An Elementary Course of Infinitesimal Calculus
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164:. Application to Rolling Curves.
When one plane curve rolls upon another, which is regarded as fixed, the instantaneous centi'e is at the point of contact.
We will suppose, in the figure, that it is the lower curve 163-164] CURVATUEE.
439 which is fixed. Let A be the point of contact, and let equal in- finitely small arcs AP, AP' (= Ss, say) be measured off along the two curves.
Let the normals at P and P' meet the common normal at A in the points and 0'. Then ultimately we have OA = E, 0'A...=B', where R, R' are the radii of curva- ture of the two curves at A. After an infinitely small displacement, P'O' will come into the same straight line with OP, the two curves being then in contact at P. Hence the angle (8^) through which the rolling curve will turn, being equal to the acute angle between OP and P'O', is equal to the sum of the angles at and 0', so that ^^-M »■ Fig. 135.
ultimately. Again, the chords AP, AP' are ultimately equal, and they include an infinitely small angle at A.
What to read after An Elementary Course of Infinitesimal Calculus?
You can find similar books in the "Read Also" column, or choose other free books by Lamb, Horace, Sir, 1849-1934 to read onlineMoreLess
To quickly assess the difficulty of the text, read a short excerpt:
164:. Application to Rolling Curves.
When one plane curve rolls upon another, which is regarded as fixed, the instantaneous centi'e is at the point of contact.
We will suppose, in the figure, that it is the lower curve 163-164] CURVATUEE.
439 which is fixed. Let A be the point of contact, and let equal in- finitely small arcs AP, AP' (= Ss, say) be measured off along the two curves.
Let the normals at P and P' meet the common normal at A in the points and 0'. Then ultimately we have OA = E, 0'A...=B', where R, R' are the radii of curva- ture of the two curves at A. After an infinitely small displacement, P'O' will come into the same straight line with OP, the two curves being then in contact at P. Hence the angle (8^) through which the rolling curve will turn, being equal to the acute angle between OP and P'O', is equal to the sum of the angles at and 0', so that ^^-M »■ Fig. 135.
ultimately. Again, the chords AP, AP' are ultimately equal, and they include an infinitely small angle at A.
What to read after An Elementary Course of Infinitesimal Calculus?
You can find similar books in the "Read Also" column, or choose other free books by Lamb, Horace, Sir, 1849-1934 to read onlineMoreLess
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