## College Physics (4th Edition)

We can rank the strings according to the fundamental frequencies, from largest to smallest: $d \gt a \gt b = c \gt e$
We can write an expression for the wave speed along a string: $v = \sqrt{\frac{F}{m/L}} = \sqrt{\frac{F~L}{m}}$ We can find an expression for the fundamental frequency: $f = \frac{v}{\lambda}$ $f = \frac{\sqrt{\frac{F~L}{m}}}{2L}$ $f = \frac{1}{2}~\sqrt{\frac{F}{m~L}}$ We can find an expression for the frequency in each case: (a) $f = \frac{1}{2}~\sqrt{\frac{F}{m~L}}$ (b) $f = \frac{1}{2}~\sqrt{\frac{F}{m~(2L)}} = \frac{\sqrt{2}}{2}\times \frac{1}{2}~\sqrt{\frac{F}{m~L}}$ (c) $f = \frac{1}{2}~\sqrt{\frac{F}{(2m)~L}} = \frac{\sqrt{2}}{2}\times \frac{1}{2}~\sqrt{\frac{F}{m~L}}$ (d) $f = \frac{1}{2}~\sqrt{\frac{2F}{m~L}} = \sqrt{2}\times \frac{1}{2}~\sqrt{\frac{F}{m~L}}$ (e) $f = \frac{1}{2}~\sqrt{\frac{F}{(2m)~(2L)}} = \frac{1}{2}\times \frac{1}{2}~\sqrt{\frac{F}{m~L}}$ We can rank the strings according to the fundamental frequencies, from largest to smallest: $d \gt a \gt b = c \gt e$