Dimensional Formula of Force

Newton's Second Law of Motion: F = m × a

Where: • m = mass (SI unit: kg) • a = acceleration (SI unit: m/s²)

Dimensions of each quantity: • [m] = [M¹L⁰T⁰] = [M] • [a] = [LT⁻²] (since acceleration = velocity/time = (distance/time)/time)

Substituting into F = ma: [F] = [M] × [LT⁻²] [F] = [M¹L¹T⁻²]

Dimensional Formula of Force = [M¹L¹T⁻²]

SI unit derivation: 1 Newton = 1 kg × 1 m/s² = kg·m·s⁻² So dimensional formula matches: [M¹][L¹][T⁻²]

SI System: • Unit: Newton (N) • 1 N = 1 kg·m·s⁻² • Named after Sir Isaac Newton

CGS System: • Unit: Dyne (dyn) • 1 dyne = 1 g·cm·s⁻² • 1 N = 10⁵ dyne

FPS System: • Unit: Pound-force (lbf) • 1 lbf ≈ 4.448 N

Key conversions: • 1 N = 10⁵ dyne • 1 kN = 1000 N • 1 MN = 10⁶ N

All types of force share the same dimensional formula [MLT⁻²]:

Gravitational Force (F = mg): [mg] = [M][LT⁻²] = [MLT⁻²] ✓

Spring Force (F = kx): [kx] = [MT⁻²][L] = [MLT⁻²] ✓ (Spring constant k has dimensions [MT⁻²])

Electrostatic Force (Coulomb's Law: F = kq²/r²): [kq²/r²] = [ML³T⁻⁴A⁻²][A²T²]/[L²] = [MLT⁻²] ✓

Frictional Force: Friction = μN = μmg → same dimensions [MLT⁻²] ✓

This dimensional consistency confirms that all forces, regardless of their origin, have the same dimensional formula.

Checking equations using dimensional formula of force:

Example 1: Verify F = ma LHS: [F] = [MLT⁻²] RHS: [ma] = [M][LT⁻²] = [MLT⁻²] ✓ (Dimensionally consistent)

Example 2: Verify Work = F × d [W] = [MLT⁻²][L] = [ML²T⁻²] (This gives dimensions of Work/Energy ✓)

Example 3: Verify Pressure = F/A [P] = [MLT⁻²]/[L²] = [ML⁻¹T⁻²] (Dimensions of Pressure ✓)

Example 4: Verify Power = F × v [P] = [MLT⁻²][LT⁻¹] = [ML²T⁻³] (Dimensions of Power ✓)

Dimensional analysis helps: • Verify correctness of physical equations • Derive relations between physical quantities • Convert units between systems

Key facts to remember:

1. Force is a vector quantity — it has both magnitude and direction, but its dimensional formula [MLT⁻²] only represents magnitude.

2. Net force causes acceleration (Newton's 2nd law): F_net = ma

3. Impulse = Force × Time: [Impulse] = [MLT⁻²][T] = [MLT⁻¹] (same as momentum ✓)

4. Force constant (spring constant k): [k] = [F/x] = [MLT⁻²/L] = [MT⁻²]

5. Tension, normal force, thrust, drag — all have dimensions [MLT⁻²]

6. In CGS, the unit dyne: 1 dyne = 10⁻⁵ N

7. Dimensional formula of weight is also [MLT⁻²] since weight = mg.