Finding an Equation for Measuring Reactive Force at the Gastrocnemius

[Richard90]

Homework Statement

Ankle reactive force?

Relevant Equations

Unknown

Good morning,

I’m new to the forum, thank you in advance. I’m an Italian med student based in France.

I’m struggling to find an equation for a project that I am following.

I should measure the reactive force at the level of the gastrocnemius (shown in the figure, red arrow). I know the force applied on the plantar side of the foot in the direction of the knee (14 N), the angle of ankle dorsiflexion (30 degrees), the position of the center of rotation in cm (the medial malleolus at the red spot) and the height of the force applied on the foot (20 cm at the level of the yellow spot).

Anybody knows if with these values is it possible to measure the reactive force (red arrow) that is opposed to the plantar force?

Best Regards,
Riccardo.

 

[kuruman]

Hi @Richard90 and welcome to PF. I have two questions for you.

1. Is the 30° angle really measured from the vertical as shown in the photo? It looks like 60° to me.
2. Where is line of action of the unknown force? Is it (more or less) along the yellow dotted line shown below?

 

[Richard90]

Hi @Richard90 and welcome to PF. I have two questions for you.

1. Is the 30° angle really measured from the vertical as shown in the photo? It looks like 60° to me.
2. Where is line of action of the unknown force? Is it (more or less) along the yellow dotted line shown below?
View attachment 321099

Good afternoon Kuruman,

Thank you very much for your answer,

Yes, exactly on that line and should counteract the force applied and on the plantar side of the foot.

The image is not exact, the foot should be at 90 normally. The starting point is at 90 degrees of flexion (foot perpendicular to the leg). The angle is measured from a flexion more than 90.

Do you think there is a equation to approximate the moment of force on the yellow line you have drawn?

 

[Lnewqban]

Welcome, @Richard90 !
What are those two green sensors?

Please, see how to calculate the force excerted by the muscle at the tendon-bone point of connection:
https://en.m.wikipedia.org/wiki/Mechanical_advantage#Lever

 

[kuruman]

Do you think there is a equation to approximate the moment of force on the yellow line you have drawn?

Yes, there is an equation. However, this is a homework forum thread and our rules prevent me from just giving you the answer, but I can guide you to it. The approach is to set the moment generated by the 14 N equal to the moment generated by the unknown force and solve the equation for the unknown force. Can you do that?

 

[Richard90]

Welcome, @Richard90 !
What are those two green sensors?

Please, see how to calculate the force excerted by the muscle at the tendon-bone point of connection:
https://en.m.wikipedia.org/wiki/Mechanical_advantage#Lever

Thank you for the link,

It’s about also a force of torque right?

 

[Richard90]

Yes, there is an equation. However, this is a homework forum thread and our rules prevent me from just giving you the answer, but I can guide you to it. The approach is to set the moment generated by the 14 N equal to the moment generated by the unknown force and solve the equation for the unknown force. Can you do that?

I should add in the equation the angle and measure the torque I think,

with the ankle at 90 degrees, 20 - 8 cm = 12 cm from the center of rotation, the resulting force should be 9.3 N? (14:12 / x:8) 14x8 / 12= 9.3 N

I found this formula for the torque, I should subtract 14-9.3 = 4.7N applied at the center of rotation.

If a force of magnitude F = 4.7 N is applied
at a distance r = 12 m from the axis of rotation
in an orientation where r makes the angle θ = 30degrees
with respect to the line of action of the force,
then the lever arm = 5.9 m (r sin θ)
and the magnitude of the torque is τ = 29 N m.

Is this right? Thank you very much for the help.

Riccardo.

 

[kuruman]

with the ankle at 90 degrees, 20 - 8 cm = 12 cm from the center of rotation, the resulting force should be 9.3 N? (14:12 / x:8) 14x8 / 12= 9.3 N

I don't understand these numbers. The measurement in the photo says 7 cm not 8 cm but that's a minor problem. The major problem is that the reaction force is less than the applied force. That can't be correct because the lever arm of the reaction force is shorter than the lever arm of the applied force. I think you tried to do it in your head instead of drawing a good diagram. The diagram below is drawn to scale for the 90° ankle angle using the numbers from the photo. Can you calculate R now?

