Math Problems Vault

Problem 1602 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Section 2.14 "Tangency of Curves. Circle of Curvature. Evolute"

Find the radius of curvature of the following curve: the involute of a circle $x=a(\cos t+t\sin t), y=a(\sin t-t\cos t)$ .

?

Problem 142 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 1.3 "Loci"

A thread wound around the circle $x^{2}+y^{2}=a^{2}$ is unwound while remaining taut. Find the locus traced by its end, if the initial position of the end of the thread was at the point $(a,0)$ . (Involute of a circle.)

?

Problem 1.24.35 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Предел. Непрерывность. Дифференцируемость." [ККЧ+03a]More problems in Subsection 1.24.2 "Problems"

Write the parametric equation of the involute of a circle, i.e., the trajectory of the end of a taut thread unwinding from a fixed round flat spool.

?

Problem 2.7.27 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Интегралы. Ряды." [ККЧ+03b]More problems in Subsection 2.7.2 "Problems"

Find the area of the figure bounded by the arc of the involute of a circle

x = a(\cos t+t \sin t), \quad y = a(\sin t-t \cos t), \quad 0 \leqslant t \leqslant 2 \pi

and the segment with endpoints $(a ; 0),(a ;-2 \pi a)$ .

?

Problem 2.7.72 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Интегралы. Ряды." [ККЧ+03b]More problems in Subsection 2.7.2 "Problems"

Find the length of the arc of the curve:

?

(1)

$x = a \cos^{3} t, y = a \sin^{3} t, 0 \leqslant t \leqslant 2 \pi$ ;

(2)

$x = \left(c^{2} / a\right) \cos^{3} t, y = \left(c^{2} / b\right) \sin^{3} t, 0 \leqslant t \leqslant 2 \pi , c^{2} = a^{2}-b^{2}$ (evolute of the ellipse);

(3)

$x = a(t-\sin t), y = a(1-\cos t), 0 \leqslant t \leqslant 2 \pi$ (cycloid);

(4)

$x = a(\cos \varphi +\varphi \sin \varphi ), y = a(\sin \varphi -\varphi \cos \varphi ), 0 \leqslant \varphi \leqslant \varphi_{0}$ (involute of a circle);

(5)

$x = \operatorname {ch}^{3} t, y = \operatorname {sh}^{3} t, 0 \leqslant t \leqslant t_{0}$ ;

(6)

$x = a e^{\alpha \varphi } \cos \varphi , y = a e^{\alpha \varphi } \sin \varphi , 0 \leqslant \varphi \leqslant \varphi_{0}$ ;

(7)

$x = \frac{t}{\sqrt{a^{2}+1}} \cos (a \ln t), y = \frac{t}{\sqrt{a^{2}+1}} \sin (a \ln t), t_{1} \leqslant t \leqslant t_{2}$ ;

(8)

$x = t-(1 / 2) \operatorname {sh} 2 t, y = 2 \operatorname {ch} t, 0 \leqslant t \leqslant t_{0}$ ;

(9)

$x = a(\cos t+\ln \operatorname {tg}(t / 2)), y = a \sin t, 0 < t_{0} \leqslant t \leqslant \pi / 2$ (tractrix);

(10)

$x = t^{2} \cos t, y = t^{2} \sin t, 0 \leqslant t \leqslant 1$ ;

(11)

$x = 1-\cos 2 t, \quad y = \sin t-(1 / 3) \sin 3 t, \quad 0 \leqslant t \leqslant \pi / 2$ ;

(12)

$x = \sin^{3} t, y = \cos 2 t, 0 \leqslant t \leqslant \pi / 2$ .

Problem 2.7.74 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Интегралы. Ряды." [ККЧ+03b]More problems in Subsection 2.7.2 "Problems"

Find the length of the arc of the curve:

?

(1)

$x = \left(t^{2}-2\right) \sin t+2 t \cos t, \quad y = \left(t^{2}-2\right) \cos t-2 t \sin t, \quad 0 \leqslant t \leqslant \pi$ ;

(2)

$x = r((\alpha +1) \cos t-\cos (\alpha +1) t), \quad y = r((\alpha +1) \sin t- -\sin (\alpha +1) t), \quad 0 \leqslant t \leqslant t_{0} \leqslant 2 \pi / \alpha , \quad r > 0$ (epicycloid);

(3)

$x = r((\alpha -1) \cos t+\cos (\alpha -1) t), \quad y = r((\alpha -1) \sin t-\sin (\alpha -1) t), \quad 0 \leqslant t \leqslant t_{0} \leqslant 2 \pi / \alpha , \quad \alpha > 1, \quad r > 0$ (hypocycloid);

(4)

$x = a(2 \cos 2 t \cos t+\sin 2 t \sin t)$ , $y = a(\sin 2 t \cos t-2 \cos 2 t \sin t), \quad 0 \leqslant t \leqslant \pi ;$

(5)

$x = r(a \cos b t-b \cos a t), \quad y = r(a \sin b t+b \sin a t)$ , $0 \leqslant t \leqslant 2 \pi /(a+b), \quad a > 0, \quad b > 0, \quad r > 0$ .

Problem 252 from Смирнов Ю.М. "Сборник задач по аналитической геометрии и линейной алгебре" [Сми16]More problems in Section 2.2 "Deriving Equations of Curves in the Plane"

(Involute of a circle.) A line rolls without slipping along the circle $x^{2}+y^{2} = r^{2}$ , its initial position being $x = r$ . Set up the parametric equations of the trajectory of a point of the rolling line, if at the initial moment the point has coordinates $(r, 0)$ , taking as the parameter the angle from the positive direction of the axis $O x$ to the position vector of the point of tangency of the line, measured counterclockwise.

?

References