PHYS3511-Biological Physics
Winter 2015, Assignment #2
Assigned on Monday 19 January 2015. Due on Monday 26 January 2015.
MANDATORY READING: 1) Energy and its conservation, Chapter 7 pp. 160-179; 2)
Section 10.1, pp. 237-239 on quantifying metabolic processes.
Question 1 Problem P7.16 of chapter 7 (page 184)
Question 2 Problem P7.20 of Chapter 7 (page 184).
Question 3 Basal Metabolic Rate (BMR): Metabolism is a generic term for all chemical
reactions that break down food thereby releasing energy. Below are some data for
metabolism and gas exchange in humans.
food
kcal/g
litre O2/g
Litres CO2/g
carbohydrate
4.1
0.81
0.81
Fat
9.3
1.96
1.39
Protein
4.0
0.94
0.75
alcohol
7.1
1.46
0.97
The table gives the energy released, the oxygen consumed, and the carbon dioxide
released upon metabolizing the food per gram of food.
a) Calculate the energy yield per litre of oxygen. Note that this is roughly constant.
NOTE for students: This shows that a person’s metabolic rate can be found by
measuring his/her oxygen consumption. In contrast, the CO2/O2 ratio depends on
the food groups, and can be used to determine what is being used as an energy
source.
b) An average adult uses 16 litres of O2 per hour. The corresponding “heat released”
is called the basal metabolic rate (BMR). Calculate the BMR per hour and per
day. What power output does this corresponds to in watts?
Question 4 A Tour de France rider consumes 8000 kcal of food daily. Suppose every day
he rides his bicycle at constant speed of 25 km/ hr for six hours.
A) Suppose the “drag” force on the cyclist is Fdrag = Bv 2 , where B = 1.5 kg/m.
Calculate the work done to overcome the drag. Would this account for the
discrepancy 8000 kcal of energy consumed?
B) Can the discrepancy in part A be accounted for by riding up the mountainside? To
answer this, determine the change in altitude necessary to account for the
difference in part A.
C) Regardless of your answer in part B, assume that the unaccounted energy goes to
evaporating water. How much water would the rider have to drink? Is this
reasonable?
Question 5 For an object of “size” r moving with a speed of v in a fluid, the Reynolds
number is defined as ℜ = vr ρ / η , where ρ is the density of the fluid, and η is the
coefficient of viscosity of the fluid. A low value of ℜ ≪ 1 corresponds to a regime where
friction dominates. Stirring produces the least possible response, namely, laminar flow
(engineers use creeping flow). When ℜ is big there’s turbulent flow.
A) Calculate the ℜ for an E. Coli of “size” 1µ m moving with speed of 30 µ m / s in
water. Comment on the significance. ηwater = 10 −3 kg / m • s
B) Calculate the ℜ for a whale that weigh 27000kg swimming at 10 m/s in water.
Assume the whale is a “square” made up mainly of water. Comment.
C) Calculate the ℜ for a human swimming in vegetable shortening at speed of 0.1
m/s. For vegetable shortening use η = 1200kg / m • s , and density similar to water.
Question 6 Consider two swimming organism.
A) An E. Coli of radius about r = 0.7 µ m is observed to swim at v = 80 µ mis −1 .
Estimate the amount of ATP that it must produced per second, in order for the E.
Coli to swim at this rate. Read section 10.1.2 (page 239) on the amount of ATP
produced by E. Coli. Does an E. Coli have the capacity to produce enough ATP to
swim this fast? From the discussion in the class does an E. Coli need to produce
ATP to swim? Briefly explain.
B) Consider now a paramecium of radius r = 100 µ m that swims at v = 3mmis −1 .
Estimate the amount of ATP needed per second. An online site claims that a
paramecium can produced 20 times the amount of ATP as an E. Coli. Based on
this claim, your calculation, and on what you read in section 10.1.2, would a
paramecium be able to produce enough ATP to swim this fast. Briefly explain.
Assume the drag force is fdrag = ζ v , valid for low Reynolds number (see question 5)
with ζ = 6πηr , with the viscosity of water η = 10 −3 kg i m −1 i s −1 .
Question 7 In the class lecture on the Principle of Flagellar Propulsion of Bacteria, I
discussed how the flagellar is modeled as a rod in helical conformation rotating about the
α Rωη L
helical axis. This produced a propulsion force of Fpropulsion =
sin ( 2θ helix ) ,
6
where α is a constant, R is the radius of the helix, ω the angular velocity of the rotating
helix, η = 10 −3 kg / m • s is the viscosity of water, L is the total length of the flagellar, and
θ helix = tan −1 ( 2π R / P ) with P the pitch of the helix. The torque that must be applied to
(
)
−1
α R 2ωη L
2 + (1 + tan 2 θ helix ) .
induce and maintain and angular velocity of ω is τ =
3
Now consider a cylindrical E-coli with flagella of length L = 10 µ m , in a helical
conformation with helix radius R = 0.2 µ m and pitch P = 2.3µ m . The E. coli is observed
to swim with a speed of v = 80 µ m / s . In all calculations, take α = 3 and the “radius” of
the E. Coli to be recoli = 0.7 µ m . However, as opposed to Question 7, assume that the drag
force obeys fdrag = vς eff , with the effective friction coefficient ζ eff = ζ body + ζ flagellar , where
ζ body = 6πηrecoli is due to the body, and ζ flagellar = αη L .
A) Calculate propulsion force Fpropulsion and the output power, Pout , that propels the E.
Coli. Compare this to the answer in part A of question 6, and briefly comment.
B) Calculate the torque produced by the E. Coli, and the corresponding input power,
Pin , by the E. Coli. HINT: Begin by using fdrag = Fpropulsion to calculate ω .
C) Calculate the efficiency of the E. Coli swimming machine, eff = ( Eout / Ein ) × 100% .
Compare your answer to E.M. Purcell “hand-waving” estimate of ∼ 1% .
Interested students can download E.M. Purcell’s classic paper:
http://alinhana.lakeheadu.ca/Purcell_life_at_low_reynolds_number.pdf