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[8 In research in cardiology and exercise physiology, it is often important to know the mass of blood pumped by a person's heart in one stroke. This information can be obtained by means of a ballistocardiograph. The instrument works as follows: The subject lies on a horizontal pallet floating on a film of air. Friction on the pallet is negligible. Initially, the momentum of the system is zero. When the heart beats, it expels a mass \(m\) of blood into the aorta with speed \(v\), and the body and platform move in the opposite direction with speed \(V\). The speed of the blood can be determined independently (for example, by observing an ultrasound Doppler shift). Assume that the blood's speed is \(50.0 \mathrm{~cm} / \mathrm{s}\) in one typical trial. The mass of the subject plus the pallet is \(54.0 \mathrm{~kg}\). The pallet moves \(6.00 \times 10^{-5} \mathrm{~m}\) in \(0.160 \mathrm{~s}\) after one heartbeat. Calculate the mass of blood that leaves the heart. Assume that the mass of blood is negligible compared with the total mass of the person. This simplified example illustrates the principle of ballistocardiography, but in practice a more sophisticated model of heart function is used.

Step by step solution

01

Understanding the Problem

The problem describes a person lying on frictionless pallet which moves when the heart beats due to the expulsion of blood. Since they assume that the overall system (the subject and the pallet) was initially at rest, the net momentum before and after the heartbeat should still be zero, as per the principle of conservation of momentum.

02

Initialize Given Variables

The speed of the blood \(v\) is \(50.0 \mathrm{~cm/s}\), which converts to \(0.50 \mathrm{~m/s}\). The combined mass of the subject and the pallet \(M\) is \(54.0 \mathrm{~kg}\). The opposite velocity at which the pallet moves \(V\) can be found by dividing the distance by the time, that is \(V=6.00 \times 10^{-5} \mathrm{~m} / 0.160 \mathrm{~s} = 0.000375 \mathrm{~m/s}\). We seek the mass of blood \(m\).

03

Apply Conservation of Momentum

Conservation of momentum states that the momentum before the event must be equal to the momentum after the event. In this case, the event is the heartbeat. Before the heartbeat, the system is at rest, so the net momentum is zero. After the heartbeat, the blood is moving at velocity \(v\) and has mass \(m\), and the subject (with the pallet) is moving at opposite velocity \(V\) and has mass \(M\). Hence, the momentum of the blood, \(m * v\), plus the momentum of the subject with the pallet, \(M * (-V)\), must equal zero.

04

Solve for m

We can rearrange the equation we got from conservation of momentum, \(m * v + M * (-V) = 0\), to solve for m. This gives us \(m = M * V / v = 54.0 \mathrm{~kg} * 0.000375 \mathrm{~m/s} / 0.50 \mathrm{~m/s} = 0.0405 \mathrm{~kg}\).

05

Convert to Grams

Since the mass of blood is usually given in grams, we change the units from kilograms to grams by multiplying by 1000. Thus the mass is \(0.0405 \mathrm{~kg} * 1000 = 40.5 \mathrm{~g}\).

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Key Concepts

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

Momentum Conservation

In the context of ballistocardiography, momentum conservation is a fundamental concept. When the heart pumps blood, it generates a momentum. Since the system of the person and the pallet was initially at rest, the total momentum must remain at zero after the heart beats.

• Initially, the momentum is zero because the person and pallet are stationary.
• When the blood moves one direction, the pallet moves in the opposite direction to maintain this balance.

This reaction results in a measurable displacement of the pallet. Through this displacement, the amount of blood pumped by the heart can be calculated using the conservation of momentum principle.

Blood Velocity

Blood velocity is crucial for understanding how momentum is transferred in ballistocardiography. The speed at which blood is ejected from the heart helps determine the shift experienced by the person's body and the pallet.

• In the exercise, blood velocity is measured at a typical value of 50 cm/s or 0.50 m/s.


• This quick burst of speed is essential to displace the significant mass of the person and pallet.

Blood velocity can be estimated through various medical techniques such as ultrasound. This data becomes an integral parameter in solving how much mass gets transferred during each heartbeat.

Exercise Physiology

Understanding the significance of ballistocardiography in exercise physiology highlights its utility in measuring heart performance.

• This method can accurately gauge heart functions by evaluating blood volume pumped in each heartbeat.


• It is valuable for researching cardiac efficiency and health during exercise or at rest.
• Understanding heart muscle dynamics through this quantitative measure helps us optimize exercise routines.

Homogeny between exertion and physiological responses supports advances in exercise planning and cardiac diagnostics.

Conservation of Momentum Principle

The conservation of momentum principle states that the total momentum in an isolated system remains constant if no external forces act upon it.

• This principle is applied in ballistocardiography, where the body and blood are considered a closed system.


• The movement of the blood during a heartbeat causes an equal and opposite reaction from the body and pallet.

Mathematically, it implies that the momentum of the moving blood and the opposite momentum of the body and pallet equal zero. This allows us to set up equations to calculate the mass of blood ejected by the heart.