The dimensional formula of Energy is [M¹L²T⁻²]. This is derived from the work-energy theorem: Energy = Work = Force × displacement = F × d. Since [F] = [MLT⁻²] and [d] = [L], energy dimensions become [ML²T⁻²]. The SI unit of energy is Joule (J), where 1 J = 1 N·m = 1 kg·m²·s⁻².
Dimensional formula of Energy = [M¹L²T⁻²].
Derived from W = F × d: [W] = [MLT⁻²][L] = [ML²T⁻²].
SI unit of Energy is Joule (J); 1 J = 1 kg·m²·s⁻².
All forms of energy (kinetic, potential, thermal, electrical) have the same dimensional formula [ML²T⁻²].
1 J = 10⁷ erg (CGS unit); 1 kWh = 3.6 × 10⁶ J.
Torque also has dimensional formula [ML²T⁻²] but is not energy — it is a vector.
Einstein's E = mc² is dimensionally consistent: [M][LT⁻¹]² = [ML²T⁻²].
Energy is the capacity to do work, and Work = Force × Displacement:
W = F × d
Dimensions of each quantity: • [F] = [M¹L¹T⁻²] (force) • [d] = [L¹] (displacement)
Substituting: [W] = [M¹L¹T⁻²] × [L¹] [W] = [M¹L¹⁺¹T⁻²] [W] = [M¹L²T⁻²]
Dimensional Formula of Energy = [M¹L²T⁻²]
SI unit verification: 1 Joule = 1 N × 1 m = (1 kg·m·s⁻²) × (1 m) = 1 kg·m²·s⁻² This confirms [M¹L²T⁻²] ✓
Alternative derivation from Kinetic Energy: KE = ½mv² [KE] = [M][LT⁻¹]² = [M][L²T⁻²] = [ML²T⁻²] ✓
All forms of energy share the same dimensional formula [ML²T⁻²]:
Kinetic Energy (KE = ½mv²): [KE] = [M][LT⁻¹]² = [ML²T⁻²] ✓
Potential Energy (PE = mgh): [PE] = [M][LT⁻²][L] = [ML²T⁻²] ✓
Elastic Potential Energy (PE = ½kx²): [½kx²] = [MT⁻²][L²] = [ML²T⁻²] ✓
Electrical Energy (E = QV = CV²): [QV] = [AT][ML²T⁻³A⁻¹] = [ML²T⁻²] ✓
Thermal Energy (Q = mcΔT): [mcΔT] = [M][L²T⁻²K⁻¹][K] = [ML²T⁻²] ✓
This dimensional consistency reflects the Law of Conservation of Energy — energy can change form but its dimensional identity remains the same.
SI System: • Unit: Joule (J) • 1 J = 1 N·m = 1 kg·m²·s⁻² • Named after James Prescott Joule
CGS System: • Unit: Erg • 1 erg = 1 g·cm²·s⁻² • 1 J = 10⁷ erg
Other units of energy: • Calorie (cal): 1 cal = 4.186 J (heat energy) • Kilocalorie (kcal): 1 kcal = 4186 J (food energy) • Kilowatt-hour (kWh): 1 kWh = 3.6 × 10⁶ J (electrical energy) • Electronvolt (eV): 1 eV = 1.6 × 10⁻¹⁹ J (atomic/nuclear physics) • BTU (British Thermal Unit): 1 BTU ≈ 1055 J
Large energy units: • 1 kJ = 10³ J • 1 MJ = 10⁶ J • 1 GJ = 10⁹ J
Energy and Work: • Both have dimensional formula [ML²T⁻²] • Both have SI unit Joule (J) • Energy is capacity to do work; work is energy transferred
Power (P = Energy/Time = Work/Time): [P] = [ML²T⁻²] / [T] = [ML²T⁻³] • SI unit: Watt (W) = J/s
Torque (τ = r × F): [τ] = [L][MLT⁻²] = [ML²T⁻²] • Same dimensional formula as energy! • But torque is a vector, energy is a scalar — they are not the same quantity
Angular Momentum (L = Iω = mvr): [L] = [ML²][T⁻¹] = [ML²T⁻¹] • Different from energy
Planck's Constant h: [h] = [Energy]/[frequency] = [ML²T⁻²]/[T⁻¹] = [ML²T⁻¹]
Example 1: Calculate KE of 2 kg object moving at 5 m/s: KE = ½mv² = ½ × 2 × 25 = 25 J Dimensions: [M][LT⁻¹]² = [ML²T⁻²] = Joule ✓
Example 2: Verify Einstein's mass-energy relation E = mc²: [E] = [M][LT⁻¹]² = [ML²T⁻²] ✓ (same as energy) c = speed of light = 3 × 10⁸ m/s E = mc² → 1 kg converts to = 1 × (3×10⁸)² = 9×10¹⁶ J
Example 3: Verify energy stored in capacitor E = ½CV²: [C] = [M⁻¹L⁻²T⁴A²] (capacitance) [V²] = [M²L⁴T⁻⁶A⁻²] (voltage squared) [CV²] = [M⁻¹L⁻²T⁴A²][M²L⁴T⁻⁶A⁻²] = [ML²T⁻²] ✓
Example 4: Gravitational PE at height h: PE = mgh = 5 × 10 × 20 = 1000 J = 1 kJ [mgh] = [M][LT⁻²][L] = [ML²T⁻²] ✓
[M¹L²T⁻²]. Derived from W = F × d: [W] = [MLT⁻²] × [L] = [ML²T⁻²].
Joule (J). 1 J = 1 N·m = 1 kg·m²·s⁻².
Yes. KE = ½mv² gives [M][LT⁻¹]² = [ML²T⁻²] and PE = mgh gives [M][LT⁻²][L] = [ML²T⁻²]. Both are the same.
Erg (erg). 1 J = 10⁷ erg. 1 erg = 1 g·cm²·s⁻².
Yes, torque [ML²T⁻²] has the same dimensions as energy, but torque is a vector quantity while energy is scalar. They are physically different quantities.
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