GALILEO: SCALING What changes when we alter the physical sizes of objects other than the physical sizes themselves? Why are there no giants?

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Presentation transcript:

GALILEO: SCALING What changes when we alter the physical sizes of objects other than the physical sizes themselves? Why are there no giants?

VOLUME OF ONE MILLION GRAINS OF RICE 1 m m m m m 3

SCALING A CUBE L kL A 1 = 6L 2 V 1 = L 3 A = 6k 2 L 2 = k 2 A 1 V = k 3 L 3 = k 3 V 1 Note: L ~ V 1/3 ~ A 1/2 and A ~ V 2/3

KING KONG Multiply the dimensions of a gorilla by 10. Then its weight is multiplied by 1000, and bone cross sectional area by 100.

EXPECTED BONE SHAPE VARIATION WITH ANIMAL MASS If bone cross sections are determined by animal weight, then we expect cross sectional area A ~ m where m is the total animal mass. Bone length should be proportional to m 1/3 So skeletal mass should be proportional to (m 1.0 )*(m 0.33 ) = m 1.33

GENERAL SCALING EQUATION y = ax b y = quantity being studied x = quantity y varies with a = coefficient b = exponent log(y) log(x) Graph on a logarithmic scale is a straight line with slope equal to b

ACTUAL BONE SHAPE VARIATION WITH ANIMAL MASS

RESTING METABOLIC RATES OF MAMMALS Maintains body temperature Heat loss proportional to surface area ~ M 2/3 So we expect Metabolic Rate ~ M 2/3

RESTING METABOLIC RATES VS MAMMAL SIZE

RESTING METABOLIC RATE FOR HUMANS Metabolic Rate (Watts) = 4M 0.75 For a 70 kg person, Rate = 70 Watts

SPECIES IDEA Metabolic Rate Mass Each Species b = 0.67 All Species b = 0.75

BODY SURFACE AREA VS MASS

SKIN AREA FOR HUMANS Area = 0.1M 0.67 with area in m 2, M in kg. For M = 70 kg, Area = 1.76 m 2

ENERGY COST OF RUNNING Work done by leg muscles per step = (Force)*(Contraction distance) ~ (M 2/3 )*(M 1/3 ) ~ M Step length ~ M 1/3 so steps per km ~ M -1/3 Work per km ~ (M)*(M -1/3 ) ~ M 0.67 So work per kg per km ~ (M 0.67 )*(M -1 ) ~ M -.33

COST OF RUNNING VS BODY WEIGHT

COST OF RUNNING FOR HUMANS Energy cost = 1 Cal/kgkm = 70 Cal/km for 70 kg

HOW HIGH CAN YOU JUMP? Energy needed to jump height h = mgh ~ M Energy available from muscles also ~ M So height h is independent of body size! Jumpermassheight Flea0.5 mg20 cm Click Beetle40 mg30 cm Locust3 g59 cm Man70 kg60 cm

COULD TYRANNOSAURUS REX RUN?