The variation of the acceleration (f) of the particle executing S.H.M. with its displacement (X) is represented by the curve

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Explanation

To solve the problem regarding the variation of acceleration (f) of a particle executing Simple Harmonic Motion (S.H.M.) with its displacement (X), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the relationship between acceleration and displacement in S.H.M.**:
- The acceleration (f) of a particle in S.H.M. is given by the formula:
\[ f = -\omega^2 x \]
where \( \omega \) is the angular frequency and \( x \) is the displacement from the mean position.

2. **Rearrange the equation**:
- We can express this equation in a more familiar linear form. If we let \( m = \omega^2 \), then the equation becomes:
\[ f = -mx \]

3. **Identify the form of the equation**:
- The equation \( f = -mx \) can be compared to the standard linear equation \( y = mx + c \). Here, we can identify:
- \( y \) corresponds to \( f \) (acceleration),
- \( x \) corresponds to \( x \) (displacement),
- The slope \( m \) is equal to \(-m\) (negative slope),
- The y-intercept \( c \) is 0.

4. **Graphical representation**:
- When we plot the graph of acceleration (f) on the y-axis against displacement (x) on the x-axis, we will get a straight line.
- The line will have a negative slope (due to the negative sign in the equation) and will pass through the origin (0,0) since the y-intercept is 0.

5. **Conclusion**:
- The graph representing the variation of acceleration with displacement in S.H.M. is a straight line with a negative slope, starting from the origin.

### Final Answer:
The variation of the acceleration (f) of the particle executing S.H.M. with its displacement (X) is represented by a straight line graph with a negative slope, passing through the origin.

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