Hello, drewbear!
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 $\displaystyle \tfrac{\pi}{3}$ radians above the horizontal and rising?
(Round your answer to one decimal place.) Code:
* * *
* * P
* o
* 2 * *
*
* * θ *
* o - - - - *
* C *
* *
* *
* *
* * *
The parametric equations of point $\displaystyle P$ are: .$\displaystyle \begin{Bmatrix}x &=& 2\cos\theta & [1] \\ y &=& 2\sin\theta & [2] \end{Bmatrix}$
$\displaystyle \text{We want }\,\frac{dy}{dt}\,\text{ when }\theta = \frac{\pi}{3}$
Differentiate [2] with respect to time: .$\displaystyle \frac{dy}{dt} \;=\;2\cos\theta\,\frac{d\theta}{dt}$
We know that: .$\displaystyle \cos\frac{\pi}{3} \,=\,\frac{1}{2},\;\;\frac{d\theta}{dt} \,=\,18$
Therefore: .$\displaystyle \frac{dy}{dt} \;=\;2\left(\tfrac{1}{2}\right)(18) \;=\;18\text{ cm/sec}$