Review_Exam2 - Keys Thornton&MarionandBoas...

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Keys Thornton & Marion and Boas. Homework problems (solu<ons at culearn). Clicker ques<ons (at culearn as well).
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1) Complex number and variable z=x+iy, or rcos θ +isin θ , or re i θ Complex conjugate, z=x‐iy is conjugate to z=x +iy. Define sinz and cosz. Roots of z 3 =8.
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Solu<on of 2 nd linear ODE Homogeneous equa<on. x”+2bx’+cx=0. x(t)=Ae [‐b+sqrt(b^2‐ac)]t +Be [‐b‐sqrt(b^2‐ac)]t . For b^2‐ac<0, oscilla<on. x(t)=Ae [‐b+ i sqrt(ac‐b^2)]t +Be [‐b‐ i sqrt(ac‐b^2)]t or x(t)=Ae ‐bt cos( ω t‐ δ ). Inhomogeneous equa<on. x”+2bx’+cx=f(t) x(t)=x c (t)+x p (t) If f(t)=de nt , x p (t) can be constructed, depending on the rela<on of n to the roots of characteris<c equa<ons.
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Spring‐block Oscillator (linear, for small amplitude oscilla<ons) F=mx” or –kx=mx” or x”+(k/m)x=0. Angular frequency ω 0 =sqrt(k/m), period T, … x(t)=Asin( ω 0 t‐ δ ). v=x’=A ω 0 cos( ω 0 t‐ δ ). E=T+U, conserved.
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