| Physics Demystified, 2nd edition |
| Stan Gibilisco |
| Explanations for Quiz Answers in Chapter 0 |
| 1. Let's remember the rules of precedence for arithmetic. Exponential operations come
first, then products and quotients, then sums and differences. We start with -4 x 22 + 16 We carry out the exponentiation to obtain -4 x 4 + 16 Then we perform the product to get -16 + 16 Finally we add, ending up with 0. The correct choice is C. |
| 2. We want to find the difference between two quantities expressed in power-of-10
notation, taking significant figures into account. Here's the initial expression: 7.776 x 105 - 6.2 x 10-17 The second value, 6.2 x 10-17, is extremely small compared to the first value, 7.776 x 105. In fact, the second quantity is roughly 22 orders of magnitude smaller than the first quantity! Whenever we add or subtract two quantities and one of them has an absolute value that's vastly smaller (or larger) than the absolute value of the other, we can ignore the quantity with the smaller absolute value "for all intents and purposes." In this case, the difference equals the first quantity, 7.776 x 105. The correct choice is A. |
| 3. The plain decimal numeral 0.045 appears in the optimum format exactly as stated. We don't need to bother with power-of-10 notation. If we want to denote this quantity in scientific notation, we can express it as 4.5 x 10-2, but the exponent isn't "large enough negatively" to warrant the use of the power-of-10 format. The correct choice is D. |
| 4. One order of magnitude represents an increase or decrease in size equivalent to a factor of 10. If two quantities differ in size by exactly four orders of magnitude, then one of them is 104, or 10,000, times as large (or small) as the other. If we divide the larger quantity by the smaller one, we get 10,000. The correct choice is A. If you think that choice B might also represent a valid answer here, look more closely at the wording! Choice B says "larger than the other by 10,000." That means we get 10,000 if we subtract the smaller quantity from the larger one. However, orders of magnitude refer to ratios or proportions, not differences. |
| 5. To get a sensible answer to this question, you must decide which of the two measurements (yours or the computer engineer's) to take as the "actual value." If I were you, I'd trust the computer engineer's measurement over my own! You can therefore assume that your Internet connection speed is actually 13.88 Mbps, as the engineer said. You measured 14.05 Mbps, which is (14.05 - 13.88), or 0.17 Mbps, faster than the "actual speed." To determine your error as a percentage, you must divide 0.17 by 13.88 and then multiply by 100, getting 1.2%. Because your measurement was too large (not too small), you can place a plus sign in front of this result, getting +1.2%. The answer is B. (In this scenario, we ignore the possibility, and in fact the real-world likelihood, that your true Internet connection speed might vary from moment to moment in time, casting doubt on the practical validity of any speed test!) |
| 6. You've measured a quantity q and come up with a value of 1.5 x 104 units. If you write q out in plain decimal form you get 15,000, but you must keep in mind the fact that the value is accurate to only two significant figures (represented by the leftmost pair of digits). If your measurement of q turns out less than 15,000 so that you must round upward, you can start with anything larger than or equal to 14,500 and end up with 1.5 x 104. If your measurement of q turns out larger than 15,000 so that you must round downward, you can start with anything strictly smaller than 15,500 and end up with 1.5 x 104. The correct choice is B. |
| 7. In this situation, your measurement is more accurate than it was in the scenario of Question 6. You can claim an accuracy of three significant figures rather than two. If your measurement of q turns out less than 15,000 so that you must round upward, you can start with anything larger than or equal to 14,950 and end up with 1.50 x 104. If your measurement of q turns out larger than 15,000 so that you must round downward, you can start with anything strictly smaller than 15,050 and end up with 1.50 x 104. The correct choice is B. |
| 8. We should start by multiplying the two coefficients,
obtaining 9 x 1.494 = 13.446. (Let's wait until we've finished all of our calculations
before we round off to the appropriate number of significant figures.) Next, we add the
powers of 10. In the first expression we have 105, and in the second expression
we have 10-6. Adding the exponents yields 10[5+(-6)]
= 10(5-6) Now we can combine the coefficient and the power of 10, getting the expression 13.446 x 10-1 In standard power-of-10 format, that's 1.3446 x 100 We can delete the 100 part of this expression because 100 = 1, so we get the value 1.3446. Finally, we must round off to one significant figure! That operation produces a final answer of 1, so choice D represents the correct answer. |
| 9. When you encounter a complicated arithmetic expression involving sums, differences, and products without any grouping symbols (such as parentheses, brackets, or braces), you should execute the products first, working from left to right. The correct choice is C. |
| 10. In power-of-10 notation, the quantity 12,006,000 translates to 1.2006000 x 107 if we don't bother with significant figures. However, we're told that the quantity is accurate to only four significant figures, so we must round it off to 1.201 x 107. A computer or scientific calculator might render that value as 1.201E+07. The correct choice is B. |