11. (JEE Main 2020 (Online) 9th January Evening Slot)

A wire of length L and mass per unit length 6.0 × 10^–3 kgm^–1 is put under tension of 540 N. Two consecutive frequencies that it resonates at are : 420 Hz and 490 Hz. Then L in meters is :

A. 5.1 m

B. 2.1 m

C. 1.1 m

D. 8.1 m

Correct Answer is Option (B)

Fundamental frequency = 70 Hz.

70 = $\frac{1}{2 l} \sqrt{\frac{T}{μ}}$

$\Rightarrow$ $l$ = 2.14 m

12. (JEE Main 2020 (Online) 8th January Evening Slot)

A transverse wave travels on a taut steel wire with a velocity of v when tension in it is 2.06 × 10⁴ N. When the tension is changed to T, the velocity changed to v/2. The value of T is close to :

A. 30.5 × 10⁴ N

B. 2.50 × 10⁴ N

C. 10.2 × 10² N

D. 5.15 × 10³ N

Correct Answer is Option (D)

$v = \sqrt{\frac{T}{μ}}$

$∴$ $\frac{v_{1}}{v_{2}} = \sqrt{\frac{T_{1}}{T_{2}}}$

v₁ = v, v₂ = $\frac{v}{2}$

$\Rightarrow$ $\frac{v}{\frac{v}{2}} =$ $\sqrt{\frac{2.06 \times 10^{4}}{T_{2}}}$

$\Rightarrow$ T₂ = $\frac{2.06 \times 10^{4}}{4}$ = 5.15 × 10³ N

13. (JEE Main 2020 (Online) 7th January Morning Slot)

Speed of a transverse wave on a straight wire (mass 6.0 g, length 60 cm and area of cross-section 1.0 mm²) is 90 ms^-1. If the Young's modulus of wire is 16 $\times$ 10¹¹ Nm^-2, the extension of wire over its natural length is :

A. 0.03 mm

B. 0.04 mm

C. 0.02 mm

D. 0.01 mm

Correct Answer is Option (A)

Velocity of the wave, v = $\sqrt{\frac{T}{μ}}$

$\Rightarrow$ T = v² $μ$

We know, Youngs modulus,

Y = $\frac{\frac{F}{A}}{\frac{Δ l}{l}}$ = $\frac{\frac{T}{A}}{\frac{Δ l}{l}}$

[As here F = T]

$\Rightarrow$ $Y \frac{Δ l}{l}$ = $\frac{T}{A}$ = $\frac{v^{2} μ}{A}$

$\Rightarrow$ $Δ$ l = $\frac{v^{2} μ l}{A Y}$

= $\frac{90 \times 90 \times \frac{60 \times 10^{- 3}}{60 \times 10^{- 2}} \times 60 \times 10^{- 2}}{1 \times 10^{- 6} \times 16 \times 10^{11}}$

= 3 $\times$ 10^-5 m

= 0.03 mm

14. (JEE Main 2019 (Online) 9th April Evening Slot)

A string 2.0 m long and fixed at its ends is driven by a 240 Hz vibrator. The string vibrates in its third harmonic mode. The speed of the wave and its fundamental frequency is :-

A. 180m/s, 80 Hz

B. 180m/s, 120 Hz

C. 320m/s, 120 Hz

D. 320m/s, 80 Hz

Correct Answer is Option (D)

We have:

$f = \frac{n v}{2 l}$

$240 = \frac{3 \times v}{2 \times 2}$

$\Rightarrow$ v = 320 m/s

Fundamental frequency = $\frac{v}{2 l}$ = 80 Hz.

15. (JEE Main 2019 (Online) 9th April Morning Slot)

A string is clamped at both the ends and it is vibrating in its 4th harmonic. The equation of the stationary wave is Y = 0.3 sin(0.157x) cos(200pt). The length of the string is : (All quantities are in SI units.)

A. 60 m

B. 20 m

C. 80 m

D. 40 m

Correct Answer is Option (C)

4^th harmonic

$4 \frac{λ}{2} = l; 2 λ = l$

From equation $\frac{2 π}{λ} = 0.157$

$λ$ = 40 ; $l$ = 2 $λ$ = 80 m

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