Gary
GaryVasco
Posts: 3,352 
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Post by Gary on Apr 20, 2016 23:44:05 GMT
Vasco, I have a different answer to the last part of part (iii). Since t is allowed to vary, shouldn’t the result of fixed a and b with large $\omega$ be a cylinder? My answer to (iv) is also different, though certainly not better. For part (iv), it might be helpful to include a reminder that the trigonometric rules for cross products also work in three dimensions. nh.ch5.ex21.pdf (191.72 KB) Gary |
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Post by Admin on Apr 21, 2016 13:27:22 GMT
Vasco, I have a different answer to the last part of part (iii). Since t is allowed to vary, shouldn’t the result of fixed a and b with large $\omega$ be a cylinder? My answer to (iv) is also different, though certainly not better. For part (iv), it might be helpful to include a reminder that the trigonometric rules for cross products also work in three dimensions. View AttachmentGary Gary Yes, I agree with you that it should be a cylinder. I was led astray by the fact that the pitch gets smaller as $\omega$ increases. This led me to think that the whole helix would shrink down to a circle. I will correct this and re-publish. Your second point about the cross product is debatable because the velocity and acceleration vectors span the osculating plane, so we can consider $\mathbb{C}$ to be this osculating plane as mentioned at the end of exercise 20, and then use the definition of the cross product explained on page 28, where it says: "For the purposes of this book we will therefore redefine the cross product...". I have looked at your document and I noticed the following: In part (i) you say b is the length of the central axis of the helix, but since $Z=bt$, I think that b must be a velocity. Vasco |
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Gary
GaryVasco
Posts: 3,352
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Post by Gary on Apr 23, 2016 4:02:04 GMT
In part (i) you say b is the length of the central axis of the helix, but since ... Agreed. I also accept the caution on using the trigonometric form of the cross-product in three dimensions. (The quote selection function is a bit goofy tonight, perhaps due to my poor internet connection.) |
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