The figure shows a rotating wheel with radius 40 cm and a connecting rod AP with length 1.2 m. The pin P slides back and forth along the x-axis as the wheel rotates counterclockwise at a rate of 360 revolutions per minute.
- Find the angular velocity of the connecting rod, $\frac{d\alpha}{dt}$, in radians per second, when $\theta = \frac{\pi }{3}$.
- Express the distance $x = \left|OP \right|$ in terms of $\theta$.
- Find an expression for the velocity of the pin P in terms of $\theta$.
I have found a solution for all three questions, but my answer to QUESTION 1 is different from the given solution from the textbook and I am struggling to figure out why.
QUESTION 1
I use the law of sines to solve this.
$$\frac{\sin\alpha }{OA} = \frac{\sin\theta }{AP}$$
from which I find that:
$$\sin\alpha =\frac{OA}{AP}\sin\theta =\frac{40}{120}\sin\theta =\frac{1}{3}\sin\theta$$
$$\alpha = \arcsin\left(\frac{1}{3}\sin\theta \right)$$
$$\frac{d\alpha }{dt} = \frac{1}{\sqrt{1 - \frac{{\sin\theta }^{2}}{9}}} \cdot \frac{1}{3}\cos\theta \cdot \frac{d\theta }{dt}$$
and this is equal to $$\frac{d\alpha }{dt} = \frac{3\pi }{\sqrt{33}}$$ when $$\theta = \frac{\pi }{3}$$
The problem here is that according to the textbook the right answer is $\frac{4\pi \sqrt{3}}{\sqrt{11}}$ which I quite do not understand. Furthermore I am not supposed, at this stage, to know the derivative of inverse functions including arcsin. Is there any other way to solve this without using the arcsin funcion?