# B Ballistocardiograph and conservation of momentum

#### FranzDiCoccio

Hi,

once again I'm probably asking a question that is more about human physiology than physics (I recently asked a question that had to do with hearing).

I found a (definitely too hasty) reference to a ballistocardiograph in a high school textbook.
So I got curious about the way this apparatus works. Roughly speaking, this is an instrument to visualise heartbeat exploiting the conservation of momentum. Here you can see one in action.
A person lays still on a bed that is free to roll back and forth (head-feet direction) with little or (ideally) no friction. The person's heart pumps blood and, as a result of the action-reaction principle, the bed sways back and forth (going in the opposite direction as the blood flows).
Also, there's an old post about this apparatus, but it does not go into the details of how it works, and it is not open for discussion any more.

I'm having some trouble understanding the oscillation, because I assume that blood moves around a circuit.
Of course the blood system is not a single loop... but whatever blood going towards the head from the heart, there should be an equal amount of blood entering the heart (and in general, going the other way).

Perhaps dissipation of kinetic energy in the blood flow has a role. Possibly also the fact that the blood goes out through a big conduit (aorta) which then splits in smaller and smaller channels...

I'm assuming that blood is pumped away from the heart in one direction. If I'm right, I'm not very sure as to what causes the bed to oscillate back.
Perhaps it is the "U" shape of the aorta.

I thought of this analogy: I'm on a long platform that can slide frictionlessly on the floor. At the back end of this platform there is a wall.
If I jump towards the back wall the whole platform goes forward. Then I hit the wall and bounce back. This causes the platform to invert the direction of its motion.

I represent the blood, and the wall represents the "U" bend in the aorta, which causes the blood to invert its direction of motion? Would this be a reasonable analogy for what is happening?

There is still something I am missing... Because I think that whatever amount of blood is pumped towards the bend in the aorta, this pushes an equal amount of blood that is beyond the bend in the opposite direction...

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#### sophiecentaur

Gold Member
I assume that blood moves around a circuit.
I used to notice my bed moved in time with my pulse and tapped against the wall - so it's a 'real' effect. The plumbing is not rigid so you can get a phase delay in the timing of the pulses round the system. Hence there may be a net movement of mass in one direction and then in the other - I guess the Aorta is shifting a lot of blood (at least a fistful) when the heart (and aorta) contracts and the capillaries in the lower half will expand a bit, temporarily. No average displacement over time.
I thought of this analogy: I'm on a long platform that can slide frictionlessly on the floor.
A better analogy is standing on the platform with a dumbbell in your hand. You move the bell away from you and your body moves backwards. Momentum is conserved and the Centre of Mass doesn't move.

#### Dale

Mentor
The plumbing is not rigid
This is the key effect. With nonrigid pipes it is not true that the amount of fluid going in one direction is the same as the amount of fluid going in the other direction around the circuit.

#### FranzDiCoccio

Hi,

thanks again for bearing with my questions!

I used to notice my bed moved in time with my pulse and tapped against the wall - so it's a 'real' effect. The plumbing is not rigid so you can get a phase delay in the timing of the pulses round the system. Hence there may be a net movement of mass in one direction and then in the other - I guess the Aorta is shifting a lot of blood (at least a fistful) when the heart (and aorta) contracts and the capillaries in the lower half will expand a bit, temporarily. No average displacement over time.
A better analogy is standing on the platform with a dumbbell in your hand. You move the bell away from you and your body moves backwards. Momentum is conserved and the Centre of Mass doesn't move.
Uhm... yes I see how your analogy produces an oscillatory motion. But it seems too simplistic.

Perhaps, using the first part of your reply, and Dale's comment:

imagine that the wall in my analogy is elastic. When I crash into it, it yields and then is recalled back, projecting me backwards. That could provide the delay you're mentioning. Now a lot of people doing the same as me on the platform would "simulate" the blood flow.

Again, my question originates from a textbook exercise. But the way it is phrased makes me think that the person has been stabbed or something like that.
The wording suggests that the person and the ballistocardiograph move one way, much like a squid would swim.
I understand that the text must be kept simple, but
I would have at least mentioned that there is no average displacement over time, as sophiecentaur writes.

#### FranzDiCoccio

This is the key effect. With nonrigid pipes it is not true that the amount of fluid going in one direction is the same as the amount of fluid going in the other direction around the circuit.
Hi Dale, thanks for replying.

Would you use a ballistocardiograph in a textbook exercise about conservation of momentum?

What the exercise actually wants to know is the velocity of the person+platform system, assuming that a certain amount of blood is pumped towards the head at a certain velocity. What the reader knows is: mass of the person+platform, mass of the pumped blood, velocity of the pumped blood.
The solution is easy, but the phrasing is a bit misleading.
The solution is (practically) the same as if the pumped blood gushes out from a wound.

I'm not sure I'm explaining it correctly (sorry, I'm not a native English speaker).

#### Dale

Mentor
Would you use a ballistocardiograph in a textbook exercise about conservation of momentum?
Probably not, but it is a good exercise in that it shows that conservation of momentum applies even for very “messy” systems.

#### DrGreg

Gold Member
When your heart expands, it sucks blood in from all the veins in your body.

When your heart contracts, it pushes blood out into all the arteries in your body.

These two things don't happen simultaneously.

Furthermore, blood doesn't flow directly from the arteries into the veins. The arteries have branches that get narrower and narrower, like a tree, and push blood into capilliaries all round the body, which expand a little (under increased pressure) to accommodate the extra blood. The veins form a separate "tree" of branches that suck blood out of the capilliaries, which then contract (under decreased pressure).

So at any given time blood is either moving away from the heart, or moving towards the heart, or not moving much (in a short pause between each heartbeat).

#### FranzDiCoccio

[...]
So at any given time blood is either moving away from the heart, or moving towards the heart, or not moving much (in a short pause between each heartbeat).
This is interesting. So, after all, when the heart contracts it is indeed as if the blood is gushing from a wound.... Sort of... I mean, on account of the fact that, for a little while, it flows only in the arteries (roughly in one direction).

If this is the case, the exercise makes sense, roughly.

Furthermore, blood doesn't flow directly from the arteries into the veins. The arteries have branches that get narrower and narrower, like a tree, and push blood into capilliaries all round the body, which expand a little (under increased pressure) to accommodate the extra blood. The veins form a separate "tree" of branches that suck blood out of the capilliaries, which then contract (under decreased pressure).
Ok, I figured that the branching of the circulatory system would make some difference.
However, shouldn't the pressure decrease as the blood flows more rapidly?

#### DrGreg

Gold Member
However, shouldn't the pressure decrease as the blood flows more rapidly?
But there is a force, from the heart, increasing the pressure and pushing blood out of the heart and into the arteries and capilliaries. Then the force stops and the pressure drops. When the heart expands, there is effectively a negative force decreasing the pressure on the veins and sucking blood from the capilliaries and veins into the heart.

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