# related rates: rolling wheel

• Feb 21st 2010, 08:13 PM
drewbear
related rates: rolling wheel
A wheel with radius 2 m rolls at 18 rad/s. How fast is a point on the rim of the wheel rising when the point is pi

place.)
• Feb 21st 2010, 10:29 PM
Soroban
Hello, drewbear!

Quote:

How fast is a point on the rim of the wheel rising when the point
is $\tfrac{\pi}{3}$ radians above the horizontal and rising?

Code:

              * * *           *          *  P         *              o       *          2  *  *                     *       *          * θ    *       *        o - - - - *       *        C        *       *                *         *              *           *          *               * * *

The parametric equations of point $P$ are: . $\begin{Bmatrix}x &=& 2\cos\theta & [1] \\ y &=& 2\sin\theta & [2] \end{Bmatrix}$

$\text{We want }\,\frac{dy}{dt}\,\text{ when }\theta = \frac{\pi}{3}$

Differentiate [2] with respect to time: . $\frac{dy}{dt} \;=\;2\cos\theta\,\frac{d\theta}{dt}$

We know that: . $\cos\frac{\pi}{3} \,=\,\frac{1}{2},\;\;\frac{d\theta}{dt} \,=\,18$

Therefore: . $\frac{dy}{dt} \;=\;2\left(\tfrac{1}{2}\right)(18) \;=\;18\text{ cm/sec}$

• Feb 21st 2010, 10:35 PM
drewbear
thank you so much Soroban!!! (Clapping)