Dimensional Analysis
This simulation demonstrates dimensional analysis of the kinetic/potential energy equation, based on Example 2.15 from your Physics textbook.
½ m v² = m g h
Dimensions of m: [M]
Dimensions of v: [L T⁻¹]
Dimensions of v²: [L T⁻¹]² = [L² T⁻²]
Dimensions of m v²: [M] × [L² T⁻²] = [M L² T⁻²]
The constant ½ is dimensionless
Final dimensions of LHS: [M L² T⁻²]
Dimensions of m: [M]
Dimensions of g: [L T⁻²]
Dimensions of h: [L]
Dimensions of g h: [L T⁻²] × [L] = [L² T⁻²]
Dimensions of m g h: [M] × [L² T⁻²] = [M L² T⁻²]
Final dimensions of RHS: [M L² T⁻²]
Therefore, the equation is dimensionally correct.
Physical Interpretation
This equation represents the conversion between kinetic energy (½mv²) and potential energy (mgh).
The dimensional consistency confirms that both sides of the equation represent the same physical quantity - energy.
The fundamental dimensions of energy are:
- [M] - Mass (energy has inertia)
- [L²] - Length squared (work is force × distance)
- [T⁻²] - Inverse time squared (related to acceleration)
Example 2.15 Solution:
LHS dimensions: [M][L T⁻¹]² = [M L² T⁻²]
RHS dimensions: [M][L T⁻²][L] = [M L² T⁻²]
Both sides have identical dimensions, confirming the equation is dimensionally correct.
