Chapter 2

Convert the following measured values from scientific notation to standard notation. For each one, indicate the number of significant figures. (a) \(5.60 \times 10^{1} \mathrm{~kg}\) (b) \(2.5 \times 10^{-4} \mathrm{~m}\) (c) \(5.600 \times 10^{6}\) miles (d) \(0.02 \times 10^{2} \mathrm{ft}\)

Using scientific notation, write the measurement \(30 \mathrm{ft}\) as having an uncertainty of: (a) \(\pm 1 \mathrm{ft}\) (b) \(\pm 0.1 \mathrm{ft}\) (c) \(\pm 0.01 \mathrm{ft}\)

Using scientific notation, write the measurement \(2200 \mathrm{ft}\) as having an uncertainty of \(\pm 100 \mathrm{ft}\).

Convert the following numbers from standard notation to scientific notation: (a) 226 (b) \(226.0\) (c) \(0.00000000050\) (d) \(0.3\) (e) \(0.30\) (f) \(900,000,574\) with an uncertainty of \(\pm 1\) million (g) \(900,000,574\) with an uncertainty of \(\pm 100\)

How many significant figures are there in the following measured values, and what is the uncertainty in each measurement? (a) \(0.001 \mathrm{~kg}\) (b) \(0.00010 \mathrm{~m}\) (c) \(102 \mathrm{~L}\) (d) \(2.600 \times 10^{-3} \mathrm{~m}\) (e) \(1.1 \times 10^{6} \mathrm{~km}\)

To the correct number of significant figures, what is the result of adding the measured values 100 in. \(+2\) in. \(+0.001\) in.? What is the uncertainty in the result?

To the correct number of significant figures, what is the product of each mathematical operation? Use scientific notation when necessary. (No units shown means a number is exact.) (a) \(2.30 \mathrm{~cm} \times 2\) (b) \(2 \mathrm{~m} \times 2.000 \mathrm{~m}\) (c) \(1001 \mathrm{~J} \times 10\) (d) \(124 \mathrm{~mm} \div 0.1 \mathrm{~mm}\)

A student walks \(20,450.2 \mathrm{ft}\) to school every day. A mile is defined as \(5280 \mathrm{ft}\). Doing the division \(20,450.2 \mathrm{ft} \div 5280 \mathrm{ft}\) per mile on a calculator gives \(3.873143939\) miles. What, if anything, is wrong with this answer?

Do these calculations using a scientific calculator and report your answers in scientific notation: (a) \(\left(3.33 \times 10^{4} \mathrm{~km}\right)+\left(2.22 \times 10^{5} \mathrm{~km}\right)\) (b) \(\left(2.444 \times 10^{9} \mathrm{~J}\right) \div\left(2.444 \times 10^{-9} \mathrm{~J}\right)\) (c) \(\left(2.34 \times 10^{2} \mathrm{~m}\right)-\left(2.34 \times 10^{1} \mathrm{~m}\right)\) (d) \(\left(4.00 \times 10^{4} \mathrm{~L}\right)+\left(6.00 \times 10^{-1} \mathrm{~L}\right)\)

What is the base SI unit of length? What is the SI unit of yolume?

What are two metric but non-SI units of volume, and why are they more often used than the SI unit of volume?

When is it correct to use \(\mathrm{cm}^{3}\) instead of \(\mathrm{mL}\) ? Explain.

Why was the SI unit system developed by scientists?

Convert each length to meters. Report your answers in scientific notation and watch your significant figures. (a) \(2.31\) gigameters \((\mathrm{Gm})\) (b) \(5.00\) micrometers \((\mu m)\) (c) 1004 millimeters \((\mathrm{mm})\) (d) \(5.00\) picometers \((\mathrm{pm})\) (e) \(0.25\) kilometer \((\mathrm{km})\)

Which is larger, a Celsius degree or a Fahrenheit degree? Explain.

Of the three temperature scales, which can have negative temperatures? For the one(s) that can't, explain why not.

Convert: (a) \(22.5^{\circ} \mathrm{C}\) to Fahrenheit and Kelvin (b) \(-3.00{ }^{\circ} \mathrm{F}\) to Celsius and Kelvin (c) \(100.0^{\circ} \mathrm{C}\) to Kelvin and Fahrenheit (d) \(65.1^{\circ} \mathrm{C}\) to Fahrenheit and Kelvin

How cold does it have to be for water to freeze in \({ }^{\circ} \mathrm{F}\) ? In \({ }^{\circ} \mathrm{C}\) ? In kelvins?

What is the coldest temperature possible in Celsius and Fahrenheit? Give your answers to an uncertainty of plus or minus one-hundredth of a degree.

Using a ruler marked in centimeters and millimeters, a student measures the diameter of a ball to be \(1.5 \mathrm{~cm}\). His partner measures the same ball with the same ruler and comes up with \(1.50\) \(\mathrm{cm}\). Which student used the ruler incorrectly? How did that student use the ruler incorrectly?

The students measure another ball with the ruler in Problem \(2.93\) and determine that its diameter is \(2.55 \mathrm{~cm}\). What is the radius of the ball to the correct number of significant figures?

Define density, and explain why the unit for density is called a derived SI unit.

Using a bathroom scale, a tub, a sponge, and a measuring cup, explain how you would measure your own density.

Two students measure the density of gold. One works with a \(100-g\) bar of pure gold. The other works with a \(200-g\) bar of pure gold. Which student measures the larger density? Explain your answer.

Suppose it takes you \(0.850\) weeks to reach the moon in a space ship. How many seconds does it take for you to get there? Use unit analysis to calculate your answer, and show your work.

