Figure shown a clamped metal string of length `30 cm` and linear mass density `0.1 kg //m`. Which is taut at a tension of `40N`. A small rider `(` piece of paper `)` is placed on string at point `P` as shown. An external vibrating tuning fork is brought near this string and oscillations of rider are carefully observed.

Now if the tension in the string is made `160N`, at which of the following frequencies of turning fork , rider will not vibrate at all

Options

A. `(100)/(3)Hz`
B. `50Hz`
C. `200Hz`
D. None of these

Answer

C

Explanation

Wave velocity in string is
`v=sqrt((T)/(mu))=sqrt((40)/(0.1))=20m//s`
Fundamental frequency of string oscillations is
`n_(0)=(100)/(3)Hz,(200)/(3)Hz, 100Hz, (400)/(3)Hz, ......`
Here rider will not oscillate at all only if it is at a node of stationary wave in all other cases of resonance and non`-` resonance it will vibrate at the frequency of tuning fork. At a distance `(l)/(3)` from one end node will appear at `3^(rd),6^(th),9^(th)` or similar higher Harmonics `i.e.` at frequencies `100Hz, 200Hz,......`
If string is divided in off no. of segments, these segments can never resonate simultaneously hence at the locaton of rider, antinode is never obtained at any frequency.