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{:(,"Column I",,"Column II"),((A),"Dimen...

`{:(,"Column I",,"Column II"),((A),"Dimensional variable",(p),pi),((B),"Dimensionless variable",(q),"Force"),((C),"Dimensional constant",(r),"Angle"),((D),"Dimensionless constant",(s),"Gravitational constant"):}`

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To solve the problem of matching the items in Column I with those in Column II, we will analyze each item based on its dimensional properties. ### Step-by-Step Solution: 1. **Identify the nature of each item in Column I:** - A: Dimensional variable - B: Dimensionless variable - C: Dimensional constant - D: Dimensionless constant 2. **Analyze the items in Column II:** - p: π (pi) - q: Force - r: Angle - s: Gravitational constant 3. **Match each item:** - **For p (π):** - The dimensional formula for π is \( m^0 l^0 t^0 \), which means it has no dimensions. Therefore, π is a dimensionless constant. - **Match:** D (Dimensionless constant) with p (π). - **For q (Force):** - The dimensional formula for force is \( m^1 l^1 t^{-2} \). This shows that force has dimensions. - **Match:** A (Dimensional variable) with q (Force). - **For r (Angle):** - The dimensional formula for angle is also \( m^0 l^0 t^0 \), indicating that angle is dimensionless. - **Match:** B (Dimensionless variable) with r (Angle). - **For s (Gravitational constant):** - The dimensional formula for the gravitational constant is \( m^1 l^3 t^{-2} \). This indicates that it has dimensions and is a constant. - **Match:** C (Dimensional constant) with s (Gravitational constant). 4. **Final Matches:** - A → q (Force) - B → r (Angle) - C → s (Gravitational constant) - D → p (π) ### Summary of Matches: - A (Dimensional variable) → q (Force) - B (Dimensionless variable) → r (Angle) - C (Dimensional constant) → s (Gravitational constant) - D (Dimensionless constant) → p (π)
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