Atmospheric pressure is caused by the weight of a column of air above you pushing down. At sea level on Earth, this pressure is equal to \(10^{5}\) newtons per square meter \(\left(\mathrm{N} / \mathrm{m}^{2}\right)\) a. Estimate the total force on the top of your head from this pressure. b. Recall that the acceleration due to gravity is \(9.8 \mathrm{m} / \mathrm{s}^{2}\). If the force in part (a) were caused by a kangaroo sitting on your head, what would the mass of the kangaroo be? c. Assume a typical kangaroo has a mass of 60 kg. How many kangaroos would have to be sitting on your head to be equal to the extremely massive kangaroo in part (b)? d. Why are you not crushed by this astonishing force on your head?

Step by step solution

01

Calculate the Area of the Top of Your Head

Assume the top of the head is roughly circular with a diameter of about 0.20 meters (20 cm). First, calculate the radius: \(\text{radius} = \frac{0.20}{2} = 0.10 \text{ meters}\). Now calculate the area of the top of the head using the formula for the area of a circle: \(\text{Area} = \pi \times (\text{radius})^2 = \pi \times (0.10)^2 = 0.0314 \text{ square meters}\).

02

Calculate the Total Force from Atmospheric Pressure

Use the given atmospheric pressure at sea level (\(10^5 \text{N/m}^2\)) and the area of the top of your head from step 1 to find the total force: \(\text{Force} = \text{Pressure} \times \text{Area} = 10^5 \text{ N/m}^2 \times 0.0314 \text{ m}^2 = 3140 \text{ N}\).

03

Calculate the Mass Equivalent of the Force

Use the relationship between force, mass, and acceleration due to gravity to find the equivalent mass of the kangaroo: \(\text{Force} = \text{mass} \times \text{gravity}\) \(3140 \text{ N} = \text{mass} \times 9.8 \text{m/s}^2\). Solving for mass: \(\text{mass} = \frac{3140 \text{ N}}{9.8 \text{m/s}^2} = 320 \text{ kg}\).

04

Calculate the Number of Typical Kangaroos

Given that a typical kangaroo has a mass of 60 kg, calculate how many such kangaroos would equal the mass calculated in step 3: \(\text{Number of Kangaroos} = \frac{320 \text{ kg}}{60 \text{ kg}} = 5.33\). Because we can’t have a fraction of a kangaroo in practice, we approximate to 5 or 6 kangaroos.

05

Understand Why You Are Not Crushed

The human body is accustomed to atmospheric pressure because it is balanced by the internal pressure within the body. This equilibrium prevents us from feeling the actual force of the atmosphere pushing down on us.

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Key Concepts

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

Force Calculation

In physics, force is calculated using the formula: \[ \text{Force} = \text{Pressure} \times \text{Area} \] Pressure is defined as the amount of force exerted per unit area. In this exercise, we had to estimate the force on top of a person's head by atmospheric pressure. Given the atmospheric pressure at sea level \(10^{5} \text{ N/m}^2 \), we multiplied it by the area of the top of the head. Given the head’s area as \(0.0314 \text{ m}^2 \), we calculated the total force to be 3140 N, which surprisingly is a large force capable of almost flattening things. To ensure accuracy, remember to use the correct area and pressure values.

Gravity

Gravity is the force by which a planet or other body draws objects toward its center. On Earth, the acceleration due to gravity is \(9.8 \text{ m/s}^2 \). In the exercise, we used this value to determine the mass that would generate the same force as the calculated atmospheric pressure. By using the formula \[ \text{Force} = \text{mass} \times \text{gravity} \], we re-arranged it to solve for mass: \[ \text{mass} = \frac{\text{Force}}{\text{gravity}} \] Hence, given the force of 3140 N, we found that the mass is 320 kg. This helps to understand how gravity and mass are interconnected in force calculations.

Area Calculation

Calculating the area is a fundamental step in various physics problems. For the area of the head, considering it as a circular shape, we used the formula for the area of a circle: \[ \text{Area} = \pi \times (\text{radius})^2 \] By converting the diameter of 0.20 meters to a radius and squaring it, we obtained the area \[ \text{Area} = \pi \times (0.10)^2 = 0.0314 \text{ m}^2 \] This precise area is critical for the force calculations as a larger area would yield a larger force and vice versa.

Pressure

Pressure is the force exerted per unit area of a surface. Given by the formula: \[ \text{Pressure} = \frac{\text{Force}}{\text{Area}} \] Atmospheric pressure at sea level is about \(10^{5} \text{ N/m}^2 \). In our problem, this pressure exerted on the small area of the top of the head resulted in a surprising amount of force (3140 N). Understanding how pressure works helps in understanding why large objects exert significant forces over small areas, a crucial point when studying fluid mechanics and gas laws.

Equilibrium

A balance of forces, or equilibrium, is essential in physics. Our bodies are not crushed under atmospheric pressure because they are in a state of equilibrium. The internal pressure within our bodies balances out the atmospheric pressure. Without this equilibrium, the immense force from atmospheric pressure would be harmful. This concept extends to various fields, demonstrating the importance of understanding and maintaining balance in forces.

Physics Problem Solving

Solving a physics problem often involves understanding and applying several concepts step-by-step. This particular problem involved:

• Calculating the area of the top of the head
• Using atmospheric pressure to find the total force
• Relating this force to mass using gravity
• Comparing this mass to known objects (kangaroos)
• Exploring the concept of equilibrium

Critical thinking and methodical problem-solving are key. Each step builds on the previous one, integrating calculations and conceptual understanding to address the problem comprehensively.