Which of the following graphs best represents the variation of acceleration 'a' with displacement x?

Options

A.
B.
C.
D.

Answer

To determine which graph best represents the variation of acceleration 'a' with displacement 'x' for a body performing Simple Harmonic Motion (SHM), we can follow these steps:

### Step 1: Understand the relationship between acceleration and displacement in SHM
In SHM, the acceleration 'a' is directly related to the displacement 'x' from the equilibrium position. The formula for acceleration in SHM is given by:
\[ a = -\omega^2 x \]
where \( \omega \) is the angular frequency.

### Step 2: Analyze the equation
From the equation \( a = -\omega^2 x \), we can see that:
- The acceleration 'a' is proportional to the displacement 'x'.
- The negative sign indicates that the acceleration is directed opposite to the displacement. This means that when the displacement is positive (to the right of the equilibrium), the acceleration is negative (to the left), and vice versa.

### Step 3: Identify the graph characteristics
The equation \( a = -\omega^2 x \) represents a linear relationship between 'a' and 'x'. Specifically:
- It is a straight line with a slope of \(-\omega^2\).
- The graph will pass through the origin (0,0) because when \( x = 0 \), \( a = 0 \).
- The line will slope downwards from left to right due to the negative slope.

### Step 4: Choose the correct graph
Based on the characteristics identified:
- The correct graph will be a straight line that starts at the origin and slopes downwards.
- This corresponds to option (b) in the provided choices.

### Conclusion
Thus, the graph that best represents the variation of acceleration 'a' with displacement 'x' is option (b).

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