Physics 114 (Exam 1)

Allometric Scaling

Shape changes with size (i.e., the relative proportions of limbs, organs, etc. change). Proportions change with growth.

Isometric Scaling

When an organisms body parts maintain proportional relationships as they grow.

What function are allometric relationships represented by?

Power law functions.

Positive Allometry. Explain?

y = ax^b…b is greater than one, so as x increases y increases at a faster rate.

Negative Allometry. Explain?

y = ax^b…b is less than one but greater than 0, so as x increases, y increases but at a lesser rate.

Isometry…Explain?

y = ax^b…b is equal to 1 so as x increases y increases at the same rate.

What is the ratio of H1/H2?

H1 and H2 are the same as L1 and L2, so L2/L1 is equal to KL, so that means that H2/H1 is equal to KL

• we want H1/H2 so we do 1/(H1/H2) to flip it, so we have to do 1/KL too.

• This gives us H1/H2 = 1/KL

What are the steps in combining power laws?

1. Identify the relationship you want. (for example you want to identify V to SA^b).

2. Find relationships for the two quantities you want to relate in terms of a third quantity. (for example V = size³ and SA = size²)

3. Take the quantity you want raised to the power of b (SA) and the common third quantity (size) and solve for the third quantity related to your powered one (for example SA = size² so size = SA^1/2).

4. Plug the relationship from step 3 into your relationship for your first quantity! (Volume = size³ and size = SA^1/2 so plug in Volume = (SA^1/2)³ so V = SA^3/2

What are the three proportionalities we need to know?

Surface Area = Size²

Volume = Size³

Mass = Size³

What would length be in a relationship of like surface area and size?

Length would be size, and it is a linear dimension that scales equally if it is isometric.

Compare SA/V to mass…

1. SA/V = Size^-1

2. Mass = Size³

3. Size = Mass^1/3

4. Plug in Mass^1/3 as size in the SA/V

5. SA/V = (Mass^1/3)^-1

6. SA/V = Mass^-1/3

2 Frogs, 1 Frog has leg length of 8cm and a mass of 1kg. Frog 2 has a leg length of 4cm. What is the mass of frog 2?

1. Mass = size³

2. (Size1/Size2)³ = Mass1/Mass2

3. Plug in: (8cm/4cm)³ = 1kg/x

4. 2³ = 1/x

5. x = 1/2³

6. x = 0.125 = 1/8

A dogs mass doubles over a period of 18 months. The SMR (MR/mass) is proportional to mass^-1/4. Relate MR when dog is large vs when dog is small.

1. SMR = MR/mass, its proportional to Mass^-1/4

2. (SMR Mass) = (Mass^-1/4)* Mass^1

3. MR = Mass³/4

4. Mass = MR^4/3

5. Mass * 2 = (MR * x)^4/3

6. Mass 2 = MR^4/3 * x^4/3

7. 2 = x^4/3

8. x = 2³/4 = 1.7

9. Mass 2 (Mass large) = MR large = MR small * 1.7

10. MRlarge = MRsmall*1.7

What does the R² value tell us?

An R² value tells us how well a trend line fits the data. An R² value very close to 1 means that the trend line fits the data almost PERFECTLY, an R² value further from 1 means the trend line fails to match the data well.

What is the linear function? What graphs fit this type best?

y = mx + b

• linear graphs fit this function the best.

What is a power law function? What graphs fit this function the best?

y = ax^b

• Log to log plots fit this function the best.

What is an exponential function? What graphs fit this function the best?

y = na^kx

• Semi-log plots fit this function the best.

What does it mean when you say a plot fits the function the best?

It means that when the function is graphed on that type of plot (exponential, log-log, semi-log) the data falls into a straight line resembling a linear y=mx+b line with a high R² value and reasonable data points.

What are the three log rules we deal with?

1. Log(x*y) = Log(x) + Log(y)

2. Log_b(a) →b^log_b(a) → a

3. Log(x^b) → bLog(x)

What is the base of a natural log? What is the base of a log? What is the base of a lg?

• Natural log is base “e”.

• Log is normally base “10”.

• Lg is log base “2”.

Walk me through how you would convert a power law function (y = ax^b) into a y = mx + b type of line?

• Power law, so you would log both sides.

  • Log(y) = Log(ax^b)

  • Log(y) = Log(a) + Log(x^b)

  • Log(y) = bLog(x) + Log(a)

  • Y. =m x. + b

Walk me through how you would convert a exponential function (y = na^kx) into a y = mx+b type of line?

• Exponential so you want to do semi-log, so Log both sides but your x will not end up logged.

  • y = na^kx

  • log(y) = log(na^kx)

  • log(y) = log(a^kx) + log(n)

  • log(y) = klog(a)*x + log(n)

  • y. = m. x. + b

IMPORTANT!!! Your slope in this is kLog(a), that entire part is the slope because it is being multiplied by your x, and IT will not change, you’re x is changing.

• In power law, you have your x as a log so blog(x) and that means that b is just your slope. In this one, your x is alone so anything before it is the slope.

Convert f = 0.32b^1.28 into its best y = mx + b form…

1. log(f) = log(0.32b^1.28)

2. log(f) = log(0.32) + log(b^1.28)

3. log(f) = log(0.32) + 1.28log(b)

Convert log(h) = 0.72log(x)-0.30 into its y = ax^b form.

1. log(h) = 0.72log(x) - 0.30

2. h = log(x^0.72) + log(10^-0.30)

3. h = log(10^-0.30*x^0.72)

4. h = 0.5x^0.72

convert n=1000e^0.41t into a semi-log ln plot of y = mx + b

ln(n) = ln(1000e^0.41t)

ln(n) = ln(1000) + ln(e)^0.42t

ln(n) = ln(1000) + 0.42t

ln(n) = 0.42t + 6.9

Convert log(n) = 0.22t + 3.7 into the form y = na^kx…

n = log(10^3.7) + log(10^0.22t)

n = log(10^3.7×10^0.22t)

n = 10^3.7× 10^0.22t