# Total Angular Momentum of a Tractrix

• MHB
• Dr Edward
In summary, the speaker is discussing their attempt to calculate the angular momentum and total angular momentum of a 5.0 m tractrix at two different hitchpoint speeds (60 and 70 km/h). They found that the lower speed resulted in a higher total angular momentum, which was unexpected. They provide equations for positions, derivatives, angular velocity, angular momentum, and total angular momentum, and ask for help in spotting an error and converting the equation for instantaneous angular momentum to total or cumulative momentum. They mention that momentum is cumulative and use the analogy of adding an impulse to a linear system to explain this concept. They express gratitude for any help in solving their problem.
Dr Edward
I have a 5.0 m tractrix and am trying to work out angular momentum and total angular momentum for two hitchpoint speeds 60 & 70 km/h.
My result shows a higher total angular momentum for the lower speed.
This is not what I expected.
Here are my equations
Positions:
View attachment 8589
Derivatives
View attachment 8590
Angular velocity
View attachment 8591
Angular Momentum
View attachment 8592
TOTAL ANGULAR MOMENTUM
View attachment 8593
If anyone can help spot the error I would be extremely grateful

#### Attachments

• Positions.JPG
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• Derivatives.JPG
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• Ang Vel.JPG
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• Ang Momentum.JPG
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• Total Mom.JPG
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I have spotted the error but don't know how to fix it.
The equation for L(t) is for instantaneous angular momentum.
Momentum is cumulative. i.e. if I add an impulse (F*t) to the classic linear analogy I impart momentum. Newton's 1st law says it will go on forever.
If I now hit it with the same impulse I will double the momentum.
Is there any mathematical genius that will convert the equation for instant angular momentum (L(t)) to total or cumulative momenta.

I should be ever so grateful.

## 1. What is total angular momentum?

Total angular momentum is a physical quantity that describes the rotational motion of an object. It takes into account both the rotational speed and the distribution of mass around the axis of rotation.

## 2. What is a tractrix?

A tractrix is a curve that is created by tracing the instantaneous positions of a point on a string that is being unwound from a stationary point while being pulled by a force. It is often described as the path of a point on a rolling wheel.

## 3. How is total angular momentum related to a tractrix?

The total angular momentum of a tractrix is equal to the product of the moment of inertia and the angular velocity of the point on the string. This relationship is derived from the conservation of angular momentum principle.

## 4. What factors affect the total angular momentum of a tractrix?

The total angular momentum of a tractrix is affected by the mass distribution of the string, the speed at which it is being pulled, and the distance between the stationary point and the point on the string. Any changes in these factors will result in a change in the total angular momentum.

## 5. How is total angular momentum calculated for a tractrix?

The total angular momentum of a tractrix can be calculated using the equation L = Iω, where L is the total angular momentum, I is the moment of inertia, and ω is the angular velocity. The moment of inertia can be calculated using the mass distribution of the string and the distance between the stationary point and the point on the string.

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