Dimensional Analysis

The same

Step 1: The principle of homogeneity is a fundamental rule in dimensional analysis. Step 2: It states that every equation relating physical quantities must be dimensionally homogeneous. Step 3: This means the dimensions of each term on the left-hand side must equal the dimensions of each term on the right-hand side. Step 4: This principle is used to check the correctness of equations and to derive new relations. If dimensions differ, the equation is definitely wrong.

[ML⁻¹T⁻²]

Step 1: Pressure = Force / Area. Step 2: Force = [MLT⁻²], Area = [L²]. Step 3: Pressure = [MLT⁻²] / [L²] = [ML⁻¹T⁻²]. Step 4: Option B is force, option C is work/energy, option D has wrong power of L. Note: Stress and coefficient of elasticity also have the same dimensional formula [ML⁻¹T⁻²].

10⁵ dynes

Step 1: Force dimensional formula = [M¹L¹T⁻²], so a=1, b=1, c=−2. Step 2: MKS: M₁=1 kg=10³ g, L₁=1 m=10² cm, T₁=T₂=1 s. Step 3: n₂ = 1 × (10³g/g)¹ × (10²cm/cm)¹ × (1)⁻². Step 4: n₂ = 1 × 10³ × 10² × 1 = 10⁵. Step 5: Therefore 1 Newton = 10⁵ dynes. Common error: Students forget to substitute the correct powers from the dimensional formula.

[ML²T⁻¹]

Step 1: From E = hν, we get h = E/ν. Step 2: Dimension of E (energy) = [ML²T⁻²]. Step 3: Dimension of ν (frequency) = [T⁻¹]. Step 4: h = [ML²T⁻²] / [T⁻¹] = [ML²T⁻²] × [T] = [ML²T⁻¹]. Step 5: Option B is energy, option C is power, option D is momentum. Planck's constant is a dimensional constant.