Question

A string of length L is stretched between two points x = 0 and x = L and the endpoints are rigidly clamped. Which of the following can represent the displacement of the string from the equilibrium position?

• A. \(x\cos (\frac{\pi x}{L})\)
• B. \(x\sin (\frac{\pi x}{L})\)
• C. \(x(\frac{x}{L}-1)\)
• D. \(x(\frac{x}{L}-1)^2\)

Answer 1

The Correct Option is B, C, D

To solve this problem, we need to determine which expressions can represent the displacement of a string stretched between two rigidly clamped points at \(x = 0\) and \(x = L\).

When a string is fixed at both ends, the displacement at these endpoints must be zero. Let's evaluate each given option under these boundary conditions:

1. Option 1: \(x\cos \left(\frac{\pi x}{L}\right)\)
   • At \(x = 0\), the displacement is \(0 \cdot \cos\left(0\right) = 0\).
   • At \(x = L\), the displacement is \(L \cdot \cos\left(\pi\right) = -L\), which is not zero.
   • Thus, this option does not satisfy the boundary condition at \(x = L\).
2. Option 2: \(x\sin \left(\frac{\pi x}{L}\right)\)
   • At \(x = 0\), the displacement is \(0 \cdot \sin(0) = 0\).
   • At \(x = L\), the displacement is \(L \cdot \sin(\pi) = 0\).
   • This option satisfies both boundary conditions, making it a valid expression for displacement.
3. Option 3: \(x\left(\frac{x}{L} - 1\right)\)
   • At \(x = 0\), the displacement is \(0 \cdot \left(0 - 1\right) = 0\).
   • At \(x = L\), the displacement is \(L \cdot \left(1 - 1\right) = 0\).
   • This option satisfies both boundary conditions as well.
4. Option 4: \(x\left(\frac{x}{L} - 1\right)^2\)
   • At \(x = 0\), the displacement is \(0 \cdot \left(-1\right)^2 = 0\).
   • At \(x = L\), the displacement is \(L \cdot (0)^2 = 0\).
   • This option also satisfies the boundary conditions at both ends.

From the analysis above, the expressions that can represent the displacement of the string are:

• \(x\sin \left(\frac{\pi x}{L}\right)\)
• \(x\left(\frac{x}{L} - 1\right)\)
• \(x\left(\frac{x}{L} - 1\right)^2\)

Hence, the correct answer is options 2, 3, and 4.