Which of the following expressions corresponds to simple harmonic motion along a straight line, where x is the displacement and a, b, c are positive constants ?

Options

A. `a+bx-cx^(2)`
B. `bx^(2)`
C. `a-bx+cx^(2)`
D. `-bx`

Explanation

To determine which of the given expressions corresponds to simple harmonic motion (SHM) along a straight line, we need to analyze the characteristics of SHM.

### Step-by-Step Solution:

1. **Understanding Simple Harmonic Motion (SHM)**:
- SHM is defined as the motion of an object where the restoring force is directly proportional to the displacement from the mean position and acts in the opposite direction. Mathematically, this is expressed as:
\[ F = -kx \]
where \( F \) is the restoring force, \( k \) is a positive constant, and \( x \) is the displacement from the mean position.

2. **Analyzing the Given Expressions**:
- We have the following expressions to consider:
1. \( A + Bx - Cx^2 \)
2. \( Bx^2 + A - Cx \)
3. \( -Bx \)
- We need to identify which of these expressions can be represented in the form of a restoring force proportional to \( -kx \).

3. **Evaluating Each Expression**:
- **Expression 1**: \( A + Bx - Cx^2 \)
- This expression is not linear due to the \( Cx^2 \) term. The presence of the \( x^2 \) term indicates that it does not represent a linear restoring force.
- **Expression 2**: \( Bx^2 + A - Cx \)
- Similar to the first expression, this contains a \( Bx^2 \) term, which also indicates non-linearity.
- **Expression 3**: \( -Bx \)
- This expression is linear and can be rewritten as \( F = -k'x \) where \( k' = B \). This matches the form of the restoring force for SHM.

4. **Conclusion**:
- The expression that corresponds to simple harmonic motion along a straight line is:
\[ -Bx \]

### Final Answer:
The expression that corresponds to simple harmonic motion along a straight line is \( -Bx \).

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