# Problems for Chapter 1

These problems ask you to construct relationships using units. The resulting equations will be of use to us later in the course. Remember that you may have to put some quantities in the denominator of the right hand side of the equation, and that some quantities there may need to be squared.

1-1. The pressure underwater is related to the depth and the density of the water. It is also proportional to the acceleration due to gravity, which has dimensions of length over time squared. Write an equation for it.

This is actually a pretty hard problem for beginners. Look up the SI units for pressure (and force). And remember, the idea is to treat units as numbers, multiplying and dividing both sides of an equation by them until you get what you want.

1-2. The flow of fluid through a pipe (volume / time) is related to the pressure difference between the head and tail ends of the pipe, the length of the pipe, the fourth power of the pipe's radius, and a dimensionful constant in the denominator called the viscosity. Larger pressure differences produce greater flow; longer pipes produce less flow, and narrower pipes produce less flow. What are the dimensions of viscosity?

1-3. The distance an object falls in a given amount of time is a function of the time and the acceleration due to gravity. The constant of proportionality is 1/2. Find the equation.

1-4. We say an object's "potential energy" increases as its height increases. If energy has dimensions of mass times velocity squared, and the potential energy is also a function of the object's mass and the acceleration due to gravity, what is the equation which relates these quantities?

1-5. An object rotating around an axis experiences an acceleration which depends on its velocity and its distance from the axis. Write an equation for it.

1-6. The acceleration in problem 1-5 can also be expressed in terms of its rate of revolution (revolutions per unit time, with dimensions of one over time) instead of its velocity. Write this version of the equation.

1-7. The force which buoys an object up that is floating in a liquid depends on the mass of the object, the density of the medium, the specific volume of the object (volume per unit mass) and the acceleration due to gravity. Write an equation for the force if the force is proportional to the density of the liquid. The dimensions of force are mass times acceleration.

This set of problems will exercise your use of units and metric conversions. To make sense of the data, you only need think of a DNA molecule as a twisted ladder: each rung on the ladder corresponds to a nucleotide pair. The genome of an organism is the number of nucleotide pairs needed to hold the information necessary to reproduce the organism.

Given the following information:

OrganismShapeDiameter (mm)Length (mm)Genome Size
T4 bacteriophagecylinder.05.11.7 x 105
E. Coli cellcylinder123.5 x 106
human liver cellsphere20 3 x 109

A nucleotide pair in a DNA molecule can be thought of as a cylinder with a diameter of 1.58 x 10-3 microns and a length of 3.34 x 10-4 microns: The density of human tissue is approximately 1.071 g / cc, and the molecular weight of a nucleotide pair is about 660 (6.02 x 1023 nucleotide pairs weighs 660 g).

1-8. Compute the volumes of the organisms.

1-9. What is the volume ratio of a 70 kg person to each of the organisms?

1-10. What percentage (by volume) of each of the organisms is taken up by its DNA?

1-11. What is the ratio of DNA length to organism length for each of the organisms?

1-12. Which of the quantities do you feel best characterizes each of the organism's complexity: volume, % DNA by volume (problem #10) or ratio of DNA length to organism length (problem #11)? Why? Think in terms of searching for additional meaning in a set of numbers; look at order as well as scale. What effect would the presence of repeated or unused segments in the genome have on this question?

1-13. Compute the average molecular weight of human tissue (look up the elemental composition of the human body and the atomic weights of the elements involved).

1-14. Compute the density of DNA. Now consider the question: can you use tissue density to obtain an estimate of the percent of a human body that is water? Pretend that human tissue is made solely of water and DNA.

Hint: If the fraction of your body that is water is x, the fraction that is DNA is 1 - x.
Why is the value you would get from the quantity inaccurate? Will the value be too large or too small? Try using molecular weight instead of density, and evaluate the result.

1-15. E. Coli DNA duplicates itself during cell division in 20 minutes. To do this, each group of 10 pairs must unwind one full turn. What rate of rotation is necessary for this? What is the linear speed of movement of the unwinding pairs (how fast is the outside of a pair moving) ?

1-16. Proteins are named by function and not by their exact chemical composition. Research the various molecular weights which have been reported for immunoglobulin. Using at least 20 different values, perform the uncertainty computations for both absolute and relative error in the values. Does the average mean anything here?

1-17. The radius of the Earth is about 6.4 x 10 3 km. The radius of the orbit of the Moon around the Earth is about 3.8 x 10 5 km, and it takes about 27 days for the Moon to complete one orbit. Assume that the Sun is very far away from the Earth. How long is the Moon in the Earth's shadow during a total lunar eclipse?

## Need more problems about units?

If you have stumbled on this page, and the equations look funny (or you just want to know where you are!), see the College Physics for Students of Biology and Chemistry home page.