Involutes
DOWNLOAD
FREE
|
Involutes: Questions Exercise 1A - Problems on finding involutes
Equations are omitted for technical reasons - download the original pdf
(1) Let (C) be a curve, given by in parametric form, i.e. [Equation goes here - download the original to see it.] , where s is the natural parameter. Prove that the involutes of [Equation goes here - download the original to see it.] is [Equation goes here - download the original to see it.] (2) Find the involutes of the catenary's curve. (3) Let the astroid [Equation goes here - download the original to see it.] and its evaluate. These curves meet the line [Equation goes here - download the original to see it.] in the first quadrant at the points K and L. (i) Find the length of KL.(ii) Find the total length of the evaluate. (iii) Find the total length of the astroid.(4) (i) Find the equation of the involutes of the curve Simultaneous differential equations - Systems of differential equations, given by [Equation goes here - download the original to see it.] (ii)Find the radius of curvature of the involutes of the circle. (5) (i) Find the length of the circle, using the properties of involutes. (ii) Using your solution to question (4), find the involutes of the involutes of the circle (C) Investigation Let [Equation goes here - download the original to see it.] be the involutes of a curve Simultaneous differential equations - Systems of differential equations. (i) Prove that the difference of two arcs Simultaneous differential equations - Systems of differential equations of the involutes [Equation goes here - download the original to see it.] between two common normals (with angle ) is given [Equation goes here - download the original to see it.] , where [Equation goes here - download the original to see it.] is the distance between the involutes [Equation goes here - download the original to see it.] and [Equation goes here - download the original to see it.] (ii) Find the curvature of [Equation goes here - download the original to see it.] Solutions Exercise 1A (1) We have [Equation goes here - download the original to see it.], where [Equation goes here - download the original to see it.] and [Equation goes here - download the original to see it.] , where [Equation goes here - download the original to see it.] is the tangent vector in M. Therefore, we have [Equation goes here - download the original to see it.] (1) [Diagram goes here - download the original to see it.] Differentiating this relation by respect of s, and taking account to the Frenet formula, we have [Equation goes here - download the original to see it.] From this, we get that [Equation goes here - download the original to see it.] , i.e. [Equation goes here - download the original to see it.] , where a is constant. Substituting this in the (1), we get that [Equation goes here - download the original to see it.] , which is the vectorial equation of the involutes of C. From this, we get immediately the parametric equation, i.e. [Equation goes here - download the original to see it.] (2) The equation of the catenary is [Equation goes here - download the original to see it.] Therefore [Equation goes here - download the original to see it.] Therefore [Equation goes here - download the original to see it.] Substituting in the parametric equation of the involutes (see Question 1). We obtain [Equation goes here - download the original to see it.] Therefore [Equation goes here - download the original to see it.] For [Equation goes here - download the original to see it.] , we get [Equation goes here - download the original to see it.] , i.e. [Equation goes here - download the original to see it.] , exactly the tractrix: [Diagram goes here - download the original to see it.] (3) [Diagram goes here - download the original to see it.] (i) The equation of the evaluate in Cartesian form is [Equation goes here - download the original to see it.] (1) i.e. an astroid which is rotated by [Equation goes here - download the original to see it.] and twise the distance between opposite vertices as in the original astroid, see figure. Therefore, the point K is obtained by from the fact that in (1). Therefore, [Equation goes here - download the original to see it.] , i.e. [Equation goes here - download the original to see it.] The point L is [Equation goes here - download the original to see it.] Therefore [Equation goes here - download the original to see it.] (ii) We have (from the construction and from the definition of the involutes) that [Equation goes here - download the original to see it.] Therefore, the total length of the evolutes is [Equation goes here - download the original to see it.] (iii) From (i), we get that the total length of the astroid is the half of the total length of its evolutes, i.e. 6. (4) (i) Let [Equation goes here - download the original to see it.] [Equation goes here - download the original to see it.] . From the Question 1 we get that the equation of the involutes is [Equation goes here - download the original to see it.] [Equation goes here - download the original to see it.] (ii) Let [Equation goes here - download the original to see it.] Let [Equation goes here - download the original to see it.] . We have [Equation goes here - download the original to see it.]Therefore, from the (i), we have the equation of the involutes of circle. [Equation goes here - download the original to see it.] see below, for [Equation goes here - download the original to see it.] [Diagram goes here - download the original to see it.] Let [Equation goes here - download the original to see it.] [Equation goes here - download the original to see it.] Equation goes here - download the original to see it.] Therefore, [Equation goes here - download the original to see it.] , so Further, [Equation goes here - download the original to see it.] , since [Equation goes here - download the original to see it.] From this, the radius of curvature [Equation goes here - download the original to see it.]
|
Contents of
Involutes
1 Involutes: Questions Exercise 1A - Problems on finding involutes
|
