Problem 77 - College Physics

Question

On the earth, the pressure generated by the heart is sufficient to pump blood to a height of 1.3 \(\mathrm{m} .\) The density of blood is 1.04 \(\mathrm{g} / \mathrm{cm}^{3} . )\) If the top of your head were 0.5 \(\mathrm{m}\) above your heart, what would be the strongest gravitational acceleration that you could endure on another planet before your heart would be unable to pump blood to your brain? $$\begin{array}{l}{\text { A. } 2 g} \\ {\text { B. } 3 g} \\ {\text { C. } 5 g} \\ {\text { D. } 10 g}\end{array}$$

Step-by-Step Solution

Answer

The strongest gravitational acceleration is closest to 3g.

Step 1: Understand the problem

We need to find the gravitational acceleration on another planet for which the heart, which can pump blood to a height of 1.3 m on Earth, would be unable to pump blood to the brain if the brain is 0.5 m above the heart.

Step 2: Identify the given values

We know the following: - Height blood is pumped on Earth: 1.3 m - Density of blood: 1.04 g/cm³ (which is 1040 kg/m³ in SI units) - Height to pump blood on another planet: 0.5 m above the heart - Gravitational acceleration on Earth: 9.8 m/s².

Step 3: Use pressure formula to find gravitational acceleration

The pressure the heart generates can be considered as \[ P = \rho g h \]where \(\rho\) is the density of blood, \(g\) is the gravitational acceleration, and \(h\) is the height. On Earth, \(g_{earth} = 9.8 \, \text{m/s}^2\):\[ P_{earth} = 1040 \times 9.8 \times 1.3 \]On another planet, this same pressure would need to be:\[ P_{planet} = \rho g' \times 0.5 \]Set \(P_{earth} = P_{planet}\):\[ 1040 \times 9.8 \times 1.3 = 1040 g' \times 0.5 \]

Step 4: Solve for gravitational acceleration

By solving the above equation, we find:\[ g' = \frac{1040 \times 9.8 \times 1.3}{1040 \times 0.5} \]Simplifying gives:\[ g' = \frac{9.8 \times 1.3}{0.5} = 25.48 \, \text{m/s}^2 \]

Step 5: Convert to multiples of Earth's gravity

Since 1 g is equivalent to Earth's gravity, which is \(9.8 \, \text{m/s}^2\), find the equivalent g value:\[ \frac{25.48}{9.8} \approx 2.6 \]Round to the nearest choice number.

Key Concepts

Pressure Calculation

Pressure is a vital concept when it comes to understanding how forces are distributed across areas. In this exercise, it is important to calculate how much pressure the heart generates to pump blood to the brain. Here, pressure is calculated using the formula:

• \(P = \rho g h\)

where:

• \(P\) is the pressure.


• \(\rho\) is the density of the fluid (in this case, blood).


• \(g\) is the gravitational acceleration.


• \(h\) is the height to which the fluid needs to be pumped.

First, we calculate the pressure that the heart generates at Earth's gravity (\(g_{earth} = 9.8 \, \text{m/s}^2\)) using the height the blood can be pumped on Earth (1.3 m). By equating this to the pressure required on another planet where the pumping height is 0.5 m, we can solve for the other planet's gravitational pull. Understanding pressure calculation helps us determine how different accelerations affect blood flow.

Density of Blood

Density is the mass per unit volume of a substance, and understanding the density of blood is critical in calculations like these. Blood's density is given as 1.04 g/cm³, which translates to 1040 kg/m³ in SI units. The density of blood affects how much pressure is needed to circulate through the body's vascular system.
In the context of this exercise, the density remains constant across Earth and any other planet since density is an intrinsic property of the substance. The density is crucial to determining the pressure exerted by the heart, as it features in the pressure calculation formula, \(P = \rho g h\). This dependence on density means that any changes in blood density would directly influence the pressure generated by the heart, though in this problem, we're focused on gravitational differences.

Pressure Formula

The pressure formula \(P = \rho g h\) is applied to find out how the same heart pressure for pumping blood works differently under various gravitational conditions. The exercise probes this formula under circumstances where different gravitational accelerations affect how high blood can be pumped.
Breaking it down:

• \(\rho \): density of the blood fluid, influences how pressure relates to weight and buoyancy.
• \(g \): gravitational acceleration, varies on different celestial bodies, affecting the force exerted by gravity on any mass.
• \(h \): the height, tells how much work is needed to move the fluid to this level under the influence of gravity.

On Earth, the gravitational force is standard, and the equation allows us to derive \(P\) for a known height. However, when on a different planet, we solve for \(g'\), the new gravitational acceleration, by setting the Earth’s calculated pressure equal to the other planet's. This shows how gravitational acceleration directly alters pressure needed in biological processes such as blood circulation.