THE UNIVERSITY OF BRITISH COLUMBIA
 
Physics 438 Assignment # 1:
 
METABOLISM
 
SOLUTIONS:
 
Tue. 09 Jan. 2007 - finish by Tue. 23 Jan.

The first part of this assignment was not for credit, but most people did it anyway! We appreciate those who Updated their Profiles, especially if they gave us their preferred Email address in the database. (To those who weren't able to get around to it in the first two weeks: it's never too late! The database awaits your input/revisions for the duration of the course.)

For the rest of the Assignment (and for all subsequent Assignments), join a group of 3-5 students. 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.

  1. STAIRCASE OLYMPICS:
    1. Determine the mechanical power output3 $P_{\rm walk}$ for each team member4 walking up four flights of stairs (four floors) in the Hebb building. The height of one "floor" in the Hebb staircase is h = 7.28 m; if you do the exercise anywhere else you must measure h. By the definition of the mechanical efficiency $\eta \equiv P/\Gamma$, we have $P_{\rm walk} = \eta_{\rm walk} \times \Gamma_{\rm walk}$ where $\Gamma_{\rm walk} = b_{\rm walk} \times \Gamma_0$ is the metabolic rate while walking up the stairs and $b_{\rm walk}$ is the corresponding metabolic activity factor. Calculate $\Gamma_{\rm walk}$ and $b_{\rm walk}$ for each team member. What value should you take for $\eta$? Discuss this choice and comment upon its validity in your written report.  ANSWER Work is defined as the product of a force F times the distance x over which the force needs to be overcome. In our case the force to overcome is the gravitational force Fg = M x g over the height h. Now, the power P is the amount of energy used per time. In our case:

      \begin{displaymath}P_{\rm climbing stairs} = {M \times g \times h \over t} \; . \end{displaymath}

      This power needs to come from somewhere, which in the end is provided by the power produced by our metabolism. Since the metabolism needs also to produce power to keep your body functions working (e.g. heat), the total metabolic power produced during the climb is only partly used for the actual climb. Therefore

      \begin{displaymath}P_{\rm climbing stairs} = \eta \times \Gamma_{\rm climbing stairs} \end{displaymath}

      where $\Gamma$ is the metabolic rate and $\eta$ the efficiency, which we don't know(!) but we have to assume some value, so we assume the "typical" value given in the textbook, namely 25%. Then the metabolic activity factor $b = \Gamma/\Gamma_0$ can be determined from $\Gamma$ by assuming that $\Gamma_0$ is given by the "mouse to elephant" allometric relation.
       
      For example, to calculate the metabolic activity factor b when M = 60 kg, h = 7.3 m and t = 20 s, we first find

      \begin{displaymath}\Gamma_{\rm climbing stairs} = {M \times g \times h \over t \times 0.25}
= 858 \; {\rm W} \end{displaymath}

      and then estimate the resting metabolic rate

      \begin{displaymath}\Gamma_0 = 4 M^{0.75} = 86 \; {\rm W} \; . \end{displaymath}

      Now, $\Gamma_{\rm climbing stairs} = b \times \Gamma_0$, so the activity factor is \fbox{ $b \approx 10$\ }.

    2. Estimate your uncertainty in this measurement.5 The largest possible value, $P_{\rm max}$, is found by combining the largest likely value of your body mass, M+,6 and the shortest possible time $\Delta t_-$. Similarly combine the lowest likely body mass M- and the longest likely time $\Delta t_+$ to find a value for $P_{\rm min}$. A good estimate of the uncertainty in your experimental result is thus $\delta P = (P_{\rm max} - P_{\rm min})/2$. Express your answer for the power $P_{\rm walk}$ in the form $P \pm \delta P$ W. (Always include units.)  ANSWER You have an uncertainty in your body mass (clothes, amount of water you have drunk, . . . ) which we assume for this example to be $\delta M = \pm 2$ kg and an uncertainty in time $\delta t = \pm 1$ s. We want to calculate the uncertainty in our calculation results. Using the prescription above with the example numbers, we get $\delta P = (P_{\rm max} - P_{\rm min})/2$ = (233.7 - 197.8)/2 $\approx 18$ W so that $\delta \Gamma_{\rm climbing stairs} = \delta P/0.25$, giving \fbox{$\delta \Gamma_{\rm climbing stairs} \approx 72$~W }.7

    3. Determine $P \pm \delta P$ for each one of your team members when running up the stairs.8 ANSWER If the same person runs up the stairs in t = 10 s we get \fbox{ $P_{\rm run} = 429 \pm 57.2$~W } and \fbox{ $\Gamma_{\rm run} = 1716 \pm 228$~W }, assuming the same uncertainties as before.

