Question

In order to get in shape for mountain climbing, an avid hiker with a mass of \(60 .\) kg ascends the stairs in the world's tallest structure, the \(828 \mathrm{m}\) tall Burj Khalifa in Dubai, United Arab Emirates. Assume that she eats energy bars on the way up and that her body is \(25 \%\) efficient in converting the energy content of the bars into the work of climbing. How many energy bars does she have to eat if a single bar produces \(1.08 \times 10^{3} \mathrm{kJ}\) of energy upon metabolizing?

Short answer

The hiker needs to eat approximately 141 energy bars to have enough energy to climb the building, considering her body's 25% efficiency in converting the energy content of the bars into the work of climbing.

Step-by-step solution

Calculate work done against gravity

The work done against gravity (W) can be calculated using the formula: \[W = mgh,\] where m is the mass of the hiker, g is the acceleration due to gravity, and h is the height of the building. Given the mass of the hiker \(m = 60 \, kg\) and the height of the building \(h = 828 \, m\). The acceleration due to gravity is approximately \(g = 9.81 \, m/s^2\). Plugging the given values into the formula, we get: \[W = 60 \, kg \times 9.81 \, m/s^2 \times 828 \, m.\]

Calculate total energy required

We are given that the hiker's body is 25% efficient in converting the energy content of the bars into the work of climbing. So, the total energy (E) required to perform the work can be calculated using the formula: \[E = \frac{W}{efficiency},\] where efficiency is given as 25% or 0.25. From Step 1, we have calculated the work done against gravity, W. Now, substitute the value of W in the formula: \[E = \frac{60 \, kg \times 9.81 \, m/s^2 \times 828 \, m}{0.25}.\]

Calculate the number of energy bars needed

Now that we have calculated the total energy required to climb the building, we need to find out how many energy bars need to be eaten to provide that energy. Each energy bar produces \(1.08 \times 10^3 \, kJ\) of energy when metabolized. Let's call the number of energy bars needed as N. So, from the total energy required and the energy produced per energy bar, we can calculate N using the formula: \[N = \frac{E}{Energy\_per\_bar},\] where Energy_per_bar is the energy produced per energy bar. From Step 2, we have calculated the total energy required (E). Now, substitute the value of E in the formula and convert kJ to J: \[N = \frac{60 \, kg \times 9.81 \, m/s^2 \times 828 \, m}{0.25 \times 1.08 \times 10^3 \, kJ \times 1000 \, J/kJ}.\] After solving the expression, round the result to the nearest whole number, as you cannot have a fraction of an energy bar. This will give you the number of energy bars needed for the hiker to climb the building.

Key concepts

Work Done Against Gravity

Let's dive into the concept of work done against gravity, which is a fundamental principle in physics related to energy. When we talk about climbing a structure like the Burj Khalifa, an immense amount of energy is required to overcome the force of gravity pulling the hiker towards the Earth.

To understand the work concept, imagine you are lifting a book from the floor onto a shelf. Here, 'work' refers to the effort exerted to move the book against gravity's pull. In a scientific context, 'work' is calculated as the product of the force applied to an object and the distance over which that force is applied. Mathematically, when the force is gravity, the formula for work done becomes:
\[W = m \times g \times h,\] where W is the work done (in joules), m is the mass (in kilograms), g is the acceleration due to gravity (in meters per second squared), and h is the height (in meters) the mass is lifted.

This equation shows that lifting heavier objects or lifting over greater heights requires more work, thereby consuming more energy. For the hiker climbing Burj Khalifa, as her mass and the height of the climb are constants, and gravity remains the same at Earth's surface, the work done can be precisely calculated.

Mass-Height-Gravity Relationship

Understanding the mass-height-gravity relationship is crucial when analyzing scenarios that involve moving objects vertically in a gravitational field, like elevating a hiker to the top of the Burj Khalifa. The relationship between these three factors can be viewed through Newton's law of universal gravitation and how these factors influence the work done against gravity.

The Earth's gravity pulls objects with a certain force, depending on their mass. This force is quantified as weight in physics, given by the equation: \[weight = m \times g.\] In essence, the more massive an object, the greater its weight, and thus the more force needed to lift it. This directly impacts the work done because work is the measure of energy transfer when a force moves an object – in this case, through a distance, h, against gravity.

Therefore, to ascend the same height, a heavier hiker would require more energy than a lighter one. The mass-height-gravity relationship is pivotal in calculating the work needed for the hiker to reach the top of Burj Khalifa. By considering the hiker's mass, the height of the climb, and the constant of gravity, the total work against gravity can be determined.

Caloric Energy of Food

The energy needed for activities like climbing stairs originates from the caloric energy of food we consume. When we eat, our bodies convert the food into energy through metabolic processes, which then fuels our muscles and other systems. Calories are the units used to measure the energy in food, and when we talk about energy bars or any food items, understanding their caloric content is essential for planning activities requiring physical effort.

The caloric energy provided by food is not entirely converted into work, such as climbing – a significant portion is lost as heat or used for other bodily functions. This efficiency variation is often expressed in a percentage, displaying the amount of consumed energy that is transformed into mechanical work. In the exercise example, the hiker's body has a 25% efficiency rate, indicating that only a quarter of the energy from the hiker's food intake is available for climbing.

To calculate the number of energy bars the hiker needs, the total energy requirement for the climb must first be established based on the work done against gravity. Then, this energy requirement is divided by the amount of energy one energy bar can provide. This gives us the total number of bars the hiker must consume to supply sufficient energy for the hike, taking her body's conversion efficiency into account.