When is an equation considered dimensionally homogeneous?

When is an equation considered dimensionally homogeneous?

• A. When It Is Unitless
• B. When The Dimensions Of The Left Side Do Not Match The Right Side
• C. When The Degree On The Left Side Equals The Degree On The Right Side
• D. When The Dimensions On The Left Side Equal The Dimensions On The Right Side

Answer: (D)

The main idea being tested is that an equation is dimensionally homogeneous when both sides share the same dimensional form. In dimensional analysis you assign base dimensions such as M for mass, L for length, and T for time. For an equation to be physically meaningful, every term on each side must combine to the same overall dimensions. That means the left-hand side and the right-hand side must have identical dimensions. This is why the correct criterion is that the dimensions on the left equal the dimensions on the right. It ensures you’re not mixing incompatible quantities, like trying to equate a velocity (dimensions L T^-1) with a length (dimensions L) or a force (dimensions M L T^-2) with something that doesn’t carry the same M, L, and T powers. A concrete example is Newton’s second law, F = m a. Force has dimensions M L T^-2, and mass times acceleration has dimensions M × (L T^-2) = M L T^-2, so the two sides match and the equation is dimensionally homogeneous. This criterion does not require the equation to be unitless, nor does it rely on the idea of “degrees” on each side. Those concepts can be misleading; what matters is that the overall dimensions balance on both sides.

The main idea being tested is that an equation is dimensionally homogeneous when both sides share the same dimensional form. In dimensional analysis you assign base dimensions such as M for mass, L for length, and T for time. For an equation to be physically meaningful, every term on each side must combine to the same overall dimensions. That means the left-hand side and the right-hand side must have identical dimensions.

This is why the correct criterion is that the dimensions on the left equal the dimensions on the right. It ensures you’re not mixing incompatible quantities, like trying to equate a velocity (dimensions L T^-1) with a length (dimensions L) or a force (dimensions M L T^-2) with something that doesn’t carry the same M, L, and T powers.

A concrete example is Newton’s second law, F = m a. Force has dimensions M L T^-2, and mass times acceleration has dimensions M × (L T^-2) = M L T^-2, so the two sides match and the equation is dimensionally homogeneous.

This criterion does not require the equation to be unitless, nor does it rely on the idea of “degrees” on each side. Those concepts can be misleading; what matters is that the overall dimensions balance on both sides.