    4. Estimate the mass of the muscles $M_{\rm musc}$ used for running up the stairs and give the power to mass ratio $X_{\rm run} = P_{\rm run}/M_{\rm musc}$ of these muscles.9 Muscle mass ($M_{\rm musc}$) = muscle volume ($V_{\rm musc}$) times muscle density ($\rho$). Muscles have about the same density as water.  ANSWER The mean leg muscle mass of 60 kg women is measured to be 15 kg using X-ray absortiometry.10 Thus the power-to-weight ratio for the running muscles (assuming the leg muscles do almost all the work) is \fbox{ $X = P_{\rm run}/M_{\rm musc} = 29$~W/kg }.

    5. Make a table including all your data showing Name, M, $\Delta t_{\rm walk}$, $P_{\rm walk}$, $\Gamma_{\rm walk}$, $\Delta t_{\rm run}$, $P_{\rm run}$, $\Gamma_{\rm run}$, $\Gamma_0$, $V_{\rm musc}$ and $X_{\rm run}$, including the uncertainty in each. Explain your most significant sources of uncertainty.  ANSWER The tables will be different for each group. Uncertainties fall into two categories: measurement imprecision and theoretical imprecision. In the former category are your elapsed time (probably measured to about $\pm$1 s) and your mass (probably estimated to about $\pm$1-2 kg, including clothes etc.). In the latter category is your efficiency $\eta$, which you had to guess in order to convert P into $\Gamma$, and your resting metabolism $\Gamma_0$, which you had to assume could be estimated from the "mouse-to-elephant" allometric relation, $\Gamma_0$[W] = 4 M[kg]0.75 - which we know is not a precise law but merely a consistent trend, with typical deviations of about a factor of two! A similar "slop" can be expected for the efficiency, which is only "typically" 25%. So you should not be expect these calculations to yield "true" values of your activity factor, were you to measure same more directly using oxygen consumption etc. As with all estimations of uncertainty, a good deal of subjective judgement is necessary; just be sure you can explain why you give the estimate you do.
      Compound table for those who submitted data:

      Weight $\Delta t_{\rm run}$ [s] $\Delta t_{\rm walk}$ [s]
      51.8 kg 30 64
      74.8 kg 18 60
      79 kg 18  
      57 kg   103
      59 kg 32 65
      140 lb 25  
      135 lb 25 64
      90 kg 69 110
      135 lb 30 68
      56 kg 24 106
      120 lb 17 61

    6. Make a log-log graph for each of $P_{\rm walk}$, $P_{\rm run}$, $\Gamma_{\rm walk}$, $\Gamma_{\rm run}$, $\Gamma_0$ and $X_{\rm run}$ as functions of body mass M.  ANSWERThis part will also be different for each group.

  2. SO SWEET SO MEAN: A hummingbird weighing M = 3.9 g visits 1000 flowers daily and thereby collects nectar with an energy content of $\Delta H = 7-12$ kcal [see R. Conniff 2000]11
    1. Take an average value of $\Delta H = 9 \pm 2$ kcal. What is the [sugar and]12 honey content of the nectar? (Honey has about 14/15 of the heat of combustion of sugar.)  ANSWER Let's treat nectar as a solution of honey in water. The specific heat of combustion of glucose is hg= 4.15 kcal/g, so the specific heat of combustion of honey is $ h_h \approx h_g \times (14/15) = 3.87$ kcal/g. $\Delta H = \Delta M \times h_h$ so \fbox{ $\Delta M \approx 2.3$~g }.
    2. Determine the metabolic rate $\Gamma$ of the little bird, and estimate its mechanical power output P. Assuming the metabolic rate function $\Gamma_0 = a \; M^{3/4}$ is applicable, determine the constant a in that rate function.  ANSWER Basal metabolic rate scales as $\Gamma_0 = a \times M^{0.75}$; the average daily metabolic rate for the hummingbird can be expressed as $\Gamma_{\rm daily} = \Delta H/\Delta t$ and therefore $\Gamma_{\rm daily} = 0.44$ W. using 4185 J/kcal with $\Gamma_{\rm daily} = a \times \Gamma_0$ and $M_{\rm hummingbird} = 0.0039$ kg, we find $a \approx 28$. Therefore \fbox{ $P = \eta \times \Gamma_{\rm daily} = 0.11$~W }, using again $\eta \approx 0.25$.
    3. Compare your calculated value of a with the constant $a_0 \approx 4$ of the mouse to elephant curve [Eq. (1.5) in the textbook] and determine the ratio r=a/a0. Should this ratio be equal to the activity factor b calculated for the staircase run?  ANSWER \fbox{ $r \equiv a/a_0 \approx 28/4 = 7$\ }.
      It is reasonable to assume that metabolic activity factors vary among species and also vary among different activities. However, $b \equiv \Gamma/\Gamma_0$ refers to the $\Gamma_0$ for that animal, so r is not b.
    4. What problems can you foresee for such a high metabolic rate?  ANSWER The high metabolic rate of hummingbirds is attributed to their characteristic hovering flight and small body size. This is probably accompanied by a very large daily energetic demand. Hummingbirds also posses the greatest mass specific power for muscle: 98 W/kg.13
    5. Calculate the specific metabolic rate $\gamma \equiv \Gamma/M$ for the hummingbird and for a 5-ton elephant. Which animal makes better use of the energy resources of the environment?  ANSWER The specific metabolic rate for the hummingbird is \fbox{ $\gamma \equiv \Gamma_{\rm daily}/M = 113$~W/kg }. For an elephant, assuming M = 5000 kg and $\Gamma_0 = 4 M^{0.75}$, $\gamma_0 = \Gamma_0/M = 0.48$ W/kg at rest, which even with an activity factor of b = 10 would not get higher then roughly \fbox{ 5~W/kg }. Pound for pound, the elephant utilizes energy far more efficiently.

