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# A generator connected to the wheel or hub of a bicycle can be used to power lights or small electronic devices. A typical bicycle generator supplies 6.00 V when the wheels rotate at $\omega=20.0$ rad/s. Find the time interval between the maximum emf of +6.00 V and the minimum emf of -6.00 V.

Solution

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Maximum emf is when

\begin{align} \sin \omega t_{max} & = 1\\ \omega t_{max} & = \sin^{-1}(1)\\ \omega t_{max} & = \dfrac{\pi}{2}\\ t_{max} & = \dfrac{\pi}{40}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \bigg[\text{Given},\omega=20\ \frac{\text{rad}}{\text{s}}\bigg] \end{align}

Similarly, minimum emf is when

\begin{align} \sin \omega t_{min} & = -1\\ \omega t_{min} & = \sin^{-1}(-1)\\ \omega t_{min} & = -\dfrac{\pi}{2}\\ t_{min} & = -\dfrac{\pi}{40} \end{align}

The time interval between the maximum and minimum emf is

$t=t_{max}-t_{min}$

Substituting the values from equation (1) and (2), we get

\begin{align*} t & = t_{max}-t_{min}\\ & = \dfrac{\pi}{40}-\bigg(-\dfrac{\pi}{40}\bigg)\\ & = \dfrac{2 \pi}{40}\\ & = 0.157\ \text{s} \end{align*}

The time interval between the maximum and the minimum emf is $\boxed{t=0.157\ \text{s}}$

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