Question

You are going to lift an elephant (mass \(\left.=5.0 \cdot 10^{3} \mathrm{~kg}\right)\) over your head \((2.0 \mathrm{~m}\) vertical displacement). a) Calculate the work required to do this. You will lift the elephant slowly (no tossing of the elephant allowed!). If you want, you can use a pulley system. (As you saw in Chapter \(5,\) this does not change the energy required to lift the elephant, but it definitely reduces the force required to do so.) b) How many doughnuts ( 250 food calories each) must you metabolize to supply the energy for this feat?

Short answer

Answer: You must eat 1 doughnut to provide enough energy to lift the elephant over your head.

Step-by-step solution

Calculate the Gravitational Potential Energy Gained

Since we are lifting the elephant to a height of 2.0 meters, we need to calculate the change in gravitational potential energy gained by the elephant. The formula for gravitational potential energy (PE) is given by: PE = mgh where m is the mass of the elephant, g is the acceleration due to gravity (approximately \(9.8\,\mathrm{m/s^2}\)), and h is the height (vertical displacement). Now, plug in the given values: PE = \((5.0 \cdot 10^3\,\mathrm{kg})(9.8\,\mathrm{m/s^2})(2.0\,\mathrm{m})\)

Calculate the Work Required

The work-energy theorem states that the work done on an object is equal to its change in gravitational potential energy. Therefore, the work required to lift the elephant is equal to the potential energy gained by the elephant: Work = PE = \((5.0 \cdot 10^3\,\mathrm{kg})(9.8\,\mathrm{m/s^2})(2.0\,\mathrm{m})\) Work = 98,000 J (joules).

Convert Work to Food Calories

Since we are given the food calories in doughnuts, we need to convert the work done in joules to food calories. We know that 1 food calorie (Cal) equals 4,184 joules. To convert the work done in joules to food calories, we divide the work by 4,184: Food Calories = Work (in joules) / 4,184 joules/Cal Food Calories = 98,000 J / 4184 J/Cal Food Calories = 23.42 Cal

Calculate the Number of Doughnuts Needed

Now that we have the energy required in food calories, we can find out the number of doughnuts needed. We are given that each doughnut has 250 food calories. To find the number of doughnuts needed, we divide the energy required by the food calories per doughnut: Number of Doughnuts = Food Calories / Food Calories per Doughnut Number of Doughnuts = 23.42 Cal / 250 Cal/Doughnut Number of Doughnuts = 0.09368 However, it is not possible to consume a fraction of a doughnut, so you would need to eat 1 doughnut to have enough energy to lift the elephant. Final answers: a) The work required to lift the elephant is 98,000 J. b) You must eat 1 doughnut to provide enough energy to lift the elephant over your head.

Key concepts

Work-Energy Theorem

The work-energy theorem is a fundamental principle in physics that connects the concept of work with the energy of a system. According to this theorem, the work done on an object is equal to the change in its kinetic energy. However, when dealing with gravitational potential energy, we apply a similar approach: the work done against the Earth’s gravitational field is stored as potential energy in the object.

In our exercise, as we lift the elephant slowly, all our work is going into increasing the elephant's gravitational potential energy without increasing its kinetic energy (since there's no tossing, the elephant’s velocity isn't changing). So, the equation for work (\(W = Fd\text{, where } F \text{ is force and } d \text{ is displacement}\)) simplifies to the gravitational potential energy formula (\(PE = mgh\text{, where } m \text{ is mass, } g \text{ is gravity, and } h \text{ is height}\)).

Understanding that work and energy are interchangeable concepts provides a valuable insight into how energy is conserved and transferred within a physical system, which leads us to look at energy conservation next.

Caloric Energy Conversion

When delving into caloric energy conversion, it's all about understanding how energy from food translates into energy that can do work, such as lifting an elephant. Food energy is usually measured in Calories (with a capital C), which are actually kilocalories in terms of physics calories (cal).

When we talk about the Calories from food, we're essentially referring to the amount of energy the food can provide to our body. In physics, energy is measured in joules (J), and our exercise demonstrates how to translate food Calories into the energy used for physical work. The conversion factor is pivotal: 1 food Calorie equals 4,184 joules.

This conversion allows us to calculate how many Calories we need to ingest to perform a certain task - in this case, lifting an elephant. It's intriguing to note that the body's complex biochemical processes, such as metabolizing a doughnut, provide the energy that can be used for mechanical work like lifting weights.

Conservation of Energy

The concept of conservation of energy is the underpinning principle of physics that dictates that energy cannot be created or destroyed, only transformed from one form to another. This idea is crucial for interpreting the work-energy scenarios like the one presented in our exercise.

When we lift an object against gravity, we convert energy from one form (such as chemical energy in our muscles or calories from a doughnut) into gravitational potential energy. No energy gets lost in this process; it simply changes form. This is why the total amount of energy calculated to lift the elephant can be directly equated with the energy content of a certain number of doughnuts.

Recognizing how energy conservation plays into everyday tasks and scientific experiments alike helps us appreciate the ongoing energy exchanges occurring within and around us. Similarly, it underscores the importance of the efficient use of energy, whether it’s while doing physics problems or considering our daily consumption and activities.