# Calculus

Gastel
Posers and Puzzles 17 Jun '06 22:07
1. 17 Jun '06 22:07
A ball rolls down a chain that is held at two points P1 and P2. The ball is already in motion, but there is a friction u1 constant for the complete path.

What is the velocity function of the ball?
2. 18 Jun '06 01:15
I just signed up for calc 2...
3. XanthosNZ
Cancerous Bus Crash
18 Jun '06 06:34
Originally posted by Coconut
I just signed up for calc 2...
You'll fail.
4. 18 Jun '06 17:10
Originally posted by Gastel
A ball rolls down a chain that is held at two points P1 and P2. The ball is already in motion, but there is a friction u1 constant for the complete path.

What is the velocity function of the ball?
If P(t) = at^2+bt+c, and b = a negative constant, then V(t) = 2at + b.
5. 18 Jun '06 17:20
Originally posted by XanthosNZ
You'll fail.
Never. I'll drop out before that happens.
6. 18 Jun '06 20:00
Originally posted by Ramiri15
If P(t) = at^2+bt+c, and b = a negative constant, then V(t) = 2at + b.
I don't believe any combination of b, a or c could ever make that the proper function for P(t). Of course, I'm an electrical engineer not a mechanical one, but my memory of this problem is that it is not quadratic.

By the way, this is just to break the monotony of "What are the factors of x^2 + 3x + 2?" questions.
7. 18 Jun '06 22:15
Originally posted by Gastel
I don't believe any combination of b, a or c could ever make that the proper function for P(t). Of course, I'm an electrical engineer not a mechanical one, but my memory of this problem is that it is not quadratic.

By the way, this is just to break the monotony of "What are the factors of x^2 + 3x + 2?" questions.
True, there is no way the function can be described as a quadratic because eventually, due to the friction, the ball would reach a constant velocity.
8. 19 Jun '06 03:321 edit
Originally posted by Ramiri15
True, there is no way the function can be described as a quadratic because eventually, due to the friction, the ball would reach a constant velocity.
yes, i see it as an exponential rise-to-max type scenario, much like terminal velocity. but the probelm takes on vector-type difficulties early in the piece, since gravity is pulling the ball down, and the chain is pulling the ball sideways, with the chains force dependant on the angle between the line P1-P2 and the vertical. the combination of these will give the downward spped without friction. the friction u1 can be thought of as acting on the final sum of gravity and angle, so that should be easier to put in. i'm not fantastic at doing vector math...

Edit: ah ha, it seems galileo considered this probelm, and found for a ball rolling down a slope, the velocity at the end of the slope is v= sqrt((10g-h)/7), where g= accleration due to gravity, h= overall height dropped by ball. should give a start point anyway.
9. 19 Jun '06 11:02
Originally posted by xcomradex
yes, i see it as an exponential rise-to-max type scenario, much like terminal velocity. but the probelm takes on vector-type difficulties early in the piece, since gravity is pulling the ball down, and the chain is pulling the ball sideways, with the chains force dependant on the angle between the line P1-P2 and the vertical. the combination of these will ...[text shortened]... ccleration due to gravity, h= overall height dropped by ball. should give a start point anyway.
Thank goodness you have Galileo as a friend! Did he consider (in the solution you povided) that the weight of the ball is deforming the chain as it moves? Also, show the steps behind the calculation.

Finally, I was certain the function was a hyper-cosine function or "chain function".
10. 19 Jun '06 21:501 edit
Originally posted by Gastel
Thank goodness you have Galileo as a friend! Did he consider (in the solution you povided) that the weight of the ball is deforming the chain as it moves? Also, show the steps behind the calculation.

Finally, I was certain the function was a hyper-cosine function or "chain function".
no, his equation was for a ball on a solid surface. i missed the subtle point that the ball deforms the chain as it moves. that makes it a bit more interesting. and i think you are right, it could make a chain function.

edit: i see above a "-" has crept in, the actual function: v=sqrt(10gh/7) ... (makes a lot more sense that way)
11. sonhouse
Fast and Curious
21 Jun '06 00:37
So one thing I see is the friction is dependent on the angle, going from effectively being called infinite where the chain is parallel to the ground and zero where the chain is perpendicular to the ground. Just an observation.
12. 21 Jun '06 01:07
The force of friction is a function of the normal force and the coefficient of friction. The coefficient of friction is constant, and the normal force has a maximum at mg where m is the mass of the ball and g is the force due to gravity. This is never infinite but can approach zero. The effect of the deflection of the chain can be interpreted in various ways but one thought is to have an additional reactive force as the chain is deformed, but remember that this is counterbalanced by the fact that the chain wants to straighten itself out. If the mass of the chain is much greater than the mass of the ball, then this is eliminated.
13. sonhouse
Fast and Curious
21 Jun '06 01:20
Originally posted by Gastel
The force of friction is a function of the normal force and the coefficient of friction. The coefficient of friction is constant, and the normal force has a maximum at mg where m is the mass of the ball and g is the force due to gravity. This is never infinite but can approach zero. The effect of the deflection of the chain can be interpreted in various wa ...[text shortened]... t. If the mass of the chain is much greater than the mass of the ball, then this is eliminated.
I just said it could be called infinite because when the chain is parallel to the ground, movement of the ball is impossible so it would have the effect of inifinite friction, it might make for a simplification of the function.
14. The Plumber
Leak-Proof
21 Jun '06 02:27
I think you need to provide some information for the chain since there are friction losses in the chain itself as the various links move against each other as the ball deforms the chain.
15. AThousandYoung
All My Soldiers...
23 Jun '06 00:341 edit
Originally posted by sonhouse
I just said it could be called infinite because when the chain is parallel to the ground, movement of the ball is impossible so it would have the effect of inifinite friction, it might make for a simplification of the function.
That would be a case of the normal force being parallel to gravitational force, leading to a net force of 0 on the ball and no movement.