The oscillation of a body on a smooth horizental surface is respresented by the equation
`X = A cos (omegat)`
where one of the following graph shown correctly the variation a with `t`?

Options

A.
B.
C.
D.

Explanation

To solve the problem, we need to analyze the given displacement equation and derive the acceleration from it. Here’s a step-by-step breakdown:

### Step 1: Understand the Displacement Equation
The displacement of the body is given by the equation:
\[ x = A \cos(\omega t) \]
where:
- \( x \) is the displacement,
- \( A \) is the amplitude,
- \( \omega \) is the angular frequency,
- \( t \) is the time.

### Step 2: Find the Velocity
The velocity \( v \) is the rate of change of displacement with respect to time. We can find it by differentiating the displacement equation with respect to \( t \):
\[ v = \frac{dx}{dt} = -A \omega \sin(\omega t) \]

### Step 3: Find the Acceleration
The acceleration \( a \) is the rate of change of velocity with respect to time. We can find it by differentiating the velocity equation:
\[ a = \frac{dv}{dt} = -A \omega^2 \cos(\omega t) \]

### Step 4: Analyze the Acceleration Equation
The acceleration equation is:
\[ a = -A \omega^2 \cos(\omega t) \]
This indicates that the acceleration is proportional to the cosine function, but with a negative sign.

### Step 5: Graph the Acceleration
The graph of \( -\cos(\omega t) \) will start at its maximum negative value when \( t = 0 \) (since \( \cos(0) = 1 \)), and it will oscillate between \( -A \omega^2 \) and \( A \omega^2 \). The graph will have the following characteristics:
- It will be a cosine wave.
- It will be inverted (negative cosine).

### Step 6: Identify the Correct Graph
From the options provided, we need to identify the graph that represents the negative cosine function. The correct graph will start at the lowest point (minimum value) and will oscillate upwards and downwards, matching the behavior of \( -\cos(\omega t) \).

### Conclusion
The correct option that represents the variation of acceleration with time is the one that shows a negative cosine wave.

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