Problem 52 - College Physics

Question

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 (e.g., 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 at a speed of \(6.00 \times 10^{-5} \mathrm{~m}\) 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

Answer

The mass of the blood that leaves the heart in one beat is approximately \(0.00648 Kg\), or \(6.48 g\).

Step 1: Recognize that there is conservation of momentum

In this situation, we have conservation of momentum, which means that the total momentum before the heart beats is equal to the total momentum after the heart beats.

Step 2: Formulate the momentum conservation equation

Let's denote \(m\) as the mass of the blood. From the conservation of momentum, we get the equation: \(m \cdot v = M \cdot V\). This equation shows that the momentum of the blood is equal to the momentum of the subject and pallet.

Step 3: Solve for the mass of the blood

To find the mass of the blood, we can rearrange the equation to \(m = \frac{M \cdot V}{v}\). By plugging in given values: \(m = \frac{54Kg \cdot 6.00 \times 10^{-5} m/s}{0.5 m/s}\)

Step 4: Calculate the answer

Calculating the above equation gives us the value for the mass of the blood pumped by the heart in one stroke.

Key Concepts

Conservation of Momentum

The principle of conservation of momentum is fundamental to understanding how a ballistocardiograph operates. In physics, momentum is the product of an object's mass and its velocity. According to the law of conservation of momentum, the total momentum of a closed system remains constant if no external forces are acting upon it.

During a heartbeat, the heart exerts a force to expel blood, and as a result, blood gains forward momentum. To preserve the total momentum of the system, which was initially zero before the heartbeat, the body and pallet must move in the opposite direction, gaining an equal amount of momentum in reverse. This backward movement of the pallet, which is very slight, can be measured by the ballistocardiograph and is key to calculating the mass of blood ejected by the heart in one stroke.

It is through careful measurement and application of this principle that researchers can quantify the mass of blood without the need for invasive procedures, thus improving the safety and simplicity of cardiac analysis.

Momentum in Cardiology

Ballistocardiography stands as a prime example of how the concept of momentum plays an essential role in cardiology. The heart is essentially a pump that generates momentum each time it expels blood. By quantifying this momentum transfer from the heart to the blood, medical professionals can obtain important indicators of cardiac health.

For instance, a change in the usual momentum transfer might suggest alterations in the stroke volume, that is, the amount of blood the heart pumps with each beat. Potentially, this could indicate heart conditions that warrant further investigation. The application of momentum concepts in cardiology allows for non-invasive monitoring and analysis, which can be crucial in diagnosing and managing heart diseases.

Physics in Medicine

The use of physics principles, such as momentum conservation, goes beyond the laboratory and is widely applied in the field of medicine. Ballistocardiography is one among many diagnostics tools that leverage physics to improve our understanding of human physiology and to facilitate medical diagnoses.

Other uses of physics in medicine include imaging technologies like X-rays, MRIs, and CT scans, which are based on various physical phenomena. The integration of concepts from physics into medicine enhances the quality of patient care by providing more precise diagnostic tools and treatments, and by contributing to the development of new technologies aimed at improving health outcomes.

Understanding these principles and the way they can be applied in real-life medical scenarios is indispensable for students pursuing careers at the intersection of physics and medicine.

Heart Function Analysis

Heart function analysis is pivotal in diagnosing and managing various cardiac diseases. Ballistocardiography specifically assists in this by providing a non-invasive method to study the heart's ability to pump blood. By analyzing the heart's function through the mass of blood it propels, clinicians can deduce the strength and efficiency of each heartbeat.

In ballistocardiography, measures such as stroke volume and cardiac output can be indirectly assessed, offering valuable information about the cardiac cycle. The technique is based on intricate models of heart function, often considering variables like heart rate, blood density, and fluid dynamics of the circulatory system. The simplicity of the ballistocardiograph method described in the exercise is a stepping stone towards understanding these more complex models and the overall cardiac performance.