Q: A biologist is comparing the heart rates of a horse ($$m = 500 kg$$, $$R = 38$$ bpm) and a hamster ($$m = 0.05 kg). If $$heart rate scales as $$R \propto m$$^{-1/4}$$, what is the expected heart rate of the hamster?

$$38 \times (0.05/500$$)^{-1/4}$$ bpm

Question 1

Which of the following best describes the role of models in physics, especially when applied to biological systems?

Accepted Answer

A model is a simplified picture of reality that still captures the essence of what you want to study.

Question 2

In a random walk model of molecular diffusion, a large number of molecules start at the origin and each step is of length $$d in a $$random direction along the $$x-$$axis. After $$100$$ steps, what is the most likely final position of a molecule?

Accepted Answer

$$x = 0$$

Question 3

After $$100$$ steps of a random walk, each of length $$d$$, what is the typical diffusion distance from the origin?

Accepted Answer

$$10d$$

Question 4

The root mean square (rms) distance for a random walk of $$n$$ steps, each of length $$d$$, is given by:

Accepted Answer

$$x_{rms} = \sqrt{n}d$$

Question 5

If the maximum possible travel distance for a molecule after $$n$$ steps is $$nd$$, what happens to the ratio of the diffusion distance to the maximum possible distance as $$n$$ increases?

Accepted Answer

It decreases as $$1/\sqrt{n}.$$

Question 6

The diffusion coefficient $$D$$ for a molecule in one dimension is given by $$D = \frac{1}{2}vd$$, where $$v is $$the molecular speed and $$d is $$the step size. If the step size $$d$$ increases but $$v$$ remains the same, what happens to $$D$$?

Accepted Answer

It increases.

Question 7

Suppose the diffusion coefficient $$D$$ for small molecules in water is $$8 	imes 10$$^{-10}$$ 	ext{ m}^2/	ext{s}. $$For the same molecules in air, where the step size $$d is $$much larger, what can you say about $$D in $$air compared to water?

Accepted Answer

It is larger than in water.

Question 8

An animal's mass $$m is $$proportional to the cube of some linear dimension $$l$$ ($$m \propto l^3$), while the strength of its bones is proportional to the square of that dimension ($$	ext{Strength} \propto $$l^2). If an $$animal doubles in size ($$l$$ doubles), by what factor does its mass increase compared to its bone strength?

Accepted Answer

Mass increases by a factor of $$8$$, bone strength by $$4.$$

Question 9

The basal metabolic rate (BMR) of mammals scales with mass $$m as$$ $$	ext{BMR} \propto m^r. On a log-log $$graph of BMR versus mass, what does the slope of the line represent?

Accepted Answer

The exponent $$r in $$the scaling law.

Question 10

If the heart rate $$R of $$mammals scales with body mass $$m as$$ $$R \propto m$$^{-1/4}$$, what happens to the heart rate when the mass increases by a factor of 16$?

Accepted Answer

It decreases by a factor of $$2.$$

Question 11

The strength of the electric force between two ions is inversely proportional to the square of the distance between them ($$F \propto 1/d^2$). If the force is 1.00$ nN at distance d$, what is the force when the distance doubles?

Accepted Answer

$$0.25 nN$$

Question 12

Which of the following is NOT a major goal for students studying physics for the life sciences, according to the text?

Accepted Answer

Memorize all historical experiments in physics.

Question 13

Why do physicists often use simple models when studying complex biological systems?

Accepted Answer

Simple models help capture the essential features and allow for quantitative analysis.

Question 14

Which mathematical property of logarithms is used to linearize power-law relationships for plotting on log-log graphs?

Accepted Answer

$$\log(a^n) = n \log(a)$$

Question 15

A biologist is comparing the heart rates of a horse ($$m = 500 kg$$, $$R = 38$$ bpm) and a hamster ($$m = 0.05 kg). If $$heart rate scales as $$R \propto m$$^{-1/4}$$, what is the expected heart rate of the hamster?

Accepted Answer

$$38 	imes (0.05/500$$)^{-1/4}$$ bpm

Question 16

Which of the following is a reason why scaling laws are important in biology?

Accepted Answer

They allow us to predict how physical characteristics change with size.

Question 17

If the average position of molecules after a random walk is $$x_{avg} = 0$$, why is the root mean square (rms) distance a better measure of spread?

Accepted Answer

Because rms distance accounts for both positive and negative displacements.

Physics for the Life Sciences: Foundations and Quantitative Reasoning

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