  3. HOT DEFENCES: Giant hornets like to eat bee larvae and honey. They are so strong that they can just invade a beehive and kill the guards at the entrance and get at their favourite food. A certain strain of Japanese honey bees has found a thermodynamic defence. They can tolerate a temperature of $47.2^\circ$C ($118^\circ$F). The hornets however can only stand $46.1^\circ$C ($115^\circ$F). The bees have learned to raise their body temperature to $47.2^\circ$ ($117^\circ$F): they humm while contracting and relaxing their flight muscles, and only generate heat without producing external mechanical work . . . and thereby steam the hornets in their own juice. Take a specific muscle power of p = P/M = 150 W/kg. The specific heat of tissue is close to that of water. Assume that the bees normally have a body temperature of $43^\circ$C and that 10% of the body weight of a bee is muscle.

    1. How much heat energy must be generated by each bee to reach the killing temperature?  ANSWER Heat energy: $\Delta Q = \Delta T \times c \times M_b$. The specific heat of tissue is about that of water: $c \approx 4.186$ kJ kg-1 K-1. Thus with $\Delta T = 47^\circ - 43^\circ$C $= 4^\circ$C and $M_b \approx 0.0002$ kg, we have $\Delta Q = 4 \times 4.186 \times 10^3 \times 2 \times 10^{-4}$ or \fbox{ $\Delta Q \approx 3.35$~J }.
    2. What is the heating power of each bee?  ANSWER The mechanical power is the efficiency times the heating power: $ P_{\rm mech} = \eta \times \dot{Q}$. The specific muscle power $p = P_{\rm mech}/M_m \approx 150$ W/kg, and so if $M_m \approx 0.1 M_b$, with Mb = 0.0002 kg we get the bee's muscle weight $M_m \approx 2 \times 10^{-5}$ kg, giving a muscle power $P_{\rm mech} = p \times M_m \approx 0.003$ W. Thus \fbox{ $\dot{Q} = P_{\rm mech} / \eta \approx 0.012$~W }.
    3. How quickly do the bees reach the killing temperature?  ANSWER The flow of energy is identified with metabolic rate, in this case a heating metabolic rate: $\Delta Q = \Gamma_{\rm heating} \times \Delta t$ with $\Delta Q = 4.8 $ J. Again, $\Gamma_0 = 4 \times M^{0.75}$ and since $\Gamma_{\rm heating} = b \times \Gamma_0$, b = 15 gives $\Gamma_{\rm heating} = 0.1$ W. Thus $\Delta t = \Delta Q/ \Gamma_{\rm heating} \approx 48$ s or \fbox{ $\Delta t = \Delta Q/\dot{Q} = 275$~s }.
      Alternatively: $\dot{Q} = c \times M_b (dT/dt)$; in this case, 0.012 W = 4.186 J/kg x 0.0002 kg x (dT/dt) giving $dT/dt = 0.0143^\circ$C/s. Thus $0.0143^\circ$C/s $\times \Delta t = 4^\circ$C implies \fbox{ $\Delta t = 279$~s }.



Jess H. Brewer 2007-02-17