First slide

Stationary waves

Question

The transverse displacement of a string clamped at its both ends is given by $y (x, t) = 0 .06 \sin (\frac{2 π}{3} x) \cos (120 πt)$ where x and y are in m and t in s. The length of the string is 1.5 m and its mass is $3 \times 10^{- 2} kg$ . The tension in the string is

Moderate

A

324 N
B

648 N
C

832 N
D

972 N

Solution

The given equation is
$y (x, t) = 0 .06 \sin (\frac{2 π}{3} x) \cos (120 πt)$
Compare it with $y (x, t) = 2 asin kxcos ωt$
we get, $k = \frac{2 π}{3}, or \frac{2 π}{λ} = \frac{2 π}{3} or λ = 3 m$
and $ω = 120 π or 2 πv = 120 π or v = 60 Hz = 60 s^{- 1}$
Velocity of wave, $v = vλ = (60 s^{- 1}) (3 m) = 180 {ms}^{- 1}$
Mass per unit length of the string,
$μ = \frac{3 \times 10^{- 2} kg}{1 .5 m} = 2 \times 10^{- 2} {kgm}^{- 1}$
Velocity of transverse wave in the string,
$v = \sqrt{\frac{T}{μ}} or v^{2} = \frac{T}{μ} or T = v^{2} μ$
$T = {(180 {ms}^{- 1})}^{2} (2 \times 10^{- 2} {kgm}^{- 1}) = 648 N$

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