physics7A-spring05-mt2-Zettl-soln

physics7A-spring05-mt2-Zettl-soln - 1. An unusual spring is...

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Unformatted text preview: 1. An unusual spring is found to obey the following restoring-force law: F = − Ax – Bx 3 , with A and B both positive constants. Find the energy stored in the spring when it is stretched from its equilibrium length by an amount ∆ x. E ∆ x = ± F(x)dx (10 points) E ∆ x = − [ − Ax − Bx 3 ]dx = [Ax + Bx 3 ]dx = A x dx + B x 3 dx = A[_x 2 + B[_x 4 = _A ∆ x 2 + _B ∆ x 4 (10 points) Note: E ∆ x = − _A ∆ x 2 − _B ∆ x 4 will earn only 6 out of the 10 above points. Note: The positive answer is the only sensible one, since the energy stored in the spring must increase as the spring is stretched. This is only possible for E ∆ x = − F(x)dx. Note: Three (3) points were subtracted for final responses of the form: ± {_A ∆ x 2 + _B ∆ x 4 } + C, when C were not explicitly set to 0. Note: Integral bounds which did not match the final response resulted in 3-pnt deduction. e.g. _Ax 2 + _Bx 4 _A(x f – x i ) 2 + _B(x f – x i ) 4 ∫ ∆ x ∫ ∆ x ∫ ∆ x ] ∆ x ] ∆ x ∫ ∆ x ∫ ∆ x ∫ ∆ x | x f x i Zettl MT 2 Problem #2. Solution : First, it is important to note that the center of mass is NOT the point to which 1/2 of the mass is to the left or right. The defn. for center of mass is: Xcm= ( ∫ x dm) / ∫ dm =( ∫ x dm) /M tottal where for this problem we integrate from 0 to X max , and we calculate Ycm similarly. So, 1st we identify that due to uniform density dm= σ dA, where σ = M tottal /A. Then, taking vertical strips we find dA= (2cx 2 )dx, so dm= σ (2cx 2 )dx. Then, you may find it useful to calculate that Area= 2/3 c Xmas^3, and then we calculate Xcm = ( ∫ x σ (2cx 2 )dx) /M tottal =3/4 X max We can calculate Ycm explicitly by the same method, or simply recognize that by symmetry about the x-axis Ycm=0 Rubric: Defn. of COM 3 pts. finding dm (can involve σ , M tottal , A, etc.) 8 pts. Recognizing Ycm=0 by symmetry (or calc.) 4 pts. Xcm integration technique and correct answer 5 pts. 7A Ó ÐÙ Ø iÓÒaÒdÙb Ö iÖÓb ÐeÑ 4 Ô Ö iÒg5 Ó ÐÙ Ø iÓÒ : C ÓÒeÔ ØÙa ÐÐÝ Øh eÑ Ó ×Ø iÑ ÔÓ ÖØaÒ ØfaØ×a ÖeÓÒ ×eÖÚa Ø iÓÒÓ feÒ eÖgÝ × iÒeØh eÓ ÐÐi× iÓÒ × a Öee Ða ×Ø iaÒdÓÒ ×eÖÚa Ø iÓÒÓ fÑ ÓÑ eÒ ØÙÑ × iÒeØh eÖea ÖeÒÓeÜ ØeÖÒa ÐfÓ Öe×ÒaÓÒ ed iÑ e× iÓÒa Ð Ó ÐÐi× iÓÒÔ ÖÓb ÐeÑ ifÝÓÙkÒÓÛ Øha ØbÓ ØhÑ ÓÑ eÒ ØÙÑ aÒd eÒ eÖgÝa ÖeÓÒ ×eÖÚ ed Øh eÒÝÓÙhaÚ eØh e fÓ ÐÐÓÛ iÒgeÕÙa...
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This note was uploaded on 09/10/2008 for the course PHYSICS 7A taught by Professor Lanzara during the Spring '08 term at Berkeley.

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physics7A-spring05-mt2-Zettl-soln - 1. An unusual spring is...

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