## Generator Efficiency

Consider an off grid generator is powered by a stationary bicycle. If a volunteer pedals the bicycle with 5 N∙m of mechanical torque at 20 rpm to produce 10 W of electricity, what is the generator’s efficiency?

Expand Hint
Efficiency of a machine:
$$\eta =\frac{P_{out}}{P_{in}}$$$where $$P_{out}$$ is the machine’s output power and $$P_{in}$$ is the machine’s input power. Hint 2 Mechanical power in a rotating machine: $$P=T \omega_m$$$
where $$T$$ is the mechanical torque, and $$w_m$$ is the angular velocity.
Since angular velocity is in rad/s, let’s convert the rotational speed from the problem statement:
$$\omega_m=\frac{2\pi}{60}n$$$where $$n$$ is the speed in rpm. $$\omega_m=\frac{2\pi}{60}\cdot 20=2.093\:rad/s$$$
Mechanical power in a rotating machine:
$$P=T \omega_m$$$where $$T$$ is the mechanical torque, and $$w_m$$ is the angular velocity. $$P= (5N\cdot m)(2.093\:rad/s)=10.467\:W$$$
Efficiency of a machine:
$$\eta =\frac{P_{out}}{P_{in}}$$$where $$P_{out}$$ is the machine’s output power and $$P_{in}$$ is the machine’s input power. For a motor, $$P_{in}$$ is the active component of the electrical power input, and $$P_{out}$$ is the mechanical power output. For a generator, it’s vice versa. $$\eta =\frac{10W}{10.467W}=0.96$$$
0.96
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