A sinusoidal wave is genrated by moving the end of a string up and down, periodically. The genrated must apply the energy has ………..x and least power when the end of the string attached to genrated to genrated has ………..Y. the most suitable option which correctly fills blanks X and Y, is

Options

A. maximum displacement, least acceleration
B. maximum displacement, maximum acceleration
C. least displacement, maximum acceleration
D. least displacement, least acceleration

Explanation

To solve the question, we need to analyze the relationship between energy, power, displacement, and acceleration in the context of a sinusoidal wave generated on a string.

### Step-by-Step Solution:

1. **Understanding the Wave Equation**:
The equation for a sinusoidal wave can be expressed as:
\[ y(x, t) = A \sin(kx - \omega t) \]
where:
- \(A\) is the amplitude,
- \(k\) is the wave number,
- \(\omega\) is the angular frequency.

2. **Power in a Sinusoidal Wave**:
The power \(P\) transmitted by the wave can be expressed in terms of the wave properties. The average power is given by:
\[ P = \frac{1}{2} \mu \omega^2 A^2 \cos^2(kx - \omega t) \]
where \(\mu\) is the linear mass density of the string.

3. **Condition for Least Power**:
The power will be least when the cosine term is zero:
\[ \cos^2(kx - \omega t) = 0 \]
This occurs when \(kx - \omega t = \frac{\pi}{2} + n\pi\) for integers \(n\). Therefore, the least power corresponds to points where the wave is at its equilibrium position (displacement is zero).

4. **Energy Considerations**:
The energy in the wave is related to its displacement. The maximum energy is associated with maximum displacement (amplitude \(A\)). Conversely, when the displacement is least (zero), the energy is also at a minimum.

5. **Acceleration**:
The acceleration of a point on the string is given by:
\[ a = -A \omega^2 \sin(kx - \omega t) \]
The acceleration is maximum when the displacement is maximum (at the peaks of the wave).

6. **Filling in the Blanks**:
- For the first blank (X), the generator must apply energy at maximum displacement (which corresponds to maximum energy).
- For the second blank (Y), the least power occurs when the end of the string is at its equilibrium position (least displacement).

Thus, the correct options for the blanks are:
- X: Maximum energy
- Y: Least displacement

### Final Answer:
The most suitable option which correctly fills the blanks X and Y is:
- X: Maximum energy
- Y: Least displacement