Physics with Calculus: Units, Conversions, and Estimation – Step-by-Step Guidance

Q1. What is the current definition of the standard meter of length?

Background

Topic: SI Units and Standards

This question tests your understanding of how the standard unit of length (the meter) is defined in modern physics, which is foundational for all measurements in physics.

Key Terms:

• Meter (m): The SI base unit of length.

• Speed of light (c): A fundamental constant in physics, $c = 299,792,458$ m/s.

Step-by-Step Guidance

1. Recall that the definition of the meter has changed over time, but the current definition is based on a fundamental constant of nature.

2. Think about which physical phenomenon is used as a reference for the meter in the SI system: is it a physical object, a distance on Earth, or something else?

3. Consider the role of the speed of light in defining the meter. The meter is now defined by the distance light travels in a specific fraction of a second in a vacuum.

Try solving on your own before revealing the answer!

Final Answer: C) the distance traveled by light in a vacuum.

The meter is defined as the distance light travels in vacuum in $1/299,792,458$ of a second.

Q2. What is the current definition of the standard second of time?

Background

Topic: SI Units and Standards

This question tests your knowledge of how the SI unit of time (the second) is defined, which is crucial for all time measurements in physics.

Key Terms:

• Second (s): The SI base unit of time.

• Cesium atom: Used in atomic clocks to define the second.

Step-by-Step Guidance

1. Recall that the second is no longer defined by astronomical phenomena like Earth's rotation.

2. Think about atomic clocks and the specific atom used for the definition.

3. The definition involves the frequency of radiation emitted during a transition between two energy levels of a cesium atom.

Try solving on your own before revealing the answer!

Final Answer: A) the frequency of radiation emitted by cesium atoms.

The second is defined as the duration of 9,192,631,770 periods of the radiation corresponding to the transition between two hyperfine levels of the ground state of the cesium-133 atom.

Q3. If a woman weighs 125 lb, is her mass in kilograms less than or greater than 125?

Background

Topic: Unit Conversion (Mass)

This question tests your ability to convert between pounds (a unit of force/mass in the imperial system) and kilograms (the SI unit of mass).

Key Terms and Conversion:

• 1 lb = 0.4536 kg

Step-by-Step Guidance

1. Recall the conversion factor between pounds and kilograms.

2. Set up the conversion: $\text{mass in kg} = \text{mass in lb} \times 0.4536$

3. Think about whether multiplying by a number less than 1 will increase or decrease the value.

Try solving on your own before revealing the answer!

Final Answer: A) less than 125.

Since 1 lb is less than 1 kg, multiplying 125 by 0.4536 gives a value less than 125.

Q4. If a tree is 15 m tall, is its height in feet less than or greater than 15?

Background

Topic: Unit Conversion (Length)

This question tests your understanding of converting meters to feet and comparing the numerical values.

Key Terms and Conversion:

• 1 m = 3.281 ft

Step-by-Step Guidance

1. Recall the conversion factor between meters and feet.

2. Set up the conversion: $\text{height in ft} = \text{height in m} \times 3.281$

3. Think about whether multiplying by a number greater than 1 will increase or decrease the value.

Try solving on your own before revealing the answer!

Final Answer: B) greater than 15.

Since 1 m is more than 1 ft, multiplying 15 by 3.281 gives a value greater than 15.

Q5. When adding two numbers, is the number of significant figures in the sum equal to the number of significant figures in the least accurate of the numbers being added?

Background

Topic: Significant Figures in Addition

This question tests your understanding of how to determine the correct number of significant figures when adding numbers.

Key Concept:

• For addition and subtraction, the result should have the same number of decimal places as the number with the least decimal places.

Step-by-Step Guidance

1. Review the rule for significant figures in addition and subtraction.

2. Compare this rule to the statement in the question: does it refer to decimal places or significant figures?

3. Think about an example: add 1.2 (2 sig figs, 1 decimal place) and 1.23 (3 sig figs, 2 decimal places). What is the correct number of decimal places in the sum?

Try solving on your own before revealing the answer!

Final Answer: B) False.

For addition, the number of decimal places (not significant figures) is what matters.

Q6. Convert $1.2 \times 10^{-3}$ to decimal notation.

Background

Topic: Scientific Notation

This question tests your ability to convert a number from scientific notation to standard decimal form.

Key Concept:

• To convert $a \times 10^{-n}$, move the decimal point $n$ places to the left.

Step-by-Step Guidance

1. Identify the exponent: $-3$ means move the decimal 3 places to the left.

2. Start with 1.2 and add zeros as needed to move the decimal point.

3. Write out the number in full decimal form, making sure to include the correct number of zeros.

Try solving on your own before revealing the answer!

Final Answer: D) 0.0012

Moving the decimal 3 places to the left gives 0.0012.

Q7. Write $7.35 \times 10^{-5}$ in full decimal notation with the correct number of zeros.

Background

Topic: Scientific Notation

This question tests your ability to convert a small number from scientific notation to decimal form.

Key Concept:

• Move the decimal point 5 places to the left for $10^{-5}$.

Step-by-Step Guidance

1. Start with 7.35 and move the decimal 5 places to the left.

2. Count the number of zeros needed to fill in the places before the digits.

3. Write the number in full, making sure the decimal point is in the correct position.

Try solving on your own before revealing the answer!

Final Answer: B) 0.0000735

Moving the decimal 5 places to the left gives 0.0000735.

Q8. Express 0.0001776 in scientific notation.

Background

Topic: Scientific Notation

This question tests your ability to convert a decimal number to scientific notation.

Key Concept:

• Move the decimal point to the right until only one nonzero digit is to the left of the decimal point. Count the number of places moved for the exponent.

Step-by-Step Guidance

1. Identify the first nonzero digit (1 in this case).

2. Move the decimal point 4 places to the right to get 1.776.

3. Since you moved the decimal to the right, the exponent is negative.

Try solving on your own before revealing the answer!

Final Answer: B) $1.776 \times 10^{-4}$

0.0001776 = $1.776 \times 10^{-4}$

Q9. If π in a parallel universe is 3.14149, express π to four significant figures.

Background

Topic: Significant Figures and Rounding

This question tests your ability to round a number to a specified number of significant figures.

Key Concept:

• To round to four significant figures, keep the first four digits and round the last one based on the next digit.

Step-by-Step Guidance

1. Identify the first four significant digits: 3, 1, 4, 1.

2. Look at the fifth digit (4) to decide whether to round up or keep the fourth digit as is.

3. Apply the rounding rule: if the next digit is 5 or more, round up; if less, keep as is.

Try solving on your own before revealing the answer!

Final Answer: B) 3.142

Rounding 3.14149 to four significant figures gives 3.142.

Q10. How many significant figures does 0.003010 have?

Background

Topic: Significant Figures

This question tests your ability to count significant figures, especially with leading zeros and zeros between/after digits.

Key Concept:

• Leading zeros are not significant; zeros between nonzero digits or after a decimal point and a nonzero digit are significant.

Step-by-Step Guidance

1. Ignore the leading zeros (0.00).

2. Count all digits from the first nonzero digit (3) onward, including zeros between or after nonzero digits.

3. Determine the total number of significant digits.

Try solving on your own before revealing the answer!

Final Answer: C) 4 significant figures.

0.003010 has four significant figures: 3, 0, 1, and the final 0 (since it is after the decimal and a nonzero digit).