Question

A metal wire of linear mass density of 9.8g/m is stretched with a tension of 1kg-wt between two rigid supports 1m apart. The wire passes at its middle point between the poles of a permanent magnet and it vibrates in resonance when carrying an alternating current of frequency n. The frequency n of the alternating source is :
(A) 50Hz
(B) 100Hz
(C) 200Hz
(D) 25Hz

Answer 1

Hint:When a condition of resonance is satisfied the frequency of the string matches with the frequency of the source of the vibration. For this question we will apply the formula of the frequency of a string between two fixed supports.

Complete step by step answer:
The frequency of the string attached between supports is given by: $f\, = \,\dfrac{1}{{2l}}\sqrt {\dfrac{T}{\mu }} $, where $l$is the distance between the two rigid supports. $T$is the tension of the string and $\mu $is the linear mass density.
Putting the values given in the question of the above-mentioned parameters, the frequency of the vibrating string:
\[\begin{array}{l}
f = \,\dfrac{1}{{2 \times 1}}\sqrt {\dfrac{{10 \times 9.8}}{{9.8 \times {{10}^{ - 3}}}}} \\
\Rightarrow 50Hz
\end{array}\]
During Resonance the frequency of the string will be equal to the frequency of the alternating current.
Hence, the frequency (n) of the alternating current = 50Hz and option (A) is the correct option.

Additional Information:
When the end of a string is fixed, the displacement of the string at that end must be zero. A transverse wave travelling along the string towards a fixed end will be reflected in the opposite direction. When a string is fixed at both ends, two waves travelling in opposite directions simply bounce back and forth between the ends. The vibrational behavior of the string depends on the frequency (and wavelength) of the waves reflecting back and forth from the ends. A string which is fixed at both ends will exhibit strong vibrational response only at the resonance frequencies.

Note: The students must be cautious about the units of the values given in the question and change them accordingly. The resonance condition must be taken into consideration. Proper values should be applied in the formula.