The ends of a stretched wire of length l are fixed at x=0 and x=l . In one experiment, the displacement of the wire is y1=asin ((π x/l))(sin â¡ ω t) and energy E1 and in another experiment, its displacement is y2=asin ((3 π x/l))(sin â¡ 3 ω t) and energy is E2 then

Question

The ends of a stretched wire of length l are fixed at x=0 and x=l . In one experiment, the displacement of the wire is y1=asin ((π x/l))(sin â¡ ω t) and energy E1 and in another experiment, its displacement is y2=asin ((3 π x/l))(sin â¡ 3 ω t) and energy is E2 then (A) E2=E1 (B) E2=3E1 (C) E2=(E1/3) (D) E2=9E1

Tardigrade Admin · Accepted Answer

In first case freq. ⇒ ω 1=ω In second case freq. ω 2=3ω ∵ Energy α (f r e q .)2 (E1/E2)=(ω2/(3 ω)2) ⇒ E2=9E1

$E_{2} = E_{1}$

$E_{2} = 3 E_{1}$

$E_{2} = \frac{E _{1}}{3}$

$E_{2} = 9 E_{1}$

In first case freq. $\Rightarrow ω_{1} = ω$
In second case freq. $ω_{2} = 3 ω$
$∵ E n er g y α (f re q .)^{2}$
$\frac{E _{1}}{E _{2}} = \frac{ω ^{2}}{( 3 ω ) ^{2}}$
$\Rightarrow E_{2} = 9 E_{1}$

Q. The ends of a stretched wire of length l are fixed at x=0 and x=l . In one experiment, the displacement of the wire is y1​=asin(lπx​)(sin⁡ωt) and energy E1​ and in another experiment, its displacement is y2​=asin(l3πx​)(sin⁡3ωt) and energy is E2​ then

E2​=E1​

E2​=3E1​

E2​=3E1​​

E2​=9E1​

In first case freq. ⇒ω1​=ω In second case freq. ω2​=3ω ∵Energyα(freq.)2 E2​E1​​=(3ω)2ω2​ ⇒E2​=9E1​

$E_{2} = E_{1}$

$E_{2} = 3 E_{1}$

$E_{2} = \frac{E _{1}}{3}$

$E_{2} = 9 E_{1}$

In first case freq. $\Rightarrow ω_{1} = ω$
In second case freq. $ω_{2} = 3 ω$
$∵ E n er g y α (f re q .)^{2}$
$\frac{E _{1}}{E _{2}} = \frac{ω ^{2}}{( 3 ω ) ^{2}}$
$\Rightarrow E_{2} = 9 E_{1}$