A train traveling at \(45.0\) miles \(/ \mathrm{h}\) has to make a trip of \(100.0\) miles. How many minutes will the trip take? Use unit analysis to calculate your answer, and show your work.

You have a great job in which you earn \(\$ 25.50\) per hour. How many dollars do you earn per second? Use unit analysis to calculate your answer, and show your work.

Gold has a density of \(19.3 \mathrm{~g} / \mathrm{mL}\). Suppose you have \(100.0\) glonkins of gold. What volume in liters will the gold occupy? Here are some conversion factors to help you: \(0.911\) ounce per glonkin and \(28.35 \mathrm{~g}\) per ounce. Use unit analysis to calculate your answer, and show your work. Treat both conversion factors as exact.

One liter is equal to \(0.264\) gallon. Suppose you have \(1.000 \times 10^{3} \mathrm{~cm}^{3}\) of water. How many gallons do you have? Use unit analysis to calculate your answer, and show your work. Treat all conversion factors as exact.

The dimensions of a rectangular box are given to be \(10.2 \mathrm{~cm} \times 43.7 \mathrm{~cm} \times 9.56 \times 10^{2} \mathrm{~mm}\). What is its volume in liters? Be careful! The units are not all the same.

You measure one edge of a cube using a meterstick marked in centimeters. Unfortunately, the edge is longer than \(1 \mathrm{~m}\). You mark the \(1-\mathrm{m}\) point on the cube edge with a pen and then, using a \(15-\mathrm{cm}\) ruler marked in millimeters, measure the remaining distance to be \(1.40 \mathrm{~cm}\). (a) What is the length of the edge in centimeters? (b) What is the volume of the cube in cubic centimeters? (Remember, the lengths of all edges of a cube are equal.) Watch your significant figures. Use scientific notation if you have to. (c) The cube has a mass of \(111 \mathrm{~kg} .\) What is its density in grams per milliliter? Watch your significant figures.

A rectangular box measures \(6.00\) in. in length, \(7.00\) in. in width, and \(8.00\) in. in height. What is the volume of the box in liters? \([2.54 \mathrm{~cm}=1\) in.]

An object travels \(80.0 \mathrm{~m} / \mathrm{s}\). How fast is it traveling in miles per hour? \([1 \mathrm{~m}=3.28 \mathrm{ft}, 1\) mile \(=5280 \mathrm{ft}]\)

Why can't you multiply just one side of an equation by something when algebraically rearranging the equation?

(a) Solve the equation \(y=z / x\) for \(x\). (b) Solve the equation \(y=z / 2 x\) for \(x\).

Solve the equation \(y=z-x\) for \(x\).

Solve the equation \(5 x-6=3 x-8\) (find the value of \(x\) that makes this equation true).

The density of a certain liquid is \(1.15 \mathrm{~g} / \mathrm{mL}\). What mass in grams of the liquid is needed to fill a \(50.00\) -mL container? Do this problem by the method of algebraic manipulation, beginning with the equation density \(=\) mass/volume and showing all steps.

Define energy.

How much heat energy is 1 cal? Give your answer in terms of changing the temperature of water.

Convert: (a) \(4.50\) Cal to calories (b) \(600.0\) Cal to kilojoules (c) \(1.000 \mathrm{~J}\) to calories (d) \(50.0\) Cal to joules

Define specific heat.

A \(100.0\) -g block of iron and a \(100.0\) -g block of aluminum are both initially at \(25.0^{\circ} \mathrm{C}\). Both are then warmed to \(100.0^{\circ} \mathrm{C}\). Does one block require more heat energy than the other to reach \(100.0{ }^{\circ} \mathrm{C}\) ? If so, how much more?

How many "big \(\mathrm{C}^{\prime \prime}\) Calories does it take to raise the temperature of \(2.00 \mathrm{~L}\) of water from \(22.0{ }^{\circ} \mathrm{C}\) to \(40.0{ }^{\circ} \mathrm{C} ?\) How many kilojoules? Take the density of water to be \(1.00 \mathrm{~g} / \mathrm{mL}\).

Why is it necessary for a calorimeter to be insulated?

A \(2.50-g\) piece of wood is burned in a calorimeter that contains \(0.200 \mathrm{~kg}\) of water. The burning causes the water temperature to increase from \(22.1^{\circ} \mathrm{C}\) to \(28.7^{\circ} \mathrm{C}\). How much heat energy is released in joules? What is the energy content of the wood in joules per gram of wood?

How many joules of heat energy would it take to raise the temperature of \(2.00\) pounds of iron from \(30.0^{\circ} \mathrm{C}\) to \(90.0^{\circ} \mathrm{C} ?[453.6 \mathrm{~g}=1\) pound \(]\)

Which one of the following expresses the measured value \(1230.0 \mathrm{~m}\) with the correct number of significant figures and appropriate Greek prefix? (a) \(1.23 \mathrm{~km}\) (b) \(1.230 \mathrm{~cm}\) (c) \(1.2300 \mathrm{~km}\) (d) \(1.2300 \mathrm{~mm}\) (e) \(12.3 \mathrm{~km}\)

Convert: (a) \(7.98 \times 10^{23} \mu \mathrm{L}\) to liters (b) \(3.00 \times 10^{-3} \mathrm{mg}\) to grams (c) \(4.21 \times 10^{8} \mathrm{~mL}\) to gallons \(\left[1 \mathrm{~m}^{3}=264\right.\) gallons \(]\)

A metal sphere has a radius \(r\) of \(4.00 \mathrm{~cm}\). What is the volume \(V\) of this sphere in cubic centimeters? The formula for the volume of a sphere is \(V=(4 / 3) \pi r^{3}\), where \(\pi=3.14159 .\)