circle problems

Question

I have no idea how to even start this question.

Could you please explain your reasoning. Also, the answer is approximately 57.3 degrees.

Answer 1

Starting with computing the angle that the spool must rotate to raise the weight 10cm:

$\theta_{S} r_{S} = 10 \text{ cm}$

$\theta_{S} = \frac{10}{30} \text { radians}$

Since Gear B is connected to the spool, the angle is the same. However, the linear distance that the circumference translates through (at the point of contact with Gear A), is given by the expression:

$d=\theta_{S} r_{B} = \frac{10}{30} (90)$

Lastly, we determing the angle that Gear A rotates through from:

$d=\theta_{A} r_{A} = \frac{10}{30} (90)$

since Gear A and Gear B are in contact, the distance d is in common. Now we can solve for the angle $\theta_{A}$:

$\theta_{A} = \frac{10}{30} \frac{90}{30} = \frac{1}{3} (3) = 1 \text{ radian} $

$1 \text{ radian} = \frac{360}{2\pi} \text{ degrees} \approx 57.3 \text{ degrees} $

I hope this helps.