Question 270: Waves

Question

At the poles of earth, a stretched wire of a given length vibrates in unison with a tuning fork. At the equator of earth, for same setting, to produce resonance with same fork, the vibrating length of wire

Accepted Answer

To understand how a wire vibrates in unison with a tuning fork at different locations on Earth, we need to consider the fundamental principles affecting its frequency. The frequency $ n $ for a stretched string is given by:
$ n = \frac{1}{2l} \sqrt{\frac{T}{m}} $
where:
$ l $ is the length of the wire,
$ T $ is the tension in the wire, and
$ m $ is the mass per unit length of the wire.
If we assume the tension $ T $ is due to a weight $ Mg $ (where $ M $ is the mass and $ g $ is the acceleration due to gravity), then the frequency becomes:
$ n = \frac{1}{2l} \sqrt{\frac{Mg}{m}} $
Considering the poles and the equator, let:
$ l_1 $ and $ g_1 $ be the length and acceleration due to gravity at the poles,
$ l_2 $ and $ g_2 $ be the length and acceleration due to gravity at the equator.
For the frequency to remain the same at both the poles and the equator:
$ n = \frac{1}{2l_1} \sqrt{\frac{Mg_1}{m}} = \frac{1}{2l_2} \sqrt{\frac{Mg_2}{m}} $
From this, we obtain:
$ \frac{\sqrt{g_1}}{l_1} = \frac{\sqrt{g_2}}{l_2} \quad 	ext{or} \quad \frac{l_2}{l_1} = \sqrt{\frac{g_2}{g_1}} $
Since the acceleration due to gravity at the equator ($ g_2 $) is less than at the poles ($ g_1 $), it follows that:
$ g_2 < g_1 $
Therefore, the ratio $\frac{l_2}{l_1} = \sqrt{\frac{g_2}{g_1}} < 1$, indicating that:
$ l_2 < l_1 $
Thus, the vibrating length of the wire should be decreased when moving from the poles to the equator to maintain the same resonance with the tuning fork.

Answer

should be decreased.

Answer

should be increased.

Answer

should be same.

Answer

should be three times the original.

Waves

At the poles of earth, a stretched wire of a given length vibrates in unison with a tuning fork. At the equator of earth, for same setting, to produce resonance with same fork, the vibrating length of wire

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