Two strings with circular cross section and made of same material, are stretched to have same amount of tension. A transverse wave is then made to pass through both the strings. The velocity of the wave in the first string having the radius of cross section $R$ is $v_1$, and that in the other string having radius of cross section $R/2$ is $v_2$. Then $\frac{v_2}{v_1} =$
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Solution Steps
1
Define Wave Velocity Formula
The velocity $v$ of a transverse wave in a stretched string is given by:
$$v = \sqrt{\frac{T}{\mu}}$$
where $T$ is the tension and $\mu$ is the linear mass density (mass per unit length).
2
Express Linear Mass Density
Linear mass density $\mu$ can be expressed in terms of the material density $\rho$ and cross-sectional area $A$:
$$\mu = \frac{\text{Mass}}{\text{Length}} = \frac{\rho \times \text{Volume}}{\text{Length}} = \rho A$$
Since the cross-section is circular, $A = \pi r^2$. Thus, $\mu = \rho \pi r^2$.
3
Establish Proportionality
Substituting $\mu$ into the velocity formula:
$$v = \sqrt{\frac{T}{\rho \pi r^2}} = \frac{1}{r} \sqrt{\frac{T}{\rho \pi}}$$
Given that tension $T$ and material density $\rho$ are the same for both strings, we find that $v \propto \frac{1}{r}$.
4
Calculate the Ratio
Using the inverse relationship between velocity and radius:
$$\frac{v_2}{v_1} = \frac{r_1}{r_2}$$
Given $r_1 = R$ and $r_2 = R/2$.
5
Final Computation
Substitute the values into the ratio:
$$\frac{v_2}{v_1} = \frac{R}{R/2} = 2$$
Final Answer: 2
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Theory
1. Transverse Wave Velocity in Strings
The speed of a transverse wave on a string is determined by the restoring force (tension) and the inertial property of the string (linear mass density). The formula $v = \sqrt{T/\mu}$ highlights that waves travel faster in tighter (higher $T$) and lighter (lower $\mu$) strings. This principle is fundamental in the design of musical instruments like guitars and violins, where adjusting tension changes the pitch by altering wave speed.
2. Linear Mass Density and Cross-section
Linear mass density ($\mu$) is a measure of mass distribution along a length. For objects with a uniform cross-sectional area $A$ and material density $\rho$, $\mu$ is calculated as $\rho \times A$. In the case of cylindrical wires or strings, $A = \pi r^2$. This means that even if strings are made of the same material, a thicker string (larger radius) will have a significantly higher linear mass density, which acts to slow down wave propagation.
3. Effect of String Geometry on Speed
By combining the wave velocity formula with the geometric definition of linear mass density, we derive $v = \frac{1}{r}\sqrt{T/\rho\pi}$. This specific relationship shows that wave speed is inversely proportional to the radius of the string. doubling the radius reduces the speed by half, while halving the radius (as seen in this problem) doubles the speed. This geometric dependence is a common focal point in JEE Main Physics problems regarding acoustics and wave mechanics.
4. Restoring Force and Inertia
The physical reason behind the $\sqrt{T/\mu}$ relationship lies in Newton’s second law. Tension $T$ provides the vertical restoring force that pulls the string back to its equilibrium position when a wave passes through. Linear mass density $\mu$ represents the inertia that resists this acceleration. The square root dependence arises from the derivation using small-angle approximations in the wave equation, ensuring that speed remains constant for a given medium regardless of wave amplitude.
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FAQs
1
Why is material density $\rho$ considered the same?
The problem states both strings are ‘made of same material’, which implies identical volumetric density $\rho$.
2
What if the tensions were different?
If tensions were different, you would include them in the ratio: $\frac{v_2}{v_1} = \frac{r_1}{r_2} \sqrt{\frac{T_2}{T_1}}$.
3
Does wave velocity depend on the length of the string?
No, wave velocity is a property of the medium (tension and linear density) and does not depend on the total length.
4
How does frequency change between the strings?
Frequency depends on the source. If the source is the same, frequency remains constant, and wavelength changes according to $\lambda = v/f$.
5
What is the difference between transverse and longitudinal waves?
In transverse waves (like on a string), particles move perpendicular to wave direction. In longitudinal waves (like sound), they move parallel.
6
What if the cross-section wasn’t circular?
The ratio would depend on the square root of the ratio of the cross-sectional areas: $v_2/v_1 = \sqrt{A_1/A_2}$.
7
Can waves travel through a string without tension?
No, without tension ($T=0$), there is no restoring force to propagate the wave, so velocity $v=0$.
8
What is $\mu$ if the mass is $m$ and length is $l$?
$\mu = m/l$. It is the total mass divided by the total length of the string.
9
Does gravity affect wave speed in vertical strings?
Yes, because tension $T$ varies along the length of a vertical string due to its own weight.
10
Is this formula valid for large amplitudes?
The standard formula $v = \sqrt{T/\mu}$ is derived using a small-amplitude approximation. For very large amplitudes, the math becomes non-linear.
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