We will worry about the 30° angle later. In fact I think finding the reaction force for an arbitrary angle θ would be better.

 

[Richard90]

I don't understand these numbers. The measurement in the photo says 7 cm not 8 cm but that's a minor problem. The major problem is that the reaction force is less than the applied force. That can't be correct because the lever arm of the reaction force is shorter than the lever arm of the applied force. I think you tried to do it in your head instead of drawing a good diagram. The diagram below is drawn to scale for the 90° ankle angle using the numbers from the photo. Can you calculate R now?

We will worry about the 30° angle later. In fact I think finding the reaction force for an arbitrary angle θ would be better.

View attachment 321172

Thank you for the diagram,

It should be F * 0.14 * sen alfa = R * 0.06 * sen beta ,

R = F 0.06 sen beta / 0.14 sen alfa, but I don’t know how to mesure the angles. Is it right the equation? I did the scientific high school but I forgot almost everything. And then how to apply the 30 degrees of flexion?

I think I don’t have the capability to find and solve the equation..

 

[kuruman]

You don't need the angles. The lever arm is defined as the perpendicular distance from the axis of rotation to the line of action of the force. The torque is the force times the lever arm. The lines of action of both forces are horizontal. What are the perpendicular distances from the axis of rotation to those lines of action? Read the plot.

 

[berkeman]

Homework Statement:: Ankle reactive force?
Relevant Equations:: Unknown

I’m an Italian med student based in France.

I’m struggling to find an equation for a project that I am following.

Just as an aside, this seems like a strange schoolwork question for a med student, IMO. The first couple of years of med schoo usually are focussed on A&P and related bio subjects. Is this an extra class that you've picked up in addition to your initial medical school classes, or are you still before medical school and working on undergrad projects related to pre-med work?

It just seems like a strange question for a medical school student to be asking without basic physics class help or knowledge. We are of course fine helping you as you keep showing your work.

 

[Richard90]

You don't need the angles. The lever arm is defined as the perpendicular distance from the axis of rotation to the line of action of the force. The torque is the force times the lever arm. The lines of action of both forces are horizontal. What are the perpendicular distances from the axis of rotation to those lines of action? Read the plot.

To have the system in balance the force R should be 32.6N? And then the ankle flex just 30 degrees because is limited from gastroc contracture, how is possible to solve it?

 

[Richard90]

Just as an aside, this seems like a strange schoolwork question for a med student, IMO. The first couple of years of med schoo usually are focussed on A&P and related bio subjects. Is this an extra class that you've picked up in addition to your initial medical school classes, or are you still before medical school and working on undergrad projects related to pre-med work?

It just seems like a strange question for a medical school student to be asking without basic physics class help or knowledge. We are of course fine helping you as you keep showing your work.

It’s how I wrote, it’s few years I don’t do these kind of topics.

 

[kuruman]

To have the system in balance the force R should be 32.6N?

That is correct.

And then the ankle flex just 30 degrees because is limited from gastroc contracture, how is possible to solve it?

It can be possibly solved if you can explain to me what "gastroc contracture" is because my understanding of anatomy is pretty much non-existent. If flexing beyond 30° cannot happen, this means that either the reaction force has reached an upper limit and cannot increase any more or that there is an additional force or there is an additional force that you have not accounted for.

I understood your original question to be "Find the reaction force R when the ankle is at 30° and 14 N are pushing in a direction perpendicular to the sole at the yellow dot." Has this changed or did I misunderstand your original question?

You realize of course that 32.6 N is not the answer to the 30° ankle position because the lever arm of the reaction force changes. That would have been the path to follow once we make sure that you remember how torques are calculated.

 

[Lnewqban]

Thank you for the link,

It’s about also a force of torque right?

No need to use torque in this case.
The tendon is acting on the lever formed by the distance between its line of action passing by the point of attachment to the bone and the fulcrum formed by the joint.

The bone is forced to pivot about the fulcrum and, as a natural result, another force of smaller magnitude appears at the plantar side of the foot.
The force exerted by the ball of the foot is less than the force pulling the tendon and muscle because the distance between its line of action and the fulcrum or joint is greater.

