5.6: Summary and further problems

A quantity with dimensions has no meaning by itself. As Socrates might have put it, the uncompared quantity is not worth knowing. Using this principle, we learned to rewrite relations in dimensionless form: in terms of combinations of the quantities where the combination has no dimensions. Because the space of dimensionless relations is much smallerthan the space of all possible relations, this rewriting simplifies many problems. Like the othertwo tools in Part II, dimensional analysis discards complexity without loss of information.

Problem 5.46: Oblateness of the Earth

Oblateness of the Earth

Because the Earth spins on its axis, it is an oblate sphere. You can estimate the oblateness using dimensional analysis and some guessing. Our measure of oblateness will be \(\Delta R = R_{eq} - R_{polar}\) (the difference between the polar and equatorial radii). Find two independent dimensionless groups built from \(\Delta R\), g, R (the Earth’s mean radius), and \(v\) (the Earth’s rotation speed at the equator). Guess a reasonable relation between them in order to estimate \(\Delta R\). Then compare your estimate to the actual value, which is approximately 21.4 kilometers.

Problem 5.47: Huge waves on deep water

One of the highest measured ocean waves was encountered in 1933 by a US Navy oiler, the USS Ramapo (a 147-meter-long ship) [34]. The wave period was 14.8 seconds. Find its wavelength using the results of Problem 5.11. Would a wave of this wavelength be dangerous to the ship?

Problem 5.48: Ice Skating

For world-record ice skating, estimate the power consumed by sliding friction. (Ice skates sliding on ice have a coefficient of sliding friction around 0.005.) Then make that power meaningful by estimating the ratio

\[\frac{\textrm{power consumed in sliding friction}}{\textrm{power consumed by air resistance}}.\]

Problem 5.49: Pressure melting during ice skating

Water expands when it freezes. Thus, increasing the pressure on ice should, by Le Chatelier’s principle, push it toward becoming water—which lowers its freezing point. Based on the freezing point and the heat of vaporization of water, estimate the change in freezing point caused by ice-skate blades. Is this change large enough to explain why ice-skate blades slip with very low friction on a thin sheet of water?

Problem 5.50: Contact radius

A solid ball of radius R, density \(\rho\), and elastic modulus Y rests on the ground. Using dimensional analysis, how much can you deduce about the contact radius r?

Problem 5.51: Contact time

The ball of Problem 5.50 is dropped from a height, hits a hard table with speed \(v\), and bounces off. Using dimensional analysis, how much can you deduce about the contact time?

Problem 5.52: Floating on water

Some insects can float on water thanks to the surface tension of water. In terms of the bug size l (a length), find the scaling exponents \(\alpha\) and \(\beta\) in

\[F_{\gamma} \: \propto \: l^{\alpha}, \]

\[W \: \propto \: l^{\beta},\]

where \(F_{\gamma}\] is the surface-tension force and W is the bug’s weight. (Surface tension itself has dimensions of force per length.) Thereby explain why a small-enough bug can float on water.

Problem 5.53: Dimensionless measures of damping

A damped spring–mass system has three parameters: the spring constant k, the mass m, and the damping constant \(\gamma\). The damping constant determines the damping force through \(F \gamma = \gamma v\), where \(v\) is the velocity of the mass.

a. Use these quantities to make the dimensionless group proportional to \(\gamma\). Mechanical and structural engineers use this group to the define the dimensionless damping ratio \(\zeta\) :

\[\zeta \equiv \frac{1}{2} \times \textrm{ the dimensionless group proportional to } \gamma.\]

b. Find the dimensionless group proportional to \(\gamma^{-1}\). Physicists and electrical engineers, following conventions from the early days of radio, call this group the quality factor Q.

Problem 5.54: Steel cable under its own weight

The stiffness of a material should not be confused with its strength! Strength is the stress (a pressure) at which the substance breaks; it is denoted \(\sigma_{y}\). Like stiffness, it is an energy density. The dimensionless ratio \(\sigma_{y}/\sigma\), called the yield strain \(\epsilon_{y}\), has a physical interpretation: the fractional length change at which the substance breaks. For most materials, it lies in the range 10⁻³…10⁻²—with brittle materials (such as rock) toward the lower end. Using the preceding information, estimate the maximum length of a steel cable before it breaks under its own weight.

Problem 5.55: Orbital dynamics

A planet orbits the Sun in an ellipse that can be described by the distance of closest approach r_min and by the furthest distance r_max. The length l is their harmonic mean:

\[l = 2 \frac{r_{min}r_{max}}{r_{min}+r_{max}} = 2(r_{min} \parallel r_{max}).\]

(You will meet the harmonic mean again in Problem 8.22, as an example of a more general kind of mean.)

