The linear density of a vibrating string is 10−4kg/m. A transverse wave is propagating on the string, which is described by the equation y = 0.02 sin (x + 30 t), where x and y are in meters and time t in seconds. Then tension in the string is
Question
The linear density of a vibrating string is 10−4kg/m. A transverse wave is propagating on the string, which is described by the equation y = 0.02 sin (x + 30 t), where x and y are in meters and time t in seconds. Then tension in the string is
Solution
The wave equation for a vibrating string is given by:
y = A sin(kx - wt)
where: A is the amplitude, k is the wave number, w is the angular frequency.
From the given equation, we can see that the amplitude A = 0.02 m and the angular frequency w = 30 rad/s.
The speed of the wave v can be found from the relation:
v = w/k
But from the given equation, we can see that k = 1 (since the coefficient of x is 1). So, the speed of the wave v = w = 30 m/s.
The tension T in the string can be found from the relation:
v = sqrt(T/u)
where u is the linear density of the string. Rearranging for T gives:
T = u * v^2
Substituting the given values, we get:
T = 10^-4 kg/m * (30 m/s)^2 = 0.09 N
So, the tension in the string is 0.09 N.
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