In SHM, the plot of acceleration y at time t and displacement x for one complete oscillation will be

Options

A. Ellipse
B. Sinusoidal curve
C. Circle
D. Straight line

Answer

To solve the question regarding the plot of acceleration (y) versus displacement (x) in Simple Harmonic Motion (SHM), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Basic Equations of SHM**:
- The displacement in SHM can be expressed as:
\[ x(t) = A \sin(\omega t + \phi) \]
- Here, \(A\) is the amplitude, \(\omega\) is the angular frequency, and \(\phi\) is the phase constant.

2. **Find the Velocity**:
- The velocity \(v\) is the derivative of displacement with respect to time:
\[ v(t) = \frac{dx}{dt} = A \omega \cos(\omega t + \phi) \]

3. **Find the Acceleration**:
- The acceleration \(a\) is the derivative of velocity with respect to time:
\[ a(t) = \frac{dv}{dt} = -A \omega^2 \sin(\omega t + \phi) \]

4. **Relate Acceleration to Displacement**:
- Since \(x(t) = A \sin(\omega t + \phi)\), we can express acceleration in terms of displacement:
\[ a = -\omega^2 x \]

5. **Plotting Acceleration vs. Displacement**:
- The equation \(a = -\omega^2 x\) represents a linear relationship between acceleration and displacement, where:
- The slope \(m = -\omega^2\)
- The y-intercept is 0 (the line passes through the origin).

6. **Conclusion**:
- When we plot acceleration (y-axis) against displacement (x-axis), we get a straight line with a negative slope. Therefore, the correct answer to the question is that the plot will be a straight line.

### Final Answer:
The plot of acceleration \(y\) at time \(t\) and displacement \(x\) for one complete oscillation will be a **straight line**.

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