The table gives r_min and r_max for the planets, as well as the specific effective potential V, which is the effective potential energy divided by the planet mass m (the effective potential itself mixes the gravitational potential energy with one component of the kinetic energy). The purpose of this problem is to see how universal functions organize this seemingly messy data set.

	rmin (m)	rmax (m)	V (m^2s^-2)
Mercury	4.6001 x 10^10	6.9818 x 10^10	-1.1462 x 10^9
Venus	1.0748 x 10^11	1.0894 x 10^11	-6.1339 x 10^8
Earth	1.4710 x 10^11	1.5210 x 10^11	-4.4369 x 10^8
Mars	2.0666 x 10^11	2.4923 x 10^11	-2.9199 x 10^8
Jupiter	7.4067 x 10^11	8.1601 x 10^11	-8.5277 x 10^7
Saturn	1.3498 x 10^12	1.5036 x 10^12	-4.6523 x 10^7
Uranus	2.7350 x 10^12	3.0063 x 10^12	-2.3122 x 10^7
Neptune	4.4598 x 10^12	4.5370 x 10^12	-1.4755 x 10^7

a. On a graph of V versus r, plot all the data. Each planet provides two data points, one for r=r_min and one for r=r_max. The plot should be a mess. But you’ll straighten it out in the rest of the problem.

b. Now write the relation between V and r in dimensionless form. The relevant quantities are v, r, GM_Sun, and the length l. Choose your groups so that V appears only in one group and r appears only in a separate group.

c. Now use the dimensionless form to replot the data in dimensionless form. All the points should lie on one curve. You have found the universal function characterizing all planetary orbits

Problem 5.56: Signal propagation speed in coaxial cable

For the coaxial cable of Problem 2.25, estimate the signal propagation speed.

Problem 5.57: Meter stick under pressure

Estimate how much shorter a steel meter stick becomes due to being placed at the bottom of the ocean. What about a meter stick made of wood?

Problem 5.58: Speed of sound in water

Using the heat of vaporization of water as a measure of the energy density in its weakest bonds, estimate the speed of sound in water.

Problem 5.59: Delta-function potential

A simple potential used as a model to understand molecules is the one-dimensional delta-function potential \(V(x) = -E_{0}L\delta(x)\), where E₀ is an energy and L is a length (imagine a deep potential of depth E₀ and small width L). Use dimensional analysis to estimate the ground-state energy.

Problem 5.60: Tube flow

In this problem you study fluid flow through a narrow tube. The quantity to predict is Q, the volume flow rate (volume per time). This rate depends on five quantities:

l	the length of the tube
\(\Delta p\)	the pressure difference between the tube ends
r	the radius of the tube
\(\rho\)	the density of the fluid
\(\nu\)	the kinematic viscosity of the fluid

a. Find three independent dimensionless groups G₁, G₂, and G₃ from these six quantities—preparing to write the most general statement as

\[\textrm{group 1} = f(\textrm{group 2, group 3}).\]

Hint: One physically reasonable group is \(G_{2} = r/l\); to make solving for Q possible, put Q only in group 1 and make this group proportional to Q.

b. Now imagine that the tube is long and thin (l >>r) and that the radius or flow speed is small enough to make the Reynolds number low. Then deduce the form of f using proportional reasoning: First find the scaling exponent \(\beta\) in \(Q \: \alpha \: (\Delta p)\beta\); then find the scaling exponent \(\gamma\) in \(Q \: \alpha \: l \gamma\). Hint: If you double \(\Delta p\) and l, what should happen to Q?

Determine the form of f that satisfies all your proportionality requirements. If you get stuck, work backward from the correct result. Look up Poiseuille flow, and use the result to deduce the preceding proportionalities; and then give reasons for why they are that way

c. The analysis in the preceding parts does not give you the universal, dimensionless constant. Use a syringe and needle to estimate the constant. Compare your estimate with the value that comes from solving the equations of fluid mechanics honestly (by looking up this value).

Problem 5.61: Boiling versus boiling away

Look up the specific heat of water in the table of constants (p. xvii) and estimate the ratio

\[\frac{\textrm{energy to bring a pot of water to boiling temperature}}{\textrm{energy to boil away the boiling water}}.\]

Problem 5.62: Kepler's law for elliptical orbits

Kepler’s third law connects the orbital period to the minimum and maximum orbital radii r_min and r_max and to the gravitational strength of the Sun:

\[T = 2 \pi \frac{a^{3/2}}{\sqrt{GM_{Sun}}},\]

where the semimajor axis a is defined as \(a \equiv (r_{min}+r_{max})/2\). Write Kepler’s third law in dimensionless form, making one independent dimensionless group proportional to T and the other group proportional to r_min.

Problem 5.63: Why Mars?

Why did Kepler need data about Mars’s orbit to conclude that planets orbit the Sun in an ellipse rather than a circle? Hint: See the data in Problem 5.55.

Problem 5.64: Froude number for a ship's hull speed

For a ship, the hull speed is defined as

\[v = \equiv 1.34 \sqrt{l},\]

where \(v\) is measured in knots (nautical miles per hour), and l, the waterline length, is measured in feet. The waterline length is, as you might expect, the length of the boat measured at the waterline. The hull speed is a boat’s maximum speed before the water drag becomes very large.

Convert this unit-specific formula to an approximate Froude number Fr, the dimensionless numberintroduced in Section 5.1.1 to estimate the maximum walking speed. For the hull speed, the Froude number is defined as

\[\textbf{Fr} \equiv \frac{v^{2}}{gl}.\]

From the approximate Froude number, guess the exact value!