THE UNIVERSITY OF BRITISH COLUMBIA
 
BIOL/PHYS 438 Assignment # 2:
 
MUSCLES, TENDONS & GASES
 
Tue. 23 Jan. 2007 - finish by Tue. 06 Feb.

Please hand in one assignment per group and list the names & Email addresses of all group members at the top of each sheet. In general, if you think some necessary information is missing, make a reasonable assumption. But always write down what that assumption is. Always estimate your uncertainty in any measured quantity, and don't forget to specify all units.

If possible, justify your input. For original comments you may score bonus points!

1. ARMSTRONG:

   Consider a human biceps as shown below.

   
   1. Measure your biceps length $\ell_0$ and the distance $s_w$. Estimate the distance $s_t$ (explain how you do this). Determine your muscle contraction $\Delta \ell$ from similar triangles $\Delta h/s_w = \Delta \ell/s_t$ when moving the hand between positions (a) and (b). Does the maximum contraction agree with the often quoted value $\Delta \ell/\ell_0 = 10$%?¹ Determine the average cross sectional area and length of your biceps muscle to find its volume $V_{\rm bic}$ and mass $M_{\rm bic} = \rho V_{\rm bic}$. What value should you use for the muscle density $\rho$?

   2. Find a heavy weight, say $M = 5$-10 kg, lift it as quickly as you can $n = 10$ times,² and record the time $\Delta t_{10} = 10 \Delta t$ for these 10 cycles. Estimate the average time $\Delta t_c$ that the muscle spent contracting in each cycle, determine your normalized muscle contraction speed $V = \Delta \ell/(\ell \times \Delta t_c)$, and compare it to the normalized maximum power speed $V_{mp} \approx 0.3 V_0$ [s$^{-1}$], where $V_0$ is the intrinsic muscle velocity for humans.³ Comment on the result.
      Determine the power needed to raise the weight: $P = M g \Delta h/\Delta t$. Estimate the mass $m_a$ of your forearm and the approximate position of its center of mass (cm), so that you can get estimates of the force $F_a$ and power $P_a$ needed to move the forearm. Add the forces $F_m = F + F_a$ and the powers $P_m = P + P_a$ and give estimates of the specific muscle force $f = F_m/A$ and the specific muscle power of your biceps $p = P_m/m_a$.
      Make a table of $\Delta \ell/\ell_0$, $m_a$, contraction frequency $\nu = n/\Delta t_{10}$, $V/V_0$, $f$ and $p$ for all team members.

2. TENDON FORCES:
   
   1. Calculate the tension $T$ in the Achilles tendon of each member of your group when he or she stands on the toes of one foot on a staircase, as shown above. For that you will have to know the person's body mass $M$ and the relevant dimensions of their foot.

   2. Measure the radius $r$ of each person's Achilles tendon and determine the corresponding specific stress $\tau = T/(\pi r^2)$.

   3. Make a table of $M, T$ and $\tau$ for all team members. Email this information (in plain text, please!) to Alex and he will make a compound table of this data for the whole class.

3. BREEZING IN WHITEHORSE: On a fine winter morning in the Klondike the air is a balmy $T_0 = 30^\circ$C below the freezing point, and the pressure is at $p = 1.03$ bar $= 1.03 \times 10^5$ N/m$^2$. Robert Rednose breathes deeply before descending into his gold mine, and takes in air at an average volume flow rate $\Phi = A \times u = 6.0$ liter/min. ($u$ is defined below). The air has to pass through his nostrils, which have a total opening area of $A = 2.8$ cm$^2$. The air warms up to the body temperature $T_B = +36^\circ$C inside the lung.

   Air is a (mostly diatomic) molecular gas with specific heat $C_p = (7/2)R$, where $R = 8.31$ J/mole $\cdot^\circ$K is the gas constant. Recall the Ideal Gas Law $pV=nRT$, where the pressure $p$ is in N/m$^2$, the volume $V$ is in m$^3$, $n$ is the number of moles in the volume $V$ and the temperature $T$ is always in $^\circ$K. Differentiate the caloric energy equation, $\Delta Q = n C_p \Delta T$, to get the heat flow rate that must be provided by the lung to maintain $36^\circ$C.

   1. How many moles are in one liter at this temperature and pressure? How many moles per second does he inhale?

   2. How much heat power [Watts] does Robert lose by warming up the air?

   3. What is the average intake velocity $u$ at which the air streams through his nostrils?

   4. Is the flow laminar ( $Re \lsim 2300$) or turbulent ( $Re \gsim 2300$)?⁴

   5. Measure the open area $A$ of your own nose and provide this data along with the body mass for each individual. Plot the area $A$ as an allometric relation $A = a M^\alpha$ on a log - log graph for the members of your group. What is the scaling exponent $\alpha$ for your data? What exponent do you expect? Why?