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Chapter 10: Problem 1477

Two wires made up of same material are of equal lengths but their radii are in the ratio \(1: 2\). On stretching each of these two strings by the same tension, the ratio between their fundamental frequency is \(\ldots \ldots \ldots .\) (A) \(1: 2\) (B) \(2: 1\) (C) \(1: 4\) (D) \(4: 1\)

Short Answer

Expert verified
The ratio between the fundamental frequencies of the two wires, when stretched with the same tension, is \(2:1\). The correct option is (B).

Step by step solution

01

Write the formula for fundamental frequency

The formula for the fundamental frequency (f) of a stretched wire is given by: \[f = \dfrac{1}{2L} \sqrt{\dfrac{T}{\mu}}\] where - L: length of the wire - T: tension in the wire - μ (mu): linear mass density of the wire, which is equal to the mass (m) of the wire divided by its length (L), μ = m/L
02

Express the linear mass density in terms of radius and volume density

Since the wires are made of the same material, their volume density (ρ, rho) is the same. The volume (V) of a wire is given by: \[V = \pi r^2 L \] where r: radius of the wire The mass (m) of the wire can be expressed as: \[m = \rho V = \rho (\pi r^2 L)\] As we know μ = m/L, we can write μ in terms of r and ρ: \[\mu = \dfrac{m}{L} = \pi r^2 \rho\]
03

Substitute the expression of mu in the formula of fundamental frequency

Now, we substitute the expression for μ in the formula of f: \[f = \dfrac{1}{2L} \sqrt{\dfrac{T}{\pi r^2 \rho}}\]
04

Write the formula for both the wires

Let's denote the radii of the wires as r1 and r2. According to the problem, their ratio is given as: \[\dfrac{r_1}{r_2} = \dfrac{1}{2}\] Now, using the formula for fundamental frequency, we have: \[f_1 = \dfrac{1}{2L} \sqrt{\dfrac{T}{\pi r_1^2 \rho}}, \quad f_2 = \dfrac{1}{2L} \sqrt{\dfrac{T}{\pi r_2^2 \rho}} \]
05

Find the ratio of fundamental frequencies

Now, we will find the ratio of the fundamental frequencies: \[\dfrac{f_1}{f_2} = \dfrac{\dfrac{1}{2L} \sqrt{\dfrac{T}{\pi r_1^2 \rho}}}{\dfrac{1}{2L} \sqrt{\dfrac{T}{\pi r_2^2 \rho}}} = \dfrac{\sqrt{\dfrac{1}{r_1^2}}}{\sqrt{\dfrac{1}{r_2^2}}}\] Remembering that the ratio of r1 to r2 is \(1:2\), so \(r_2 = 2r_1\). we can substitute to get: \[\dfrac{f_1}{f_2} = \dfrac{\sqrt{\dfrac{1}{r_1^2}}}{\sqrt{\dfrac{1}{(2r_1)^2}}} = \dfrac{1}{\sqrt{\dfrac{1}{4}}} = 2\] So, the ratio between their fundamental frequencies is 2:1. The answer is (B) \(2:1\).

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Most popular questions from this chapter

Length of a steel wire is \(11 \mathrm{~m}\) and its mass is \(2.2 \mathrm{~kg}\). What should be the tension in the wire so that the speed of a transverse wave in it is equal to the speed of sound in dry air at \(20^{\circ} \mathrm{C}\) temperature? (A) \(2.31 \times 10^{4} \mathrm{~N}\) (B) \(2.25 \times 10^{4} \mathrm{~N}\) (C) \(2.06 \times 10^{4} \mathrm{~N}\) (D) \(2.56 \times 10^{4} \mathrm{~N}\)

A body of mass \(1 \mathrm{~kg}\) suspended from the free end of a spring having force constant \(400 \mathrm{Nm}^{-1}\) is executing S.H.M. When the total energy of the system is 2 joule, the maximum acceleration is $\ldots \ldots . \mathrm{ms}^{-2}$. (A) \(8 \mathrm{~ms}^{-2}\) (B) \(10 \mathrm{~ms}^{-2}\) (C) \(40 \mathrm{~ms}^{-2}\) (D) \(40 \mathrm{cms}^{-2}\)

If the kinetic energy of a particle executing S.H.M. is given by \(\mathrm{K}=\mathrm{K}_{0} \cos ^{2} \omega \mathrm{t}\), then the displacement of the particle is given by \(\ldots \ldots\) (A) $\left\\{\mathrm{K}_{0} / \mathrm{m} \omega^{2}\right\\} \sin \omega \mathrm{t}$ (B) $\left\\{\left(2 \mathrm{~K}_{0}\right) /\left(\mathrm{m} \omega^{2}\right)\right\\}^{1 / 2} \sin \omega t$ (C) \(\left\\{2 \omega^{2} / \mathrm{mK}_{0}\right\\} \sin \omega t\) (D) $\left\\{2 \mathrm{~K}_{0} / \mathrm{m} \omega\right\\}^{1 / 2} \sin \omega t$

For the following questions, statement as well as the reason(s) are given. Each questions has four options. Select the correct option. (a) Statement \(-1\) is true, statement \(-2\) is true; statement \(-2\) is the correct explanation of statement \(-1\). (b) Statement \(-1\) is true, statement \(-2\) is true but statement \(-2\) is not the correct explanation of statement \(-1\). (c) Statement \(-1\) is true, statement \(-2\) is false (d) Statement \(-1\) is false, statement \(-2\) is true (A) a (B) \(\mathrm{b}\) (C) \(\mathrm{c}\) (D) \(\mathrm{d}\) Statement \(-1:\) The periodic time of a S.H.O. depends on its amplitude and force constant. Statement \(-2:\) The elasticity and inertia decides the frequency of S.H.O. (A) a (B) \(b\) (C) c (D) \(\mathrm{d}\)

A string of length \(70 \mathrm{~cm}\) is stretched between two rigid supports. The resonant frequency for this string is found to be \(420 \mathrm{~Hz}\) and \(315 \mathrm{~Hz}\). If there are no resonant frequencies between these two values, then what would be the minimum resonant frequency of this string ? (A) \(10.5 \mathrm{~Hz}\) (B) \(1.05 \mathrm{~Hz}\) (C) \(105 \mathrm{~Hz}\) (D) \(1050 \mathrm{~Hz}\)
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