Q16CQ - College Physics (Urone)

Question

A device used since the 1940 s to measure the kick or recoil of the body due to heart beats is the “ballistocardiograph.” What physics principle(s) are involved here to measure the force of cardiac contraction? How might we construct such a device?

Step-by-Step Solution

Answer

The ballistocardiograph works on the principles of impulsive force and conservation of momentum, which are the consequences of Newton’s second and third laws of motion, respectively.

Step 1: Describe how to calculate the force of a cardiac contraction.

A force that acts on a body for a short period of time is called cardiac contraction force. An impulsive force is the name for this type of force. This force is defined as the ratio of momentum change to time interval. $F = \frac{△P}{△t}$

Here, P_i = 0 is the force, $△P$  is the change in momentum and  $△t$ is the time interval. This is a consequence of Newton’s second law of motion. The time interval is equal to the pulsation time, and you need to calculate the rate of change of momentum.

Step 2: Arrive at an expression to calculate the force of cardiac contraction.

As per Newton's third law of motion, momentum is conserved in the body. Momentum is the product of mass and velocity. The initial momentum will be zero because the body is immobile.

P_i = 0

The final momentum is equal to the sum of the momentum of the heart pump and the momentum of the body that recoils in the direction opposite to the heart pump.

$P_f = m{v}_1 - \left(M - m\right)v_2$

Here, m is the mass of blood, M is the mass of the body, v₁ is the velocity of blood and v₂ is the velocity of body.  Therefore, change in momentum is calculated as follows:

$$\begin{gathered} △P = P_f - P \\ △P =m{v}_1 - \left(M-m\right)v_2 - 0 \\ △P =m\left(v_1-v_2\right) - M{v}_2 \end{gathered}$$

Substitute this value in the expression of force.

$F=\frac{m\left(v_1+v_2\right)-M{v}_2}{△t}$

Thus, the force can be calculated.

Step 3: (c) Describe how to construct a ballistocardiograph.

The ballistocardiograph can be constructed by taking a highly mobile table that is capable of sensing the movements of the body that is placed on the table, and the recording system can be attached to it. It gives the frequency at which the heart beats.

Therefore, the ballistocardiograph works on the principle of Newton’s second and third laws of motion.