Force ratios in Parametric Kinematics

Suppose you have you a part of a machine is following a parametric curve in space...

And another part is following a different curve in space...

The force ratio is just the ratio of the magnitude of the derivatives of both curves. 

Starting with a simple example... 

Say something is connecting a part of a machine moving from A to B and another part moving from C to D. The machine happens to be a ramp and a rope but a ramp is just probably the simplest machine. The two motions can be parameterized like so:

The distance from C to A is 30.06 and we'd like that distance between the two points on the curves to stay constant so...

Now we can solve for s in terms of t. 

Now R becomes:

The magnitude of the derivative of this equation is:

And the magnitude of the derivative of Q is 5. 

Since the output speed is slower, at 96%, the force on the object on the ramp has increased, via the action of the ramp as a machine. This means that it would require more force to stop the part of the machine from going along R then it would to stop the part moving along Q. 

Another example:

In the last blog post I calculated the motion of P4 for a crank length of 1 and a Coupler length of 5 to be:

Of course R2 is parameterized as:

The derivative of the first is: 

in the y direction and 0 in the x direction. so this is also the total magnitude of the derivative. 

The magnitude of the derivative of B is:

So the ratio of the two is:

A graph of it looks like: