Problems

Use the orders of magnitude you found in the previous problem to answer the following questions to within an order of magnitude. (a) How many electrons would it take to equal the mass of a proton? (b) How many Earths would it take to equal the mass of the Sun? (c) How many Earth–Moon distances would it take to cover the distance from Earth to the Sun? (d) How many Moon atmospheres would it take to equal the mass of Earth’s atmosphere? (e) How many moons would it take to equal the mass of Earth? (f) How many protons would it take to equal the mass of the Sun?

A generation is about one-third of a lifetime. Approximately how many generations have passed since the year 0?

Calculate the approximate number of atoms in a bacterium. Assume the average mass of an atom in the bacterium is 10 times the mass of a proton.

Assuming one nerve impulse must end before another can begin, what is the maximum firing rate of a nerve in impulses per second?

Roughly how many floating-point operations can a supercomputer perform in a human lifetime?

The following times are given in seconds. Use metric prefixes to rewrite them so the numerical value is greater than one but less than 1000. For example, $7.9\times {10}^{\mathrm{-2}}\text{s}$ could be written as either 7.9 cs or 79 ms. (a) $9.57\times {10}^5\text{s;}$ (b) 0.045 s; (c) $5.5\times {10}^{\mathrm{-7}}\text{s;}$ (d) $3.16\times {10}^7\text{s.}$

The following lengths are given in meters. Use metric prefixes to rewrite them so the numerical value is bigger than one but less than 1000. For example, $7.9\times {10}^{\mathrm{-2}}\text{m}$ could be written either as 7.9 cm or 79 mm. (a) $7.59\times {10}^7\text{m;}$ (b) 0.0074 m; (c) $8.8\times {10}^{\mathrm{-11}}\text{m;}$ (d) $1.63\times {10}^{13}\text{m.}$

The following masses are given in kilograms. Use metric prefixes on the gram to rewrite them so the numerical value is bigger than one but less than 1000. For example, $7\times {10}^{\mathrm{-4}}\text{kg}$ could be written as 70 cg or 700 mg. (a) $3.8\times {10}^{\mathrm{-5}}\text{kg;}$ (b) $2.3\times {10}^{17}\text{kg;}$ (c) $2.4\times {10}^{\mathrm{-11}}\text{kg;}$ (d) $8\times {10}^{15}\text{kg;}$ (e) $4.2\times {10}^{\mathrm{-3}}\text{kg.}$

The speed limit on some interstate highways is roughly 100 km/h. (a) What is this in meters per second? (b) How many miles per hour is this?

In SI units, speeds are measured in meters per second (m/s). But, depending on where you live, you’re probably more comfortable of thinking of speeds in terms of either kilometers per hour (km/h) or miles per hour (mi/h). In this problem, you will see that 1 m/s is roughly 4 km/h or 2 mi/h, which is handy to use when developing your physical intuition. More precisely, show that (a) $1.0\text{m/s}=3.6\text{km/h}$ and (b) $1.0\text{m/s}=2.2\text{mi/h}.$

Soccer fields vary in size. A large soccer field is 115 m long and 85.0 m wide. What is its area in square feet? (Assume that 1 m = 3.281 ft.)

Mount Everest, at 29,028 ft, is the tallest mountain on Earth. What is its height in kilometers? (Assume that 1 m = 3.281 ft.)

Tectonic plates are large segments of Earth’s crust that move slowly. Suppose one such plate has an average speed of 4.0 cm/yr. (a) What distance does it move in 1.0 s at this speed? (b) What is its speed in kilometers per million years?

The density of nuclear matter is about 10¹⁸ kg/m³. Given that 1 mL is equal in volume to cm³, what is the density of nuclear matter in megagrams per microliter (that is, $\text{Mg/}\mu \text{L}$)?

A commonly used unit of mass in the English system is the pound-mass, abbreviated lbm, where 1 lbm = 0.454 kg. What is the density of water in pound-mass per cubic foot?

It takes $2\pi$ radians (rad) to get around a circle, which is the same as 360°. How many radians are in 1°?

A light-nanosecond is the distance light travels in 1 ns. Convert 1 ft to light-nanoseconds.

A fluid ounce is about 30 mL. What is the volume of a 12 fl-oz can of soda pop in cubic meters?

Consider the physical quantities s, v, a, and t with dimensions $[s]=\text{L},$ $[v]={\text{LT}}^{\mathrm{-1}},$ $[a]={\text{LT}}^{\mathrm{-2}},$ and $[t]=\text{T}.$ Determine whether each of the following equations is dimensionally consistent. (a) $v^2=2as;$ (b) $s=v{t}^2+0.5a{t}^2;$ (c) $v=s\text{/}t;$ (d) $a=v\text{/}t.$

Suppose quantity $s$ is a length and quantity $t$ is a time. Suppose the quantities $v$ and $a$ are defined by v = ds/dt and a = dv/dt. (a) What is the dimension of v? (b) What is the dimension of the quantity a? What are the dimensions of (c) ${\displaystyle \int v}dt,$ (d) ${\displaystyle \int a}dt,$ and (e) da/dt?

The arc length formula says the length $s$ of arc subtended by angle $Ɵ$ in a circle of radius $r$ is given by the equation $s=rƟ.$ What are the dimensions of (a) s, (b) r, and (c) $\text{Ɵ?}$

Assuming the human body is primarily made of water, estimate the number of molecules in it. (Note that water has a molecular mass of 18 g/mol and there are roughly 10²⁴ atoms in a mole.)

Estimate the number of molecules that make up Earth, assuming an average molecular mass of 30 g/mol. (Note there are on the order of 10²⁴ objects per mole.)

Roughly how many solar systems would it take to tile the disk of the Milky Way?

The average density of the Sun is on the order 10³ kg/m³. (a) Estimate the diameter of the Sun. (b) Given that the Sun subtends at an angle of about half a degree in the sky, estimate its distance from Earth.

A floating-point operation is a single arithmetic operation such as addition, subtraction, multiplication, or division. (a) Estimate the maximum number of floating-point operations a human being could possibly perform in a lifetime. (b) How long would it take a supercomputer to perform that many floating-point operations?

Suppose your bathroom scale reads your mass as 65 kg with a 3% uncertainty. What is the uncertainty in your mass (in kilograms)?

An infant’s pulse rate is measured to be 130 ± 5 beats/min. What is the percent uncertainty in this measurement?

A can contains 375 mL of soda. How much is left after 308 mL is removed?

(a) How many significant figures are in the numbers 99 and 100.? (b) If the uncertainty in each number is 1, what is the percent uncertainty in each? (c) Which is a more meaningful way to express the accuracy of these two numbers: significant figures or percent uncertainties?

(a) A person’s blood pressure is measured to be $120\pm 2\text{mm Hg}.$ What is its percent uncertainty? (b) Assuming the same percent uncertainty, what is the uncertainty in a blood pressure measurement of 80 mm Hg?

What is the area of a circle 3.102 cm in diameter?

Perform the following calculations and express your answer using the correct number of significant digits. (a) A woman has two bags weighing 13.5 lb and one bag with a weight of 10.2 lb. What is the total weight of the bags? (b) The force F on an object is equal to its mass m multiplied by its acceleration a. If a wagon with mass 55 kg accelerates at a rate of 0.0255 m/s², what is the force on the wagon? (The unit of force is called the newton and it is expressed with the symbol N.)