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Magazine Problem 32
(a) How many square centimeters are in 1 square foot? $(1 \text { in. }=2.54
\mathrm{cm} .)$ (b) How many square centimeters are in 1 square meter? (c)
Using your answers to parts (a) and (b), but without using your calculator,
roughly how many square feet are in one square meter?
Problem 33
A snail crawls at a pace of \(5.0 \mathrm{cm} / \mathrm{min.}\) Express the
snail's speed in (a) \(\mathrm{ft} / \mathrm{s}\) and (b) $\mathrm{mi} /
\mathrm{h}$.
Problem 34
An average-sized capillary in the human body has a cross-sectional area of
about \(150 \mu \mathrm{m}^{2} .\) What is this area in square millimeters
\(\left(\mathrm{mm}^{2}\right) ?\)
Problem 35
An equation for potential energy states \(U=m g h .\) If \(U\) is in joules, with
\(m\) in \(\mathrm{kg}, h\) in \(\mathrm{m},\) and \(g\) in $\mathrm{m} /
\mathrm{s}^{2},$ find the combination of SI base units that are equivalent to
joules.
Problem 36
One equation involving force states that \(F_{\mathrm{net}}=m a\) where \(F_{\text {net }}\) is in newtons, \(m\) is in \(\mathrm{kg}\), and \(a\) is in \(\mathrm{m} \cdot \mathrm{s}^{-2}\) Another equation states that \(F=-k x,\) where \(F\) is in newtons, \(k\) is in \(\mathrm{kg} \cdot \mathrm{s}^{-2},\) and \(x\) is in \(\mathrm{m} .\) (a) Analyze the dimensions of \(m a\) and \(k x\) to show they are equivalent. (b) What are the dimensions of the force unit newton?
Problem 37
An equation for the period \(T\) of a planet (the time to make one orbit about
the Sun) is \(4 \pi^{2} r^{3} /(G M),\) where \(T\) is in \(\mathrm{s}, r\) is in
\(\mathrm{m}, G\) is in $\mathrm{m}^{3} /\left(\mathrm{kg} \cdot
\mathrm{s}^{2}\right),\( and \)M$ is in kg. Show that the equation is
dimensionally correct.
Problem 38
The relationship between kinetic energy \(K\) (SI unit $\left.\mathrm{kg} \cdot
\mathrm{m}^{2} \cdot \mathrm{s}^{-2}\right)\( and momentum \)p\( is \)K=p^{2} /(2
m),\( where \)m$ stands for mass. What is the SI unit of momentum?
Problem 39
An expression for buoyant force is \(F_{\mathrm{B}}=\rho g V,\) where
\(F_{\mathrm{B}}\) has dimensions \(\left[\mathrm{MLT}^{-2}\right], \rho\)
(density) has dimensions \(\left[\mathrm{ML}^{-3}\right],\) and \(g\)
(gravitational field strength) has dimensions \(\left[\mathrm{LT}^{-2}\right]\).
(a) What must be the dimensions of \(V ?\) (b) Which could be the correct
interpretation of \(V:\) velocity or volume?
Problem 40
Use dimensional analysis to determine how the linear speed $(v \text { in }
\mathrm{m} / \mathrm{s}$ ) of a particle traveling in a circle depends on
some, or all, of the following properties: \(r\) is the radius of the circle;
\(\omega\) is an angular frequency in \(\mathrm{s}^{-1}\) with which the particle
orbits about the circle, and \(m\) is the mass of the particle. There is no
dimensionless constant involved in the relation.
Problem 41
What is the approximate distance from your eyes to a book you are reading?
