. Differential and integral calculus, an introductory course for colleges and engineering schools. When the radius vector turns about the pole with a constant angularvelocity k, what are the velocity components along the tangent and along the radius vector? Note what these become when d = x, ^, 0. 3. A bicyclist is riding at the rate of k feet per minute, and the radiusof his wheel is a feet. Find the velocity of the projection upon theground of a point on the rim of the wheel. Where is this velocity thegreatest? the least? Does the point on the rim ever move backward? 4. A point moves around
. Differential and integral calculus, an introductory course for colleges and engineering schools. When the radius vector turns about the pole with a constant angularvelocity k, what are the velocity components along the tangent and along the radius vector? Note what these become when d = x, ^, 0. 3. A bicyclist is riding at the rate of k feet per minute, and the radiusof his wheel is a feet. Find the velocity of the projection upon theground of a point on the rim of the wheel. Where is this velocity thegreatest? the least? Does the point on the rim ever move backward? 4. A point moves around an ellipse with a constant tangential velocityk. Find its velocity components parallel to the principal axes of theellipse. In the following curves the radius vector turns about the pole with aconstant angular velocity k. Find the velocity components along theradius vector and along the tangent. 5. The logarithmic spiral, p — aew. 6. The spiral of Archimedes, p = ad. 7. The lemniscate of Bernoulli, p2 = a2 cos 2 d. Note what the tangen-tial velocity becomes when e = 0, ^, ^, tt. CHAPTER XVI. CURVATURE. EVOLUTES AND INVOLUTES 108. Curvature. The curvature of an arc is, in everydayphrase, its deviation from a straight line. A straight line haseverywhere the same direction,while a curve changes its directionfrom point to point. We maytherefore define the total or abso-lute curvature of an arc to be itstotal change in direction. It ismeasured by the angle throughwhich the tangent line turns asthe point of contact moves fromone end of the arc to the other. Thus the absolute curvature of the arc PQ in the figure isa = 6 — 6. It is evident that the absolute curvature is alsomeasured by the angle between the normals at the ends of the arc. The mean or average curvature is the ratio of the total curvatureto the length of the arc. It is what the total curvature of eachlinear unit of arc would be if the curvature of the arc were the OL same throughout. The mean curvature of the arc PQ
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Keywords: ., bookcentury1900, bookdecade1910, booksubjectcalculu, bookyear1912
