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Chapter 13: Problem 77

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? $$\begin{array}{l}{\text { A. } 2 g} \\ {\text { B. } 3 g} \\ {\text { C. } 5 g} \\ {\text { D. } 10 g}\end{array}$$

Short Answer

Expert verified
The strongest gravitational acceleration is closest to 3g.

Step by step solution

01

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.
02

Identify the given values

We know the following: - Height blood is pumped on Earth: 1.3 m - Density of blood: 1.04 g/cm鲁 (which is 1040 kg/m鲁 in SI units) - Height to pump blood on another planet: 0.5 m above the heart - Gravitational acceleration on Earth: 9.8 m/s虏.
03

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 \(P_{earth} = P_{planet}\):\[ 1040 \times 9.8 \times 1.3 = 1040 g' \times 0.5 \]
04

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:\[ g' = \frac{9.8 \times 1.3}{0.5} = 25.48 \, \text{m/s}^2 \]
05

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.

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Key Concepts

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

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/cm鲁, which translates to 1040 kg/m鲁 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鈥檚 calculated pressure equal to the other planet's. This shows how gravitational acceleration directly alters pressure needed in biological processes such as blood circulation.

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Most popular questions from this chapter

A liquid is used to make a mercury-type barometer, as described in Section \(13.2 .\) The barometer is intended for spacefaring astronauts. At the surface of the earth, the column of liquid rises to a height of 2185 \(\mathrm{mm}\) , but on the surface of Planet \(\mathrm{X},\) where the acceleration due to gravity is one-fourth of its value on earth, the column rises to only 725 \(\mathrm{mm} .\) Find (a) the density of the liquid and (b) the atmospheric pressure at the surface of Planet \(\mathrm{X}\) .

A barrel contains a 0.120 \(\mathrm{m}\) layer of oil of density 600 \(\mathrm{kg} / \mathrm{m}^{3}\) floating on water that is 0.250 \(\mathrm{m}\) deep. (a) What is the gauge pressure at the oil-water interface? (b) What is the gauge pressure at the bottom of the barrel?

Blood pressure. Systemic blood pressure is expressed as the ratio of the systolic pressure (when the heart first ejects blood into the arteries) to the diastolic pressure (when the heart is relaxed): systemic blood pressure \(=\frac{\text { systolic pressure }}{\text { diastolic pressure }}\) Both pressures are measured at the level of the heart and are expressed in millimeters of mercury (or torr), although the units are not written. Normal systemic blood pressure is \(\frac{120}{80}\) . (a) What are the maximum and minimum forces (in newtons) that the blood exerts against each square centimeter of the heart for a person with normal blood pressure? (b) As pointed out in the text, blood pressure is normally measured on the upper arm at the same height as the heart. Due to therapy for an injury, a patient's upper arm is extended 30.0 \(\mathrm{cm}\) above his heart. In that position, what should be his systemic blood pressure reading, expressed in the standard way, if he has normal blood pressure? The density of blood is 1060 \(\mathrm{kg} / \mathrm{m}^{3}\) .

(a) Calculate the buoyant force of air (density 1.20 \(\mathrm{kg} / \mathrm{m}^{3} )\) on a spherical party balloon that has a radius of 15.0 \(\mathrm{cm}\) . (b) If the rubber of the balloon itself has a mass of 2.00 \(\mathrm{g}\) and the balloon is filled with helium (density 0.166 \(\mathrm{kg} / \mathrm{m}^{3}\) ), calculate the net upward force (the "lift") that acts on it in air.

You are designing a diving bell to withstand the pressure of seawater at a depth of 250 \(\mathrm{m}\) (a) What is the gauge pressure at this depth? (You can ignore the small changes in the density of the water with depth.) (b) At the 250 \(\mathrm{m}\) depth, what is the net force due to the water outside and the air inside the bell on a circular glass window 30.0 \(\mathrm{cm}\) in diameter if the pressure inside the diving bell equals the pressure at the surface of the water? (You may ignore the small variation in pressure over the surface of the window.)
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