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Let us investigate the properties of this type of driven oscillation. Suppose that our system is subject to an external force of the form fext (t) = m ω02 X1 cos(ω t). 32). 39), we obtain d2 x dx + 2ν + ω02 x = ω02 X1 cos(ω t). 40), and represent the right-hand side of the above equation as ω02 X1 exp(−i ω t). It is again understood that the physical solutions are the real parts of these expressions. Note that ω is now a real parameter. We obtain a −ω2 − i 2 ν ω + ω02 e−i ω t = ω02 X1 e−i ω t .

7. A particle of mass m moves (in one dimension) in a medium under the influence of a retarding force of the form m k (v3 + a2 v), where v is the particle speed, and k and a are positive constants. Show that for any value of the initial speed the particle will never move a distance greater than π/(2 k a), and will only come to rest as t → ∞. 8. Two light springs have spring constants k1 and k2 , respectively, and are used in a vertical orientation to support an object of mass m. Show that the angular frequency of oscillation is [(k1 + k2 )/m]1/2 if the springs are in parallel, and [k1 k2 /(k1 + k2 ) m]1/2 if the springs are in series.

14) Thus, Now, the planet’s acceleration is written a= dv d2 r ¨ eθ + r θ˙ e ˙r + (˙r θ˙ + r θ) ˙θ . 15) Again, eθ has a non-zero time-derivative because its direction changes as the planet moves around. 10) with respect to time yields ˙θ = θ˙ (− cos θ, − sin θ) = −θ˙ er . 16) ˙ eθ . 3), can be written ˙ eθ = − G M er . 19) and a tangential equation of motion, r θ¨ + 2 ˙r θ˙ = 0. 6 Conic Sections The ellipse, the parabola, and the hyperbola are collectively known as conic sections, since these three types of curve can be obtained by taking various different plane sections of a right cone.

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