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A particle performs SHM along a straight...

A particle performs SHM along a straight line and its position is `vec(R )`, acceleration is `vec(a)`, velocity is `vec(v)` is and force on particle is `vec(f)`. Then which of the following statement are true?
(i) `vec(v). vec(a)` is always + ve
(ii) `vec(v).vec(R )` may be -ve
(iii) `vec(f).vec(R )` is always -ve
(iv) `vec(v)` is parallel to `vec(f)` sometimes

A

ii, iii & iv

B

i, ii & iv

C

i, iii & iv

D

ii & iii

Text Solution

AI Generated Solution

The correct Answer is:
To solve the problem, we need to analyze the four statements given about a particle performing Simple Harmonic Motion (SHM) and determine which ones are true. ### Step-by-step Solution: 1. **Understanding SHM**: In SHM, the position vector \( \vec{R} \), velocity vector \( \vec{v} \), acceleration vector \( \vec{a} \), and force vector \( \vec{f} \) are related. The acceleration is directly proportional to the negative of the displacement, which can be expressed as: \[ \vec{a} = -\omega^2 \vec{R} \] where \( \omega \) is the angular frequency. 2. **Analyzing Statement (i)**: The statement claims that \( \vec{v} \cdot \vec{a} \) is always positive. - In SHM, when the particle moves away from the mean position, \( \vec{v} \) and \( \vec{a} \) are in opposite directions (as \( \vec{a} \) is directed towards the mean position). Thus, \( \vec{v} \cdot \vec{a} < 0 \) when the particle is moving away from the mean position. - Therefore, this statement is **false**. 3. **Analyzing Statement (ii)**: The statement claims that \( \vec{v} \cdot \vec{R} \) may be negative. - When the particle is moving towards the mean position (i.e., returning from the extreme position), \( \vec{v} \) is directed towards the mean position while \( \vec{R} \) is directed away from it. This results in \( \vec{v} \cdot \vec{R} < 0 \). - Therefore, this statement is **true**. 4. **Analyzing Statement (iii)**: The statement claims that \( \vec{f} \cdot \vec{R} \) is always negative. - Since \( \vec{f} = -k \vec{R} \) (where \( k \) is a positive constant), the force vector is always directed opposite to the position vector. Thus, \( \vec{f} \cdot \vec{R} < 0 \). - Therefore, this statement is **true**. 5. **Analyzing Statement (iv)**: The statement claims that \( \vec{v} \) is parallel to \( \vec{f} \) sometimes. - When the particle is at the mean position, both \( \vec{v} \) and \( \vec{f} \) are directed towards the mean position (in the same direction). Hence, they can be parallel at this point. - Therefore, this statement is **true**. ### Conclusion: The true statements are: - (ii) \( \vec{v} \cdot \vec{R} \) may be negative. - (iii) \( \vec{f} \cdot \vec{R} \) is always negative. - (iv) \( \vec{v} \) is parallel to \( \vec{f} \) sometimes. Thus, the correct answer is that statements (ii), (iii), and (iv) are true.
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