# Homework 10 Question 7

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A bicycle is turned upside down while its owner repairs a flat tire on the rear wheel. A friend spins the front wheel, of radius 0.342 m, and observes that drops of water fly off tangentially in an upward direction when the drops are at the same level as the center of the wheel. She measures the height reached by drops moving vertically (see figure below). A drop that breaks loose from the tire on one turn rises h = 53.0 cm above the tangent point. A drop that breaks loose on the next turn rises 51.0 cm above the tangent point. The height to which the drops rise decreases because the angular speed of the wheel decreases. From this information, determine the magnitude of the average angular acceleration of the wheel.

Use energy conservation, we can find the initial vertical velocity of the water droplets with $$\frac 12mv^2=mgh$$
For the adjacent two water droplets, we find $v_1$ and $v_2$, which also gives $\omega_1=\frac {v_1}{R}$ and $\omega_2=\frac {v_2}{R}$.
Between the two droplets, the wheel rotates for a complete circle so it will experience a rotation covering $\Delta \theta=2\pi$.
Then using the kinematic equation of $\omega_2^2-\omega_1^2=2\alpha\Delta\theta$, we can find the angular acceleration.