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Chapter 1: Problem 48

A person taking a reading of the temperature in a freezer in Celsius makes two mistakes: first omitting the negative sign and then thinking the temperature is Fahrenheit. That is, the person reads \(-x^{\circ} \mathrm{C}\) as \(x^{\circ} \mathrm{F}\). Oddly enough, the result is the correct Fahrenheit temperature. What is the original Celsius reading? Round your answer to three significant figures.

Short Answer

Expert verified
The original Celsius reading is approximately \(-11.429^\circ \mathrm{C}\), rounded to three significant figures.

Step by step solution

01

Write the temperature conversion formula

We know that the formula to convert Celsius to Fahrenheit is: \[F = \frac{9}{5}C + 32\] We need to use this formula to find the correct Celsius reading. #Step 2: Plug in the incorrect reading and correct them#
02

Plug in the incorrect reading and correct them

The person read the temperature as \(x^\circ \mathrm{F}\) while it was actually \(-x^\circ \mathrm{C}\). So, we have: \[x = \frac{9}{5}(-x) + 32\] #Step 3: Solve for x#
03

Solve for x

Now we need to solve for x: \begin{align*} x &= \frac{9}{5}(-x) + 32 \Rightarrow \\ x + \frac{9}{5}x &= 32 \Rightarrow \\ \frac{14}{5}x &= 32 \Rightarrow \\ x &= \frac{32\cdot{5}}{14} \end{align*} #Step 4: Calculate the value of x and round to 3 significant figures#
04

Calculate the value of x and round to 3 significant figures

Now we calculate the value of x and round it to three significant figures: \[x \approx \frac{160}{14} \approx 11.429\] So, the original Celsius reading is approximately \(-11.429^\circ \mathrm{C}\), rounded to three significant figures.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Celsius to Fahrenheit
Understanding how to convert temperatures from Celsius to Fahrenheit is vital in various scientific contexts as well as in everyday life, particularly when dealing with international data or travel. The formula for converting Celsius to Fahrenheit is is one of the most basic formulas in thermodynamics and is given by: In this equation, ' The key takeaway in using this formula is remembering that the scale for Celsius involves incrementation by of one degree per increment, while the Fahrenheit scale progresses by increments of 1.8 degrees for each degree Celsius. Additionally, there is an offset of 32 degrees, which accounts for the differences in the zero points of the two scales.
Significant Figures in Physics
In physics, the precision of measured values is conveyed through the use of significant figures. These are the digits in a number that contribute to its accuracy, beginning with the first non-zero digit and extending to the last digit that is reliably known. For instance, the number 0.0036 has two significant figures 鈥� 3 and 6 鈥� because the zeros merely serve as placeholders for decimal position, not as accurate measurements.

Using Significant Figures

When rounding numbers in physics, significant figures play a crucial role. This is evident in our temperature conversion example, where the person's reading initially lacks precision due to the omitted negative sign and mistaken unit. After correcting for the errors, we must ensure that the final Celsius temperature is expressed with the appropriate number of significant figures. This means rounding off our calculated value to reflect the precision of the data. Remember, the number of significant figures in the result of a calculation is determined by the original number with the fewest significant figures. In the given problem, the result is rounded to three significant figures to conform to the precision of the thermometer readings or the given data.
Problem-Solving in Physics
Problem-solving is an essential skill in physics, helping students and professionals alike to apply theoretical concepts to practical situations. Using a structured approach allows for a thorough understanding and accurate solutions to complex problems.

Steps in Problem-Solving

  • Identify: Determine what is given and what needs to be found.
  • Formulate: Transform the problem into mathematical terms using formulas or equations.
  • Compute: Solve the mathematics involved.
  • Verify: Check if the solution makes sense in terms of physical principles.
  • Communicate: Express the solution clearly with appropriate significant figures and units.
In our temperature conversion scenario, the problem-solving steps begin by identifying the error made in reading the temperature. Next, we formulated the problem using the Celsius to Fahrenheit conversion formula, then computed the correct temperature, and finally, verified that the answer makes sense with the provided context. As a student or physicist, mastering these steps not only helps you solve problems efficiently but also ensures that your results are accurate and reliable.

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Most popular questions from this chapter

A pressure cooker contains water and steam in equilibrium at a pressure greater than atmospheric pressure. How does this greater pressure increase cooking speed?

On a trip, you notice that a 3.50 -kg bag of ice lasts an average of one day in your cooler. What is the average power in watts entering the ice if it starts at \(0^{\circ} \mathrm{C}\) and completely melts to \(0^{\circ} \mathrm{C}\) water in exactly one day?

Let's stop ignoring the greenhouse effect and incorporate it into the previous problem in a very rough way. Assume the atmosphere is a single layer, a spherical shell around Earth, with an emissivity \(e=0.77\) (chosen simply to give the right answer) at infrared wavelengths emitted by Earth and by the atmosphere. However, the atmosphere is transparent to the Sun's radiation (that is, assume the radiation is at visible wavelengths with no infrared), so the Sun's radiation reaches the surface. The greenhouse effect comes from the difference between the atmosphere's transmission of visible light and its rather strong absorption of infrared. Note that the atmosphere's radius is not significantly different from Earth's, but since the atmosphere is a layer above Earth, it emits radiation both upward and downward, so it has twice Earth's area. There are three radiative energy transfers in this problem: solar radiation absorbed by Earth's surface; infrared radiation from the surface, which is absorbed by the atmosphere according to its emissivity; and infrared radiation from the atmosphere, half of which is absorbed by Earth and half of which goes out into space. Apply the method of the previous problem to get an equation for Earth's surface and one for the atmosphere, and solve them for the two unknown temperatures, surface and atmosphere. a. In terms of Earth's radius, the constant \(\sigma,\) and the unknown temperature \(T_{s}\) of the surface, what is the power of the infrared radiation from the surface? b. What is the power of Earth's radiation absorbed by the atmosphere? c. In terms of the unknown temperature \(T_{e}\) of the atmosphere, what is the power radiated from the atmosphere? d. Write an equation that says the power of the radiation the atmosphere absorbs from Earth equals the power of the radiation it emits. e. Half of the power radiated by the atmosphere hits Earth. Write an equation that says that the power Earth absorbs from the atmosphere and the Sun equals the power that it emits. f. Solve your two equations for the unknown temperature of Earth.

A pendulum is made of a rod of length \(L\) and negligible mass, but capable of thermal expansion, and a weight of negligible size. (a) Show that when the temperature increases by \(d T,\) the period of the pendulum increases by a fraction \(\alpha L d T / 2 .\) (b) A clock controlled by a brass pendulum keeps time correctly at \(10^{\circ} \mathrm{C}\). If the room temperature is \(30^{\circ} \mathrm{C}\), does the clock run faster or slower? What is its error in seconds per day?

Give an example in which \(A\) has some kind of nonthermal equilibrium relationship with \(B\), and \(B\) has the same relationship with \(C,\) but \(A\) does not have that relationship with \(C\).
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