Low cost tachometer

A tachometer is a device that measures the rotational speed of an engine, like in a car. Here we analyze a particularly simple tachometer in which the height of a bead corresponds to the speed of the engine.

A frictionless wire segment (with one end fixed at the origin) rotates around the $y$ -axis with constant angular velocity $\omega\si[per-mode=symbol]{\radian\per\second}$ . The wire makes an angle $\theta=\SI{60}{\degree}$ with the $x$ axis. There is a massive bead attached to the wire in such a way that it can slide freely along the length of the wire. When the tachometer is used to measure the speed of a lawnmower engine, the bead maintains a constant distance of $\sqrt{3}\si{\meter}$ from the $y$ axis.

Find the value of $\omega$ (in $\si[per-mode=symbol]{\radian\per\second})$ , the speed of the lawnmower.

Details

• $g = \SI[per-mode=symbol]{10}{\meter\per\second\squared}$ .