Suppose a force F is applied to a cylinder of elastic material. Thecylinder has

Question

Suppose a force F is applied to a cylinder of elastic material. Thecylinder has cross-sectional area A0 and initial length L0. Recallthat Young's modulus E is defined as E=stress/strain=(F/A0)/(L/L0) = (F/L0)/(A0/L)
The Young's modulus of muscle fiber is 3.12×107 Pa. A musclefiber 2.00 cm in length and 0.100 cm in diameter is suspended witha mass M hanging on its end.

a)Given that L/L0=0.10, calculate M.
b)Derive a formula for the work, w, as a function of change inlength, L.
Suppose the muscle is stretched reversibly and adiabatically,starting from its initial 2.00 cm length and a temperature of310.0ºC. The heat capacity of the muscle may be taken to be C= 4.0×106 J/(m3·K).
c)Calculate w and U.
d)Estimate the change in temperature.

Any help on any parts a thru d would be awesome! Thanks!

Explanation / Answer

a) E = (F/A0) / (L/L0) = F*L0 / [A0*L] F = E*A0*L/L0 F = M*g diamter= 0.1 cm = 0.001 m M = E*A0*(L/L0)/g = (3.12 e7 Pa)*(*[0.001m]2 / 4) *(0.1) / (9.8 m/s2) M = 0.250 kg b) W = integral [F*d[L] = Integral [ E*A0*L/L0]*d[L] W = (E*A0/2L0)*L2 c) L/L0 = 0.1 L0 = 2 cm = 0.02 m L = 0.002 m W = (3.1e7 Pa)*(*[0.001 m]2 / 4) /[2*0.02 m]*[0.002 m]2 W = 0.00243473431 J = 2.43 x 10-3 J d) b/c adiabatic, there is no heat U = q + W = 0 + W = W U =C*V*T           where V = volume = height**d2/4 T = W/ [C*V] V = (0.02 m)*()*(0.001 m)2/4 = 1.57079633e-8m3 T = 0.00243473431 J / [ 4e6 J/m3/K * 1.57079633e-8m3] T = 0.03875 K Tfinal - Tinitl = T Tfinal = 310 K + 0.03875 = 310.04 K

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