A particle is executing SHM. Then the graph of acceleration as a function of displacement is

Options

A. straight line
B. circle
C. ellipse
D. hyperbola

Explanation

To solve the problem of determining the graph of acceleration as a function of displacement for a particle executing simple harmonic motion (SHM), we can follow these steps:

### Step 1: Understand the relationship between force, acceleration, and displacement
In SHM, the restoring force acting on the particle is directly proportional to the displacement from the equilibrium position and is directed towards that position. Mathematically, this can be expressed as:
\[ F = -kx \]
where \( F \) is the force, \( k \) is the spring constant, and \( x \) is the displacement.

### Step 2: Apply Newton's second law
According to Newton's second law, the force can also be expressed in terms of mass and acceleration:
\[ F = ma \]
where \( m \) is the mass of the particle and \( a \) is its acceleration.

### Step 3: Set the two expressions for force equal to each other
Since both expressions represent the same force, we can set them equal:
\[ ma = -kx \]

### Step 4: Solve for acceleration
Rearranging the equation to solve for acceleration \( a \):
\[ a = -\frac{k}{m} x \]
This shows that acceleration is directly proportional to the displacement \( x \) but in the opposite direction (hence the negative sign).

### Step 5: Identify the nature of the graph
The equation \( a = -\frac{k}{m} x \) indicates that acceleration \( a \) varies linearly with displacement \( x \). The slope of the line is \( -\frac{k}{m} \), which is a constant. Therefore, the graph of acceleration as a function of displacement will be a straight line passing through the origin with a negative slope.

### Conclusion
Thus, the graph of acceleration as a function of displacement for a particle executing SHM is a straight line with a negative slope.

### Final Answer
The answer is that the graph of acceleration as a function of displacement is a straight line.

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