For a particle executing SHM, if x, v, a and F represent dispacement, velocity, acceleration and restoring force, then
(A) `vec(F).vec(x)` is always negative
(B) `vec(v).vec(x)` is always negative
(C) `vec(v) xx vec(a)` is always zero
(D) `vec(a) xx vec(v)` is always positive

Options

A. all A, B, C, D are correct
B. only A, C are correct
C. only B, C are correct
D. only C, D are correct

Answer

To solve the problem, we need to analyze each of the given statements about a particle executing Simple Harmonic Motion (SHM).

1. **Understanding SHM**: In SHM, the displacement \( x \), velocity \( v \), acceleration \( a \), and restoring force \( F \) are related as follows:
- The restoring force \( F \) is given by \( F = -kx \), where \( k \) is the spring constant.
- The acceleration \( a \) is given by \( a = \frac{d^2x}{dt^2} = -\omega^2 x \), where \( \omega \) is the angular frequency.
- The velocity \( v \) is the derivative of displacement, \( v = \frac{dx}{dt} \).

2. **Analyzing Each Option**:

**Option (A)**: \( \vec{F} \cdot \vec{x} \) is always negative.
- Since \( \vec{F} = -k\vec{x} \), the dot product \( \vec{F} \cdot \vec{x} = (-k\vec{x}) \cdot \vec{x} = -k|\vec{x}|^2 \).
- This is always negative (since \( k > 0 \) and \( |\vec{x}|^2 \) is always positive).
- **Conclusion**: This statement is **true**.

**Option (B)**: \( \vec{v} \cdot \vec{x} \) is always negative.
- The velocity \( \vec{v} \) can be positive or negative depending on the direction of motion. When the particle is moving towards the mean position, \( \vec{v} \) and \( \vec{x} \) can have the same direction, making the dot product positive.
- **Conclusion**: This statement is **false**.

**Option (C)**: \( \vec{v} \times \vec{a} \) is always zero.
- The velocity \( \vec{v} \) and acceleration \( \vec{a} \) are always perpendicular in SHM. Therefore, the cross product \( \vec{v} \times \vec{a} \) is not always zero; it is zero only when they are parallel, which is not the case in SHM.
- **Conclusion**: This statement is **false**.

**Option (D)**: \( \vec{a} \times \vec{v} \) is always positive.
- The cross product \( \vec{a} \times \vec{v} \) is not defined in terms of being positive or negative. It depends on the direction of both vectors. Since \( \vec{a} \) is always directed towards the mean position and \( \vec{v} \) can be in either direction, this statement is ambiguous.
- **Conclusion**: This statement is **false**.

3. **Final Answer**: The only correct statement is (A): \( \vec{F} \cdot \vec{x} \) is always negative.