Converting Forces in a Circular Motion: Is It Possible and How?

In summary, the conversation discusses converting force components from Cartesian coordinates to radial and tangential components in a 2D Cartesian coordinate system. This can be done using rotation matrices and is relevant for studying central nervous system control strategies in handwriting. However, there may be difficulties in transforming displacement and stress tensors from Cartesian to cylindrical coordinates when the point lies on the Z axis.
  • #1
Lxhook
3
0
How would I go about the following:

A set of forces are pushing an object in a circle. These forces are recorded in the x and y direction in a 2D Cartesian coordinate system. I would like to convert the x and y force components to radial and tangential components.

Is this possible? If so, how?

All of the information I can find deals with using particle kinematics in such a conversion but I want a kinetic conversion.
Thanks!
 
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  • #2
Sorry if the above question seems like a homework problem or trivially possible/impossible. I study biomechanics and human motor control and and using the conversion to study central nervous system control strategies in handwriting. It's been awhile since I've looked at this sort of thing, so any help is much appreciated.
 
  • #3
Say [tex]f_x[/tex] and [tex]f_y[/tex] are your force components in cartesian coordinates (along x and y respectively), [tex]f_r[/tex] is the radial component of the force and [tex]f_t[/tex] is the tangential component of the force. And say that [tex]\theta[/tex] is the angle of the line segment between the center of the circle and the point where the force is applied (see attached diagram). (the angle is measured from the X axis (horizontal))

Your problem consists simply in expressing your force vector in another reference frame, which can be done by multiplying it by a rotation matrix. Using the definitions given above (and in the diagram), you have the equations:
[tex]
\left(\begin{array}{c} f_r \\ f_t \\ \end{array} \right)
=
\left( \begin{array}{cc}
cos \theta & sin \theta \\
-sin \theta & cos \theta \\
\end{array} \right)
\left(\begin{array}{c} f_x \\ f_y \\ \end{array} \right)
[/tex]
and, inversly,
[tex]
\left(\begin{array}{c} f_x \\ f_y \\ \end{array} \right)
=
\left( \begin{array}{cc}
cos \theta & -sin \theta \\
sin \theta & cos \theta \\
\end{array} \right)
\left(\begin{array}{c} f_r \\ f_t \\ \end{array} \right)
[/tex]

I hope this helps!

If you want to know more about rotation matrices, I suggest taking a look at Wolfram's page on the subject (mathworld.wolfram.com/RotationMatrix.html), which is presented in a more elegant fashion than its Wikipedia counterpart (en.wikipedia.org/wiki/Rotation_matrix).
(Sorry, apparently I'm too new here to be allowed to put direct URL links in my replies :confused:)
 

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  • #4
Thanks. I was way over thinking it. I appreciate your help a lot!
 
  • #5
Hi, I have some problem in transformation, if the point is lying on the r=0 axis. (i.e. Z axis), can anyone have any idea for this?
 
  • #6
The magnitude F of the centripetal force is equal to the mass m of the body times its velocity squared v[tex]^{2}[/tex] divided by the radius r of its path: F=mv[tex]^{2}[/tex]/r.

The x and y components of centripetal force within an X/Y plane are:

Fx[tex]^{2}[/tex] + Fy[tex]^{2}[/tex] = centripetal force[tex]^{2}[/tex]
 
  • #7
Perhaps by "set of forces are pushing an object in a circle," you are looking for torque. Foot-pounds of torque is the number of pounds force applied to the end of a handle times the length of the handle.

Torque is a tangential force with no radial component, centripetal force is a radial force with no tangential component.

The influence of a force that has both a tangential and a radial component depends on the details of the system. For example: If only circular motion is possible, application of radial force will not influence angular velocity of the target object.
 
  • #8
jayakkumar said:
Hi, I have some problem in transformation, if the point is lying on the r=0 axis. (i.e. Z axis), can anyone have any idea for this?

I'm not certain I understand your question. Maybe you could explain the context of your problem.

Nonetheless, let me just say that expressing a force in terms of radial and tangential components only makes sense with respect to a specific point (i.e. the center of rotation). In other words, radial and tangential components of an applied force are only relevant in the contexte of a rotation around a given point.

However, if a force is applied AT the center of rotation, it will not induce any rotation. This is why I do not understand why you would want to express such a force in radial/tangential components.
 
  • #9
WhiteFox said:
I'm not certain I understand your question. Maybe you could explain the context of your problem.

