬁes the equation
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Unformatted text preview: ¨ x + 10 ˙ x + 16 x = 0 (a) Find the damping constant b and the spring constant k . Is the harmonic oscillator under, critical, or over-damped? (b) At time t = 0 the particle is projected from the point x = 1 towards the origin with a speed u . Find an expression for x ( t ). (c) At what time t will the particle pass through the origin? How large must u be to ensure that this happens? 5. The equation of motion for a driven harmonic oscillator given by m ¨ x + b ˙ x + kx = F cos( ωt ) has the steady state solution x = D cos( ωt-φ ) where the amplitude D and the phase φ are completely determined for the system. (a) Find the value of ω where the average potential energy h U i is maximum. (b) Find the value of ω where the average kinetic energy h K i is maximum. (c) Are your answers in part (a) and (b) the same? Discuss. (d) Find an expression for the average total energy h E i . 2...
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S08 AM261 MT Q - ¨ x 10 ˙ x 16 x = 0(a Find the damping...

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