In an SHM, x is the displacement and a is the acceleration at time t. the plot of a against x for one complete oscillation will be

Options

A. a straight line
B. a circle
C. an ellipse
D. a sinusoidal curve

Explanation

To solve the problem of determining the plot of acceleration (a) against displacement (x) for simple harmonic motion (SHM), we can follow these steps:

### Step 1: Understand the relationship between acceleration and displacement in SHM
In simple harmonic motion, the acceleration (a) is related to the displacement (x) by the equation:
\[ a = -\omega^2 x \]
where \( \omega \) is the angular frequency.

### Step 2: Analyze the equation
From the equation \( a = -\omega^2 x \), we can see that acceleration is directly proportional to displacement but with a negative sign. This indicates that as the displacement increases in one direction, the acceleration acts in the opposite direction.

### Step 3: Determine the nature of the graph
Since the relationship is linear (i.e., it can be expressed in the form \( y = mx + c \)), we can conclude that the plot of acceleration (a) against displacement (x) will be a straight line. The slope of this line will be \(-\omega^2\), and it will pass through the origin because when \( x = 0 \), \( a \) is also 0.

### Step 4: Identify the maximum and minimum values
The maximum displacement in SHM is the amplitude (A), and at this point, the maximum acceleration is given by:
\[ a_{\text{max}} = -\omega^2 A \]
Thus, the graph will extend from \( -\omega^2 A \) (when \( x = A \)) to \( \omega^2 A \) (when \( x = -A \)).

### Step 5: Sketch the graph
On a graph where the x-axis represents displacement (x) and the y-axis represents acceleration (a), the line will slope downwards from the point \( (A, -\omega^2 A) \) to \( (-A, \omega^2 A) \).

### Conclusion
Thus, the plot of acceleration against displacement for one complete oscillation in SHM is a straight line.

### Final Answer
The correct option is: **A straight line.**

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