Ruzzle/Riddle Thread
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melodicbard
Registered Senior Member
fo3 said:
Place the rope on the table. Tilt the table until the rope just begin to fall. Measure and calculate the tangent tan(x) of the angle made by the surface of the table and the floor by using the ruler. Coef. of friciton = tan(x).
geodesic, data,
I had no doubt that you, guys, will understand what is going on in my problem (after you read it and enjoy it!).
But the problem needs to have the answer on its question, is not it true? And the question was: "What is wrong with Golden Theorem"?
So, teaching Math, how we should formulate the Golden Theorem to avoid such situations? geodesic's "It should have "for n > 0", then it is correct." is very good, but there is more: it is possible some generalisation.
More generally: how we should formulate the basic stuff in the theory of equations to avoid the same situation with examples as
sin²x +cos²x - 1 = 0?
I had no doubt that you, guys, will understand what is going on in my problem (after you read it and enjoy it!).
But the problem needs to have the answer on its question, is not it true? And the question was: "What is wrong with Golden Theorem"?
So, teaching Math, how we should formulate the Golden Theorem to avoid such situations? geodesic's "It should have "for n > 0", then it is correct." is very good, but there is more: it is possible some generalisation.
More generally: how we should formulate the basic stuff in the theory of equations to avoid the same situation with examples as
sin²x +cos²x - 1 = 0?
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Measure the length of the rope. (Lt = Total length of rope)
Hang the rope over the edge of the table. Measure the length remaining on the table.
Find by trial and error how little rope is needed on the table to prevent the rope from sliding right off. (Lr = length remaining on the table)
mu = Lt / (Lt - Lr)
Hang the rope over the edge of the table. Measure the length remaining on the table.
Find by trial and error how little rope is needed on the table to prevent the rope from sliding right off. (Lr = length remaining on the table)
mu = Lt / (Lt - Lr)
Well, the first generalisation would be n>=0 instead of n>0. A 0-polynomial is a nonzero constant, and a=0 has 0 solutions for a!=0. The degree of the 0 polynomial (not to be confused with "a 0-polynomial") is undefined, so the theorem says nothing about it. To generalise further:
Define "F(x) is indentically zero" to mean that F(x)=0 for every x in the domain of F, for a real function F.
Then we get
x is a solution to F(x) = 0 for every x in the domain of F if and only if F(x) is identically zero.
That's trivial though. We can come up with more specific cases, like:
Let {y_n(x)} be a sequence of real functions with the same domain, with the property that if S is the set of all y_i(x) in {y_n(x)} that aren't identically zero, then S is a linearly independent set. Let F(x) be the sum of the sequence {a_n * y_n(x)} where the a_i are real constants. Then
F(x) = 0 for every x in the domain of F (which is, of course, the same as the domain of each of the y_i(x)) if and only if a_i = 0 whenever y_i(x) is not identically zero,
or in other words, F(x) is identically zero if and only if a_i*y_i(x) is identically zero for every i.
Define "F(x) is indentically zero" to mean that F(x)=0 for every x in the domain of F, for a real function F.
Then we get
x is a solution to F(x) = 0 for every x in the domain of F if and only if F(x) is identically zero.
That's trivial though. We can come up with more specific cases, like:
Let {y_n(x)} be a sequence of real functions with the same domain, with the property that if S is the set of all y_i(x) in {y_n(x)} that aren't identically zero, then S is a linearly independent set. Let F(x) be the sum of the sequence {a_n * y_n(x)} where the a_i are real constants. Then
F(x) = 0 for every x in the domain of F (which is, of course, the same as the domain of each of the y_i(x)) if and only if a_i = 0 whenever y_i(x) is not identically zero,
or in other words, F(x) is identically zero if and only if a_i*y_i(x) is identically zero for every i.
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Pete said:
That is right. But due to the fact that we are ignoring the friction against the edge of the table, this results in a too big answer. About 1,1-1,3 depending on the table and the rope.
The rope is not under a 90 degree angle on the edge of the table, it is doing a slight turn, so that at the touch point, it is under a approx. 45 degree angle. This could ofcourse be taken into account too.
melodicbard
Registered Senior Member
fo3 said:
Actually, my method is MUCH more difficult to carry out than Pete's.
Would it not affect my hair? For example on my arm.
Otherwise, would not the non-magnetic stick be attracted to the magnetic one, and therefore observe its "path" to see which one is the magnetic one. (This one could be hard though.)
(I mean that the end of the non-magnetic one would turn and be attracted to the magnetic one (if the magnetic one is hold behind the thumd and another finger for instance))
Otherwise, would not the non-magnetic stick be attracted to the magnetic one, and therefore observe its "path" to see which one is the magnetic one. (This one could be hard though.)
(I mean that the end of the non-magnetic one would turn and be attracted to the magnetic one (if the magnetic one is hold behind the thumd and another finger for instance))