Nonetheless, let me just say that expressing a force in terms of radial and tangential components only makes sense with respect to a specific point (i.e. the center of rotation). In other words, radial and tangential components of an applied force are only relevant in the contexte of a rotation around a given point.

However, if a force is applied AT the center of rotation, it will not induce any rotation. This is why I do not understand why you would want to express such a force in radial/tangential components.

Thanks for the response.

Let me explain my question in detail.

My question about transformation is not for force components but displacements and stresses at a point in cylindrical coordinate frame.

I have displacement and stress tensors in Cartesian coordinate frames (XYZ) for some points and I would like to transfrom those data in Cylindrical coordinate frame (RTZ). I used the same formula given above for force transformation for R and Theta components and z component is same as cartesian Z component.

Here the formula has the cosine and sine theta, where the points lying in the z axis, singularity occurs in finding the RTZ of a point in Cylindrical Coordinate from Cartesian coordinate XYZ. I used the following formula to convert the XYZ to RTZ. R = SQRT (x*x + y*y) and theta = tan-1(x/y) and Z=Z.

Here I have problem with the points lying on Z axis where R=0 and theta is indeterminate.

Can anyone help me.?
 
  • #10
jayakkumar said:
Thanks for the response.

Let me explain my question in detail.

My question about transformation is not for force components but displacements and stresses at a point in cylindrical coordinate frame.

I have displacement and stress tensors in Cartesian coordinate frames (XYZ) for some points and I would like to transfrom those data in Cylindrical coordinate frame (RTZ). I used the same formula given above for force transformation for R and Theta components and z component is same as cartesian Z component.

Here the formula has the cosine and sine theta, where the points lying in the z axis, singularity occurs in finding the RTZ of a point in Cylindrical Coordinate from Cartesian coordinate XYZ. I used the following formula to convert the XYZ to RTZ. R = SQRT (x*x + y*y) and theta = tan-1(x/y) and Z=Z.

Here I have problem with the points lying on Z axis where R=0 and theta is indeterminate.

Can anyone help me.?

Ok, this clarifies a few things.

First, for the sake of clarity for anyone who may be reading this, I'd like to point out that my original answer (and the original question) did not involve polar coordinates (which describe a point in terms of angle 'theta' and arc length 'r'). It was rather a matter of rotating cartesian coordinates for a given angle 'theta'.

Now, on the subject of converting from XYZ to RTZ, there is in fact an indetermination for 'theta' at R = 0, and unfortunatly I do not have a magic answer.

I am by no means an expert on the subject and I have never worked with displacement and stress tensors (therefore I can't really grasp the context of your problem), but if it can help at least a bit, my advices would be:
1 - If this is for an analytical problem, and if you can work in cartesian coordinates (or generalized coordinates (not an expert at that either) or another singularity-free coordinate system) it might allow you to have 'cleaner' equations, albeit possibly more complex;
2 - If you are working on an algorithm, you probably will have to resort to a 'hack' for the R=0 case (e.g. if 'theta' is of little importance, set it to 0 or to a random value)

You might also want to check the litterature on your subject (although you probably already did). Chances are someone faced the same problem and found a way around.

Finally, you can always hope that someone more knowledgeable will drop by! ;)
 
  • #11
Thanks for the suggestion White Box. I'll study the literature and get it clarified. Thanks again for responsible suggestion.
 

1. What is Circular Force Conversion?

Circular Force Conversion is the process of converting a force acting in a circular motion into a linear force. This is commonly seen in situations where a circular force is applied to an object, but the desired outcome is a linear movement.

2. How is Circular Force Conversion calculated?

Circular Force Conversion is calculated using the formula F=mv^2/r, where F is the linear force, m is the mass of the object, v is the velocity, and r is the radius of the circular path.

3. What are some real-world examples of Circular Force Conversion?

One example of Circular Force Conversion is seen in centrifuges, where a circular force is used to separate substances of different densities. Another example is in car steering systems, where a circular force from the steering wheel is converted into linear motion to turn the wheels.

4. Can Circular Force Conversion be reversed?

Yes, Circular Force Conversion can be reversed. This is known as Linear Force Conversion and is the process of converting a linear force into a circular force. It is commonly seen in systems such as pulleys and gears.

5. How important is Circular Force Conversion in the field of physics?

Circular Force Conversion is a fundamental concept in physics and is used in many fields such as engineering, mechanics, and astrophysics. It is crucial in understanding and predicting the motion of objects in circular paths.

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