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16 CHAPTER 1. PROBLEM-SOLVING STRATEGIES

1.5 Problems

1.1. Furlongs per fortnight squared
2
2
Convert g = 9.8 m/s into furlongs/fortnight . A fortnight is two weeks, a furlong is 220
yards, and there are 1.09 yards in a meter.
1.2. Miles per gallon
The efficiency of a car is commonly rated in miles per gallon, the dimensions of which are
length per volume, or equivalently inverse area. What is the value of this area for a car that
gets 30 miles per gallon? What exactly is the physical interpretation of this area? Note:
There are 3785 milliliters (cubic centimeters) in a gallon, and 1609 meters in a mile.
y
1.3. Painting a funnel
Consider the curve y = 1/x, from x = 1 to x = ∞. Rotate this curve around the x axis to
create a funnel-like surface of revolution, as shown in Fig. 1.5. By slicing up the funnel
x into disks with radii r = 1/x and thickness dx (and hence volume (πr ) dx) stacked side
2
by side, we see that the volume of the funnel is

∫ ∞
∞ π π
V = 2 dx = − = π, (1.6)
Figure 1.5 1 x x 1
which is finite. The surface area, however, involves the circumferential area of the disks,
√
′2
which is (2πr) dx multiplied by a 1 + y factor accounting for the tilt of the area. The
surface area of the funnel is therefore
√
∫ ∫
∞ 2π 1 + y ′2 ∞ 2π
A = dx > dx, (1.7)
1 x 1 x
which is infinite. (The square root factor is irrelevant for the present purposes.) Since the
volume is finite but the area is infinite, it appears that you can fill up the funnel with paint
but you can’t paint it. However, we then have a problem, because filling up the funnel with
paint implies that you can certainly paint the inside surface. But the inside surface is the
same as the outside surface, because the funnel has no thickness. So we should be able to
paint the outside surface too. What’s going on here? Can you paint the funnel or not?

1.4. Planck scales
2
Three fundamental physical constants are Planck’s constant, ~ = 1.05 · 10 −34 kg m /s; the
2
3
8
gravitational constant, G = 6.67·10 −11 m /(kg s ); and the speed of light, c = 3.0·10 m/s.
These constants can be combined to yield quantities with dimensions of length, time, and
mass (known as the Planck length, etc.). Find these three combinations and the associated
numerical values.
1.5. Capillary rise
If the bottom end of a narrow tube is placed in a cup of water, the surface tension of the
water causes the water to rise up in the tube. The height h of the column of water depends
on the surface tension γ (with dimensions of force per length), the radius of the tube r, the
mass density of the water ρ, and g. Is it possible to determine from dimensional analysis
alone how h depends on these four quantities? Is it possible if we invoke the fact that h
is proportional to γ? (This is believable; doubling the surface tension γ should double the
height, because the surface tension is what is holding up the column of water.)
1.6. Fluid flow
Poiseuille’s equation gives the flow rate Q (volume per time) of a fluid in a pipe, in the case
where viscous drag is important. How does Q depend on the following four quantities:
the pressure difference ∆P (force per area) between the ends of the pipe, the radius R
and length L of the pipe, and the viscosity η (with units of kg/(m s)) of the fluid? To