A force F=-4x+8 (in N) is acting on a block where x is the position of the block in metres. The energy of oscillation is 32 J. The block oscillates between two points, out of which the value of position of one point (in metres) is an integer from 0 to 9. Find it.

Explanation

To solve the problem step by step, we can follow these instructions:

### Step 1: Understand the Force Equation
The force acting on the block is given by the equation:
\[ F = -4x + 8 \]
This indicates that the force is a function of the position \( x \).

### Step 2: Find the Mean Position
The mean position (equilibrium position) occurs when the force \( F \) is zero. Setting the force equation to zero:
\[ -4x + 8 = 0 \]
Solving for \( x \):
\[ 4x = 8 \]
\[ x = 2 \]
Thus, the mean position is \( x = 2 \) meters.

### Step 3: Identify the Spring Constant
From the force equation, we can identify the spring constant \( k \):
\[ F = -kx \]
Comparing with our equation:
\[ k = 4 \, \text{N/m} \]

### Step 4: Use the Energy Equation
The total mechanical energy \( E \) in simple harmonic motion is given by:
\[ E = \frac{1}{2} k A^2 \]
where \( A \) is the amplitude. Given that the energy \( E \) is 32 J, we can set up the equation:
\[ 32 = \frac{1}{2} \times 4 \times A^2 \]

### Step 5: Solve for Amplitude
Rearranging the energy equation to solve for \( A^2 \):
\[ 32 = 2A^2 \]
\[ A^2 = \frac{32}{2} = 16 \]
Taking the square root:
\[ A = 4 \, \text{m} \]

### Step 6: Determine the Oscillation Points
The block oscillates around the mean position \( x = 2 \) with an amplitude of \( 4 \) meters. Therefore, the maximum and minimum positions are:
- Maximum position:
\[ x_{max} = 2 + 4 = 6 \, \text{m} \]
- Minimum position:
\[ x_{min} = 2 - 4 = -2 \, \text{m} \]

### Step 7: Identify Valid Positions
Since the problem specifies that one of the positions must be an integer between 0 and 9, we can see that:
- The valid position within this range is \( 6 \, \text{m} \).

### Final Answer
The position of one point where the block oscillates is:
\[ \boxed{6} \, \text{m} \]

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