For `SHM` to take place force acting on the body should be proportional to `-x` or `F =- kx`. If `A` be the amplitude then energy of oscillation is `1//2KA^(2)`.
Force acting on a block is `F = (-4x +8)`. Here `F` is in newton and `x` in the position of block on x-axis in meters

Options

A. Motion of the block is periodic but not simple harmonic
B. Motion of the block is not period
C. Motion of the block is simple harmonic about the origin, `x =0`
D. Motion of the block is simple harmonic about `x =2m`

Explanation

To solve the problem, we need to analyze the given force equation and determine the characteristics of the motion of the block.

### Step 1: Identify the given force equation
The force acting on the block is given by:
\[ F = -4x + 8 \]

### Step 2: Rearrange the force equation
We can rewrite the force equation as:
\[ F = -4(x - 2) \]
This shows that the force can be expressed in the form of a restoring force, where the equilibrium position (mean position) is at \( x = 2 \).

### Step 3: Identify the spring constant (k)
From the rearranged equation, we can see that the coefficient of \( x \) is \(-4\). Therefore, we can identify:
\[ k = 4 \]

### Step 4: Determine the mean position (x₀)
From the equation \( F = -k(x - x₀) \), we can see that the mean position \( x₀ \) is:
\[ x₀ = 2 \]

### Step 5: Determine the amplitude (A)
In simple harmonic motion (SHM), the amplitude is the maximum displacement from the mean position. Since the force equation indicates a linear restoring force, the motion can be periodic. The amplitude can be considered as the distance from the mean position to the maximum displacement. If we assume the maximum displacement is \( A \), then:
\[ A = 4 \] (as inferred from the force equation)

### Step 6: Determine the type of motion
Since the force is not purely proportional to \(-x\) (due to the constant term +8), the motion is not simple harmonic motion. The motion is periodic but not simple harmonic because the restoring force does not follow the form \( F = -kx \) around the origin.

### Conclusion
The motion of the block is periodic but not simple harmonic due to the presence of the constant force term.

### Final Answer
The motion of the block is periodic but not simple harmonic.

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