Problem 7 - Fundamentals of Physics

Question

In 1654 Otto von Guericke, inventor of the air pump, gave a demonstration before the noblemen of the Holy Roman Empire in which two teams of eight horses could not pull apart two evacuated brass hemispheres. (a) Assuming the hemispheres have (strong) thin walls, so that \(R\) in Fig. 14-29 may be considered both the inside and outside radius, show that the force \(\vec{F}\) required to pull apart the hemispheres has magnitude \(F=\pi R^{2} \Delta p\), where \(\Delta p\) is the difference between the pressures outside and inside the sphere. (b) Taking \(R\) as \(30 \mathrm{~cm}\), the inside pressure as \(0.10 \mathrm{~atm}\), and the outside pressure as \(1.00 \mathrm{~atm}\), find the force magnitude the teams of horses would have had to exert to pull apart the hemispheres. (c) Explain why one team of horses could have proved the point just as well if the hemispheres were attached to a sturdy wall.

Step-by-Step Solution

Answer

(a) The force is \( F = \pi R^2 \Delta p \). (b) Force magnitude is about 25782 N. (c) One team suffices if one hemisphere is attached to a wall.

Step 1: Understand the Problem and Definitions

The goal is to find the force required to pull apart two hemispheres with a pressure difference between their inside and outside. The outside pressure is higher than the inside pressure, creating a net force holding them together due to the vacuum created inside.

Step 2: Apply the Concept of Pressure and Force

Pressure is defined as force per unit area, given by the formula \( P = \frac{F}{A} \). Rearranging this, the force is given by \( F = P \cdot A \). In this case, the force acts across the effective area over the evacuated hemispheres.

Step 3: Determine the Effective Area

When pulling the hemispheres apart, consider only the circular area over which the pressure difference acts. This area is the circular cross-section with radius \( R \), and its area is \( A = \pi R^2 \).

Step 4: Relate the Force to Pressure Difference

The force magnitude is the product of the pressure difference \( \Delta p \) and the area \( A = \pi R^2 \). Thus, \( F = \Delta p \cdot \pi R^2 \).

Step 5: Calculate Pressure Difference

Given that the outside pressure \( P_{\text{outside}} = 1.00 \) atm and the inside pressure \( P_{\text{inside}} = 0.10 \) atm, the pressure difference \( \Delta p = P_{\text{outside}} - P_{\text{inside}} = 0.90 \) atm.

Step 6: Determine Force Magnitude

Convert the pressure difference from atm to N/m²: 1 atm = 101325 N/m², so \( \Delta p = 0.90 \times 101325 \). With \( R = 0.30 \) m, calculate \( F = \pi (0.30)^2 \times (0.90 \times 101325) \).

Step 7: Calculation

Substitute values: \( F = \pi (0.09) \times 91192.5 \approx 25782 \) N. This is the force required to pull apart the hemispheres.

Step 8: Explanation for Part (c)

If one side of the hemispheres is attached to a sturdy wall, there is no need for opposing teams because the wall provides the reaction force. Thus, only one team of horses is enough to pull the other hemisphere away.

Key Concepts

Pressure Difference

The concept of pressure difference plays a crucial role in understanding many physical phenomena. Pressure itself is defined as the force per unit area exerted in a direction perpendicular to the surface of an object. When there is a variance in pressure between two regions, it leads to a pressure difference. In the case of Otto von Guericke's experiment, the pressure difference is the difference between the atmospheric pressure outside the hemispheres and the lower pressure inside, created by evacuating air through a pump.

This pressure difference results in a net force that can act inwards or outwards on a surface. When the inside pressure is significantly lower, like in this demonstration, it creates a strong inward force. This force is what makes it difficult to separate the hemispheres, as the outside higher pressure pushes them together, making it a stunning demonstration of the effects of atmospheric pressure.

Force Calculation

In physics, force calculation is derived from fundamental principles such as Newton's laws and basic definitions of pressure. To calculate the force necessary to pull the hemispheres apart, we start with the relationship between force, pressure, and area. Pressure is defined with the formula: \( P = \frac{F}{A} \), where \( P \) is the pressure, \( F \) is the force, and \( A \) is the area the pressure acts upon.

Rearranging this equation, we find that the force can be calculated as \( F = P \times A \). In the context of von Guericke's spheres, the effective area \( A \) is the area of the circular cross-section, namely \( \pi R^2 \). The force, then, relates directly to the pressure difference, \( \Delta p \), across this area. Thus, the calculation of force becomes: \( F = \Delta p \times \pi R^2 \), neatly tying together the concept of pressure difference and force calculation.

Vacuum Pressure

Vacuum pressure refers to the pressure of a gas that is lower than the atmospheric pressure. In the experiment conducted by Otto von Guericke, creating a vacuum inside the steel hemispheres was crucial. The vacuum pump he invented made it possible to remove air from the inside, significantly reducing the internal pressure and creating a vacuum.

This reduction in pressure leads to a significant difference between the inside and outside pressures, establishing a vacuum pressure scenario. Inside the hemispheres, the pressure was only 0.10 atm, while outside the pressure was the normal 1.00 atm, making the inside nearly a vacuum relative to the outside. Such a large pressure differential magnifies the force required to separate the hemispheres, as the external atmospheric pressure dominates, pressing the hemispheres together.

Otto von Guericke

Otto von Guericke was a pioneer in the field of physics and engineering during the 17th century. He made significant contributions with his invention of the air pump, which was revolutionary at the time. His most famous experiment involved the evacuated brass hemispheres, which demonstrated the power and force associated with atmospheric pressure.

Von Guericke's demonstration was not just a testament to the strength of air pressure but also an early exploration into the properties of vacuums. By showing that teams of horses could not pull the hemispheres apart, von Guericke effectively illustrated both the potency of pressure difference and the importance of understanding the vacuum pressure. His experiments laid important groundwork for future studies in thermodynamics and pressure which remain relevant today, highlighting the interconnectivity between experiments and theoretical advancements.