Question

As the captain of the scientific team sent to Planet Physics, one

of your tasks is to measure g. You have a long, thin wire labeled

1.00 g/m and a 1.25 kg weight. You have your accurate space cadet

chronometer but, unfortunately, you seem to have forgotten a

meter stick. Undeterred, you first find the midpoint of the wire by

folding it in half. You then attach one end of the wire to the wall

of your laboratory, stretch it horizontally to pass over a pulley at

the midpoint of the wire, then tie the 1.25 kg weight to the end

hanging over the pulley. By vibrating the wire, and measuring

time with your chronometer, you find that the wire’s second harmonic

frequency is 100 Hz. Next, with the 1.25 kg weight still

tied to one end of the wire, you attach the other end to the ceiling

to make a pendulum. You find that the pendulum requires 314 s to

complete 100 oscillations. Pulling out your trusty calculator, you

get to work. What value of g will you report back to headquarters?

Short answer

The solution also provides

a long, thin wire labelled

1.00 g/m and a 1.25 kg weight and

the wire’s second harmonic

frequency is 100 Hz

Step-by-step solution

Description of the wire and its accelaration

It also deals with the relation between the lengths and the acceleration

Speed and sound will be

$$\begin{gathered} v=\sqrt{\frac{t}{\mu }}=\sqrt{\frac{mg}{\mu }} \\ {f}_m=m\frac{v}{2S} \\ T=2\mathrm{\pi }\sqrt{\frac{\mathrm{L}}{\mathrm{g}}}\underset{}{\to }\frac{\mathrm{L}}{\mathrm{g}}=\frac{{\mathrm{T}}^2}{4{\mathrm{\pi }}^2} \end{gathered}$$