midterm2_solutions - PHYS521 Biological Physics Midterm...

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PHYS521. Biological Physics. Midterm Exam #2 Solutions November 18, 2010 Each problem is graded out of 10 points. It is enough to solve two problems to get full credit for the exam. If you solve more problems, you will have extra credit. Please give understandable explanations. No partial credit will be given for wrong results without explanations. Problem 1 A muscle fiber is placed in an apparatus that periodically stretches it. That is, the fiber’s length is forced to change in time as ) sin( ) ( 0 t A x t x . The force exerted back on the apparatus by the fiber is then also periodic function of time. For small amplitude A , it is a wave with a phase shift )] sin( [ ) ( 0 t B f t f . Here A and B are positive constants. If the phase shift is greater than zero, we say that “the displacement lags the force”. (a. 5 points) Work out the rate at which the apparatus does work on the fiber (i.e., the power). Find the average of this rate over one full cycle. (b. 5 points) The figure shows two curves. One gives the phase shift for a living muscle fiber; the other curve is for a dead fiber. Which is which? Which unusual feature of one of the curves tells you the answer? Solution: (a) The instantaneous power of the muscle is given by ) ( ) ( t x t f P . We can write
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) cos( )] sin( [ 0 t A t B f P The averaged over the period power is given by ) cos( ) sin( ) sin( 0 t t AB t A f P In this expression the first term is zero, but the second one would be equal to zero only for 0 . In general sin 2 1 sin ) ( cos ) cos( cos ) sin( ) cos( ) sin( 2 t t t t t Substituting into the expression for power we get sin 2 1 AB P (b) In the figure is positive everywhere for trace 1, but trace 2 has intervals of negative . Since
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midterm2_solutions - PHYS521 Biological Physics Midterm...

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