Picture copied from
https://mass4d.com/blogs/clinicians-blog/foot-biomechanical-theories-explained

 

[Richard90]

That is correct.

It can be possibly solved if you can explain to me what "gastroc contracture" is because my understanding of anatomy is pretty much non-existent. If flexing beyond 30° cannot happen, this means that either the reaction force has reached an upper limit and cannot increase any more or that there is an additional force or there is an additional force that you have not accounted for.

I understood your original question to be "Find the reaction force R when the ankle is at 30° and 14 N are pushing in a direction perpendicular to the sole at the yellow dot." Has this changed or did I misunderstand your original question?

You realize of course that 32.6 N is not the answer to the 30° ankle position because the lever arm of the reaction force changes. That would have been the path to follow once we make sure that you remember how torques are calculated.

Thank you very much for the answer and the explanation,

Yes, exactly what I was meaning. It’s hard to explain what I wanted to find because the project is at the first stage of development.

τ = rFsin(θ)

If I use the torque should be : τ = 0.14 x (32.6-14) sin(30) ?

We want to measure the reactive force at the level of the Gastrocnemius, for 14N. And this case example the ankle flexed from 0 to 30 degrees. In other cases this angle for the same force is different, according to the intrinsic elasticity of the tendon of the subject. For a higher force the measured angle was greater, till the reaching of the maximum dorsiflexion permitted by the specific anatomy of the subject.

 

[Richard90]

No need to use torque in this case.
The tendon is acting on the lever formed by the distance between its line of action passing by the point of attachment to the bone and the fulcrum formed by the joint.

The bone is forced to pivot about the fulcrum and, as a natural result, another force of smaller magnitude appears at the plantar side of the foot.
The force exerted by the ball of the foot is less than the force pulling the tendon and muscle because the distance between its line of action and the fulcrum or joint is greater.

Picture copied from
https://mass4d.com/blogs/clinicians-blog/foot-biomechanical-theories-explained

View attachment 321297

Thank you for your point of view,

But is this applied to walking? Or for a muscle relaxed and a force applied at a certain distance from the fulcrum as in this experiment?

 

[Lnewqban]

Thank you for your point of view,

But is this applied to walking? Or for a muscle relaxed and a force applied at a certain distance from the fulcrum as in this experiment?

Walking is a more complicated process in which forces and directions constantly change.

As I understand your experiment or research, you are testing how far the feet can be rotated (counter-clockwise in the previous picture) until the muscle is fully stretched and offering enough resistance (to be calculated) to stop deflection over 30 degrees from relaxed position once a force is applied to the ball of the foot.

Is that accurate?
If so, how is the force applied onto the foot?
A constant vertical force, or one that changes direction with the foot?

 

[kuruman]

Thank you very much for the answer and the explanation,

Yes, exactly what I was meaning. It’s hard to explain what I wanted to find because the project is at the first stage of development.

τ = rFsin(θ)

If I use the torque should be : τ = 0.14 x (32.6-14) sin(30) ?

We want to measure the reactive force at the level of the Gastrocnemius, for 14N. And this case example the ankle flexed from 0 to 30 degrees. In other cases this angle for the same force is different, according to the intrinsic elasticity of the tendon of the subject. For a higher force the measured angle was greater, till the reaching of the maximum dorsiflexion permitted by the specific anatomy of the subject.

What you use depends on where the action line of reactive force is. Shown below is the way one could approach the problem. In the picture h = 7 cm, w = 6 cm, L = 20 cm and θ is some arbitrary angle. I have assumed that, as the ankle flexes from θ = 0 to some other value, the distance w from the center of rotation (red dot) to the point of application of R remains the same. Is that correct? If it is correct, then the lever arm of the torque generated by R is $w~\cos\!\theta$ and changes as the angle changes.

I have a question for you to satisfy my own curiosity. Is it necessary to measure the reactive force when the foot is at some angle? As you say, there is the problem of intrinsic elasticity that varies from subject to subject. Why not measure it when when the foot is vertical?