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Concepts Of Physics MCQ Edition [Volume 1]PhysicsWave Motion and Waves on a String

A string of length L secured at both ends vibrates in its fundamental mode with a frequency and a maximum amplitude A . Choose the origin at one end of the string and the X -axis along the string. Let the Y -axis be along the direction of displacement. Set t = 0 at the instant the middle point of the string crosses its mean position travelling towards the positive y -direction. Determine the equation describing the s

Options

  1. Ay = A ( x L ) (2 t)
  2. By = A ( x L ) (2 t)
  3. Cy = A ( 2 x L ) (2 t)
  4. Dy = A ( x L ) (2 t)

Correct answer

A. y = A ( x L ) (2 t)

Step-by-step solution

The general equation of a standing wave is y(x, t) = f(x) g(t) . Since the string is fixed at x = 0 and x = L , the displacement at these points must be zero. This requires the spatial part to be a sine function: f(x) = A (kx) . For the fundamental mode, the string vibrates in a single loop, which means L = 2 = 2L . The wave number is k = 2 = 2 2L = L . Thus, the spatial part is f(x) = A ( x L ) . The temporal part is given by g(t) = ( t + ) . At t = 0 , the middle point of the string ( x = L 2 ) crosses the mean p

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