- #1
zosterae
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I have been working on this for a few days and can't seem to find the right direction to go in. Here's the problem:
Two beads, A and B, slide along frictionless wires. Bead A moves along a straight horizontal wire. Bead B moves along a wire with a dip in it, but it returns to its starting height by the finish. Both are moving with velocity v initially and both beads pass the “Start” position at the same time. Two observations should be obvious:
1) At the “Finish” position both beads will have the same speed. They began with speed v, and they will finish with speed v.
2) While bead B was in the dip, it had a higher speed than A. At all times, bead B was either moving faster than bead A or at the same speed as bead A.
So the question is, “Which bead arrives at the finish first?”
1) Bead A which traveled a shorter distance.
2) Bead B which traveled a longer distance, but faster.
Or
3) Do they arrive at the same time?
So far:
My first instinct was that they arrive at the same time. It seems reasonable to me that the greater velocity would be 'canceled out' by the longer distance, but I can also imagine it would be possible for B to arrive there first. If the height of the curve is relatively small, then I think it would make sense that B would get there first because it has a higher velocity the whole distance, and the distance is only slighty longer that that for A. The only thing I can think of is that it depends on the height of the curve whether B will get there first or at the same time (I'm pretty sure B will not be last)
I understand how energy is conserved, and for bead B acceleration is positive until it hits the lowest height, then is negative because force= mgcos(*) and the angle increases from <90 to >90. I realize that I have to use work/energy. I originally thought I should use the equations I know to solve for t (the time it takes for B to reach the finish) and compare it to that of A (for which t=v/r, where r is the distance from start to finish). To do this I think I need to find an equation for the arc length in order to find r for B (I started out thinking it should be a parabolic arc, but that seemed too complicated, and I'm not sure how I would apply the arc length of a circle equation to this). The only problem is that I can't seem to figure out how to do it. I am pretty sure I need to use calculus somewhere, but I'm not sure about that. If I could get a push in the right direction it would help immensely. I'm sure I'm just overlooking something small that is preventing me from making connections... that's usually what it comes down to for me! Thanks!
Two beads, A and B, slide along frictionless wires. Bead A moves along a straight horizontal wire. Bead B moves along a wire with a dip in it, but it returns to its starting height by the finish. Both are moving with velocity v initially and both beads pass the “Start” position at the same time. Two observations should be obvious:
1) At the “Finish” position both beads will have the same speed. They began with speed v, and they will finish with speed v.
2) While bead B was in the dip, it had a higher speed than A. At all times, bead B was either moving faster than bead A or at the same speed as bead A.
So the question is, “Which bead arrives at the finish first?”
1) Bead A which traveled a shorter distance.
2) Bead B which traveled a longer distance, but faster.
Or
3) Do they arrive at the same time?
So far:
My first instinct was that they arrive at the same time. It seems reasonable to me that the greater velocity would be 'canceled out' by the longer distance, but I can also imagine it would be possible for B to arrive there first. If the height of the curve is relatively small, then I think it would make sense that B would get there first because it has a higher velocity the whole distance, and the distance is only slighty longer that that for A. The only thing I can think of is that it depends on the height of the curve whether B will get there first or at the same time (I'm pretty sure B will not be last)
I understand how energy is conserved, and for bead B acceleration is positive until it hits the lowest height, then is negative because force= mgcos(*) and the angle increases from <90 to >90. I realize that I have to use work/energy. I originally thought I should use the equations I know to solve for t (the time it takes for B to reach the finish) and compare it to that of A (for which t=v/r, where r is the distance from start to finish). To do this I think I need to find an equation for the arc length in order to find r for B (I started out thinking it should be a parabolic arc, but that seemed too complicated, and I'm not sure how I would apply the arc length of a circle equation to this). The only problem is that I can't seem to figure out how to do it. I am pretty sure I need to use calculus somewhere, but I'm not sure about that. If I could get a push in the right direction it would help immensely. I'm sure I'm just overlooking something small that is preventing me from making connections... that's usually what it comes down to for me! Thanks!
