Rolling Around

Calculus Level 4

A unit circle rolls around the circumference of a large circle of radius 4, as shown in the figure above. The epicycloid traced by a point on the circumference of the smaller circle is given by

$x= 5\cos t- \cos 5t$ and $y= 5 \sin t-\sin 5t$ .

Find the distance traveled by the point in one complete trip about the larger circle.

Problem from Calculus by Ron Larson.

30 40 60 20

2 solutions

Hana Wehbi
Jun 1, 2016

To find the total distance traveled by the point, you can find the arc length of the portion lying in the first quadrant and multiplying it by $4$ .

$\LARGE\S=\Large 4 \int_{0}^{\pi/2}$ $\Large\sqrt{(x')^{2}+(y')^{2}}dt$

$\LARGE\S=\Large 4 \int_{0}^{\pi/2}$ $\Large\sqrt{(-5sint+5sin5t)^{2}+(5cost-5cost)^{2}}dt$ =

$\LARGE\S=\Large 20 \int_{0}^{\pi/2}$ $\Large\sqrt{2-2sintsin5t-2costcos5t} dt$ =

$\LARGE\S=\Large 20 \int_{0}^{\pi/2}$ $\Large\sqrt{2-2cos4t }dt$ =

$\LARGE\S=\Large 20 \int_{0}^{\pi/2}$ $\Large\sqrt{4sin^{2}2t} dt$ =

$\LARGE\S=\Large 40 \int_{0}^{\pi/2}$ $\Large\sin2t dt$ = $\Large -20 [cos2t]_{0}^{\pi/2}$ = $\Large 40$

For the epicycloid shown in the figure, an arc length of $40$ seems about right because the circumference of a circle of radius $6,\ is\ 2\pi\times r = 12 \pi= 37.7$ $\Large\diamond$

Iliya Hristov
Dec 7, 2020

Rolling Around

Problem from Calculus by Ron Larson.

2 solutions

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