The graph between instantaneous acceleration and angualr displacement of a particle performing S.H.M. is

Options

A. parabola
B. straight line
C. sinusoidal
D. circle

Answer

To determine the graph between instantaneous acceleration and angular displacement of a particle performing Simple Harmonic Motion (SHM), we can follow these steps:

### Step 1: Understand the relationship in SHM
In SHM, the displacement \( x \) of a particle can be expressed as:
\[ x(t) = A \sin(\omega t) \]
where:
- \( A \) is the amplitude,
- \( \omega \) is the angular frequency,
- \( t \) is time.

### Step 2: Derive the expression for acceleration
The instantaneous acceleration \( a \) is the second derivative of displacement with respect to time. First, we find the velocity \( v(t) \):
\[ v(t) = \frac{dx}{dt} = A \omega \cos(\omega t) \]
Now, differentiating the velocity to find acceleration:
\[ a(t) = \frac{dv}{dt} = -A \omega^2 \sin(\omega t) \]

### Step 3: Relate angular displacement to acceleration
The angular displacement \( \theta \) can be represented as:
\[ \theta(t) = \omega t \]
Thus, we can express the acceleration in terms of angular displacement:
\[ a(\theta) = -A \omega^2 \sin(\theta) \]

### Step 4: Analyze the graph of acceleration vs. angular displacement
The function \( a(\theta) = -A \omega^2 \sin(\theta) \) indicates that:
- The acceleration is proportional to the negative sine of the angular displacement.
- The graph of \( \sin(\theta) \) oscillates between -1 and 1, hence \( a(\theta) \) will oscillate between \(-A \omega^2\) and \(A \omega^2\).

### Step 5: Sketch the graph
- The graph of \( a(\theta) \) will be a sine wave that is inverted due to the negative sign.
- It will cross the horizontal axis at \( \theta = n\pi \) (where \( n \) is an integer) and reach its maximum and minimum values at \( \theta = (2n+1)\frac{\pi}{2} \).

### Conclusion
The graph between instantaneous acceleration and angular displacement of a particle performing SHM is a sine wave that is inverted, oscillating between \(-A \omega^2\) and \(A \omega^2\).

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