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, we need to analyze the relationship between acceleration (a) and displacement (x) in Simple Harmonic Motion (SHM). Let's go through the solution step by step.

### Step 1: Understanding SHM
In SHM, the acceleration (a) of an object is directly related to its displacement (x) from the mean position. The relationship can be expressed mathematically as:
\[ a = -\omega^2 x \]
where \( \omega \) is the angular frequency of the motion.

**Hint:** Recall that in SHM, acceleration is always directed towards the mean position and is proportional to the displacement.

### Step 2: Analyzing the Equation
From the equation \( a = -\omega^2 x \), we can see that acceleration is proportional to displacement but with a negative sign. This means that when displacement is positive (object is on one side of the mean position), acceleration is negative (pulling it back towards the mean position), and vice versa.

**Hint:** The negative sign indicates that acceleration acts in the opposite direction to displacement.

### Step 3: Plotting the Graph
If we plot acceleration (a) on the y-axis and displacement (x) on the x-axis, the equation \( a = -\omega^2 x \) represents a straight line with:
- Slope = \(-\omega^2\)
- Y-intercept = 0 (since when \( x = 0 \), \( a = 0 \))

**Hint:** A linear relationship means the graph will be a straight line, and the slope will determine the steepness of the line.

### Step 4: Conclusion
Since the graph of acceleration against displacement is a straight line that passes through the origin, the correct answer to the question is that the plot of acceleration (a) against displacement (x) for one complete oscillation will be a straight line.

**Final Answer:** The plot of acceleration (a) against displacement (x) for one complete oscillation will be a straight line.