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Chapter 18: Problem 26

\(.\) What mass of butter, which has a usable energy content of \(6.0\) \(\mathrm{Cal} / \mathrm{g}(=6000 \mathrm{cal} / \mathrm{g})\), would be equivalent to the change in gravitational potential energy of a \(73.0 \mathrm{~kg}\) man who ascends from sea level to the top of Mt. Everest, at elevation \(8.84 \mathrm{~km}\) ? Assume that the average \(g\) for the ascent is \(9.80 \mathrm{~m} / \mathrm{s}^{2}\)

Short Answer

Expert verified
The mass of butter equivalent to the energy change is approximately 253.0 g.

Step by step solution

01

Calculate the Gravitational Potential Energy Change

The change in gravitational potential energy \( \Delta U \) is given by the formula \( \Delta U = mgh \), where \( m \) is mass in kilograms, \( g \) is the gravitational acceleration in \( \mathrm{m/s^2} \), and \( h \) is the height change in meters. For the given problem, the values are \( m = 73.0 \ \mathrm{kg} \), \( g = 9.80 \ \mathrm{m/s^2} \), and \( h = 8840 \ \mathrm{m} \). Calculating: \[ \Delta U = 73.0 \times 9.80 \times 8840 = 6351472 \ \mathrm{J} \]
02

Convert Joules to Calories

To convert the gravitational potential energy from joules to calories, use the conversion factor \( 1 \ \mathrm{cal} = 4.184 \ \mathrm{J} \). Calculating calories:\[ \Delta U = \frac{6351472}{4.184} \approx 1517972 \ \mathrm{cal} \]
03

Calculate Mass of Butter Required

Given that butter has a usable energy content of \( 6000 \ \mathrm{cal/g} \), calculate the mass of butter required to provide \( 1517972 \ \mathrm{cal} \) of energy. Calculating mass:\[ \text{mass} = \frac{1517972}{6000} \approx 253.0 \ \mathrm{g} \]

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Key Concepts

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

Energy Conversion
Energy conversion is the process of changing energy from one form to another. In the context of our exercise, we focus on the conversion of gravitational potential energy, which is the energy an object possesses due to its position in a gravitational field, to a form of energy that can be used or consumed.

As a person climbs a mountain, their body converts biochemical energy, from calories, into mechanical energy, and finally to gravitational potential energy as they gain height. This exercise involves the conversion of potential energy (due to the man's ascent) into a consumable energy equivalent, which in this case, is the energy content of butter.

By understanding energy conversion, we can calculate how much of one form (like food calories) is required to accomplish a task (like climbing). The main takeaway is how energy can be transformed and not necessarily created or destroyed, highlighting the law of energy conservation.
Calories to Joules
Converting calories to joules involves understanding that both units measure energy, but they are used in different contexts. Calories are often used in food energy contexts, while joules are commonly used in scientific contexts.

The conversion factor between these two units is fundamental:
  • 1 calorie (cal) is equal to 4.184 joules (J).
This means when you have energy measured in calories and you need it in joules (or vice versa), you multiply or divide by this conversion factor, respectively.

In our exercise, the gravitational potential energy calculated in joules was converted to calories to see how this energy equates in more familiar food energy terms. This step is crucial when relating physical activities to food energy, as it helps in understanding how much energy consumption is needed for certain tasks.
Mass Calculation
Mass calculation involves determining the amount of a substance needed to achieve or match a certain energy requirement, based on its energy content. In many practical scenarios, such as food consumption, it is essential to calculate mass to understand how much food is needed to equal certain energy expenditure.

For instance, in this problem, the mass of butter required to match the energy used during the climb was determined. Butter's energy content is a key factor here. It gives 6000 calories per gram, allowing us to calculate how much butter equates to the energy expended.

The formula used is straightforward:
  • The mass of butter = Total calories needed / Calories per gram of butter
This involves dividing the total calories calculated from the potential energy by butter's energy content per gram. By understanding this concept, you can relate energy expenditure to tangible items, like food.

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Most popular questions from this chapter

A lab sample of gas is taken through cycle abca shown in the \(p\) - \(V\) diagram of Fig. \(18-42 .\) The net work done is \(+1.2 \mathrm{~J}\). Along path \(a b\), the change in the internal energy is \(+3.0 \mathrm{~J}\) and the magnitude of the work done is \(5.0 \mathrm{~J} .\) Along path \(c a\), the energy transferred to the gas as heat is \(+2.5\) J. How much energy is transferred as heat along (a) path \(a b\) and \((\mathrm{b})\) path \(b c ?\)

\({ }^}\) "as The giant hornet Vespa mandarinia japonica preys on Japanese bees. However, if one of the hornets attempts to invade a beehive, several hundred of the bees quickly form a compact ball around the hornet to stop it. They don't sting, bite, crush, or suffocate it. Rather they overheat it by quickly raising their body temperatures from the normal \(35^{\circ} \mathrm{C}\) to \(47^{\circ} \mathrm{C}\) or \(48^{\circ} \mathrm{C}\), which is lethal to the hornet but not to the bees (Fig. 18-43). Assume the following: 500 bees form a ball of radius \(R=2.0 \mathrm{~cm}\) for a time \(t=20 \mathrm{~min}\), the primary loss of energy by the ball is by thermal radiation, the ball's surface has emissivity \(\varepsilon=0.80\), and the ball has a uniform temperature. On average, how much additional energy must each bee produce during the 20 min to maintain \(47^{\circ} \mathrm{C}\) ?

ssM A sample of gas expands from an initial pressure and volume of \(10 \mathrm{~Pa}\) and \(1.0 \mathrm{~m}^{3}\) to a final volume of \(2.0 \mathrm{~m}^{3}\). During the expansion, the pressure and volume are related by the equation \(p=a V^{2}\), where \(a=10 \mathrm{~N} / \mathrm{m}^{8} .\) Determine the work done by the gas during this expansion.

(c) Evaporative cooling of beverages. A cold beverage can be kept cold even on a warm day if it is slipped into a porous ceramic container that has been soaked in water. Assume that energy lost to evaporation matches the net energy gained via the radiation exchange through the top and side surfaces. The container and beverage have temperature \(T=15^{\circ} \mathrm{C}\), the environment has temperature \(T_{\text {env }}=32^{\circ} \mathrm{C}\), and the container is a cylinder with radius \(r=2.2 \mathrm{~cm}\) and height \(10 \mathrm{~cm}\). Approximate the emissivity as \(\varepsilon=1\), and neglect other energy exchanges. At what rate \(d m / d t\) is the container losing water mass?

A certain substance has a mass per mole of \(50.0 \mathrm{~g} / \mathrm{mol}\). When \(314 \mathrm{~J}\) is added as heat to a \(30.0 \mathrm{~g}\) sample, the sample's temperature rises from \(25.0^{\circ} \mathrm{C}\) to \(45.0^{\circ} \mathrm{C}\). What are the (a) specific heat and (b) molar specific heat of this substance? (c) How many moles are in the sample?
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