1. Introduction to Dimensions
Dimensions are used to verify the correctness of equations. They describe the nature and properties of physical quantities. Dimensions help identify the units (e.g., SI units) in which quantities are measured.
2. How to Write Dimensions
Dimensions are represented within square brackets using three fundamental physical quantities:
- Mass (M)
- Length (L)
- Time (T)
Most formulas use these three quantities; others are derived from them.
3. Example: Dimensions of Velocity
Formula: \( v = \frac{\text{Displacement}}{\text{Time}} \)
- SI Units:
- Displacement → meters (m)
- Time → seconds (s)
- Derived Dimension: \( v = [M^0L^1T^{-1}] \)
Explanation:
- \( M^0 \): Mass not involved
- \( L^1 \): Length involved with power 1
- \( T^{-1} \): Time inverse
4. Verifying the Correctness of Equations
Example: Second Equation of Motion
Equation: \( s = vt + \frac{1}{2}at^2 \)
- Each term on both sides has consistent dimensions.
- The equation is dimensionally correct.
5. Practical Use of Dimensional Analysis
- Ensures formulas are correct.
- Helps derive units of unknown quantities.
- Simplifies the verification of physical laws and equations.
6. Conclusion
Dimensional analysis proves correctness and identifies relationships between quantities. Example: Verified the second equation of motion using dimensions.
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