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Preamble
Once my friend and classmate - Oleg Yakovlev - suggested to consider "advanced" version of well known school task.
School case:
A block slides down an inclined plane from a height H and (possibly) continues to move horizontally. Find where the block will stop using known values of: the angle of inclination and the coefficient of friction k .
"Advanced" case:
The task is the same, but with an additional condition: the inclined plane passes into a horizontal way smoothly (without a jump in the derivative) along an arc of a circle of radius R.
The task turned out to be interesting.
But this is not about that 😉
Along work with this task, I discovered a couple of funny "paradoxes". Of course, it cannot be compared with how Achilles was catching up with the tortoise, but still ...
So, this is about that 😉
What we remember from school
Let's start with school again. The movement of a point on a circle.
Imagine a small body that is moving around a certain center on a weightless inextensible thread. For simplicity, we assume that there are no external forces. Without the thread, the body would move in a straight line with a constant speed. The thread does not allow the body to move away from the center: the thread tension force N creates centripetal acceleration to the body, and changes the direction of movement (direction of velocity). In this case, the tensile force does not change the (kinetic) energy of the body, since the force is always perpendicular to the (elementary) displacement. And this means that the speed of the body does not change in magnitude.
Now let's slightly change the conditions: let the body move in a circle not due to the thread, but "leaning" on the inner wall of the cylinder. Now the centripetal acceleration of the body is due to the reaction of the support N. Everything is as before. The only force (support reaction) is always perpendicular to the elementary displacement, which means that the energy of the body does not change, and its velocity remains constant in magnitude.
So, now about paradoxes
1. Polygons and circle.
Consider the approximation of a circle by polygons. Both ancient mathematicians and modern programmers did this. 😉
However, the task of the motion of a body along the inner surface of a cylinder, some questions arise when considering this approximation...
1.1. Movement without friction.
1.1.1. Inside polygon (along edges): Consider two neighboring segments S1 and S2 of a regular polygon (inscribed in a circle). While the body is moving along one of the edges of the polygon (the straight line), nothing acts on the body ( even the support, since there are no external forces and there is nothing to react to ).
Only at the point of moving to the next edge of the polygon the body hits surface...
If the body moves along S1 with the speed V1 , then at the vertex A (when moving to S2) the speed of the body decreases: V2= V1 *cos(a) . The velocity component V1 *sin(a) is lost (inelastic heat/deformation losses; elastic interaction is not considered - it is more complicated). At the next vertex, another hit, then another... *)
That is, when moving along the inner surface of the polygon , the speed of the body decreases at each and every vertex even without friction (!)
1.1.2. However, the speed of the body remains constant (there are no losses) when moving along a circle without friction.
How can you approximate a circle with a polygon?
*) Obviously, if the polygon is a square, then the angle a = 90 degrees, and the speed is completely lost ( V2 = 0 ) at the first vertex / transition. By increasing the number of edges of the polygon, the angle a decreases. So, the loss at the vertex (transition from segment to segment) decreases. However, the total number of vertices/transitions grows with number of edges.
1.2. Let's add friction ( F = -kN, k > 0 ) in consideration
1.2.1. Moving along the inner surface of the polygon:
If there are no any external forces, the body does not "feel" any support reaction ( N = 0 ) while moving along a straight line segment (the edge of the polygon). This means that there is no friction force (F = -kN = 0) even with a non-zero coefficient of friction (!) The body still loses speed at each vertex due to "hit", but it is not related to friction.
1.2.2. However, at moving along a circle, centripetal acceleration appears precisely due to the non-zero reaction of the support N. And that means there is friction force F = -kN.
Again: How to "conjugate" such differences? How does a polygon transformed into a circle?
2. Perpetuum mobile?
Now let's consider the friction of the body on the (inner) surface of the cylinder in more detail. The force of friction is directed (always) against the direction of motion. So the speed of the body is decreasing. And a reasonable question arises: when the body stops?
According to the Newton's second law, we have (along trajectory):
Here m is the mass of the body, a is the acceleration along the trajectory, kN is the friction force (proportional to the support reaction force), k is the friction coefficient, N is the support reaction force.
The support reaction N provides centripetal acceleration, that is:
Here V is the velocity of the body, R is the radius of the circle (trajectory).
Because
we get a simple differential equation:
or
, where
Integrating over time t , we get:
At
we have
and
Then, the expression for the velocity is:
That is, the velocity decreases over time, reaching zero at infinite time(!).
Let's see what happens to the length distance traveled L. Let's remember that : dL/dt = V ...
Integrating again, we get:
At
we have
, so
Thus, expression for the length distance traveled is:
That is, the distance traveled L grows with time to infinity for any (!) value of the friction coefficient ( k > 0 ).
Friction does not stop the moving body (!)
What is this? A perpetual motion machine?
Hints
1.1. ...when moving along the inner surface of the polygon the speed of the body decreases from vertex to vertex even without friction (!) While when moving along a circle (without friction), the speed of the body remains constant (there are no losses).
2. Perpetuum mobile?
The length of distance traveled really grows with time as logarithm of (1+ct). Try to estimate (and compare) velocities and/or distances passed for the first and for the thousandth unit of time. All is clear now 😉
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