Units & Dimensions Handwritten Notes PDF

# Units & Dimensions ## Handwritten Notes ### Unit and Dimension 1. **Units**: Standard of representation. * **CGS System** * Mass (M): gm = 1 kg * Length (L): cm = 1 m * Time (T): sec * **MKS System** * Mass (M): kg = 10³ gm...

# Units & Dimensions ## Handwritten Notes ### Unit and Dimension 1. **Units**: Standard of representation. * **CGS System** * Mass (M): gm = 1 kg * Length (L): cm = 1 m * Time (T): sec * **MKS System** * Mass (M): kg = 10³ gm * Length (L): m = 10² cm * Time (T): sec 2. **Physical quantity** * PQ = numerical value * unit * B = ny = const * On changing u, n changes but Q = const. 3. **Dimension**: Power of M, L and T needed to represent any physical quantity. * Velocity: \[v] = \[MLT^-1] * Acceleration: \[a] = \[MLT^-2] * Force: \[F] = \[MLT^-2] * Pressure/Stress: \[p] = \[ML^-1T^-2] * Work/Energy/Torque: \[W] = \[ML²T^-2] * Power: \[P] = \[ML²T^-3] * Angular momentum: \[L] = \[ML²T^-1] * Impulse: \[J] = \[MLT^-1] * Plank's constant: \[h] = \[ML²T^-1] * Gravitational constant: \[G] = \[ML³T^-2] * **NOTE**: Dimensions of any physical quantity does not depends on formula to be used for dimension. 4. **Rule 1 for Dimension**: * Pressure - x =   +   P     → must be a kind of pressure 5. **Rule 2 for Dimension**: * Dimension of = Dimension of LH8 RHS * **Principle of Homogeneity** * A given equation can be dimensionally correct, but not mathematically correct. * A given is mathematically correct, it will be dimensionally correct too, with some exceptions. 6. θ → Dimensionless, but has unit (radian). * sin θ → Dimensionless. * cos θ * etc. * Log(...) → Dimensionless * e(...) 7. Q = ny = constant * D¹/₄/₁ = D²/₄/₂ 8. **Absolute error**: True value - Measured value * In case true value is not given, than mean value is used as true value. * Arithmetic mean of all values. a_m = a₁ + a₂ + a₃ +.... n 9. **Mean Absolute error**: Δa_m = Δa₁ + Δa₂ + Δa₃ +.... n 10. **Relative error**: Δa/a or Δa_m/a_m 11. **Percentage error**: Relative error * 100. 12. **Propagation of error in addition/subtraction and multiplication/Division** * X = x + x * Z = X + Y * Z = X·Y   absolute error in x or * Z = X - Y   Z =   X   Y * and Y =   y + y   absolute evor in y * ΔZ = ± (ΔX + ΔY)   ΔZ = ± (ΔX + ΔY)   ΔZ = ± $(\frac{ΔX}{X} + \frac{ΔY}{Y})$ 13. If any PQ is represented as * x = a^p b^q c^r d^s/_e^gf^b: Δx/x = ± (Δp/p + &916;q/q + Δr/r + Δs/s + Δg/g + Δh/h) 14. **Vernier Caliper**: Total Reading = MSR + VSR - Zero error * **Main scale reading:** * Mark on main scale just before zero of Vernier scale. * **Vernier scale reading:** * Coinciding vs division with any MS mark. * (Least count) LC = IMSD - IVSD LC = x (1 - IMSD IVSD) * **no zero error:** * MS:  * VS: * O(MS): * O VS): * **+ve zero error:** * O(MS): * O(VS): * tve * MS:→ zero * The zero error is the difference between the actual zero and the zero mark. * error = C.V.S.D x L.C.   O(MS):   O(VS):   - ve   MS:   zero = (total USD - CVSD) x L.C.   error   O(VS) 15. **Screw Gage**: * MSR = 5.5 * VSR = 0 X .5 → P * SD: No. of CSD * O(MS) * "No zero error" * when zero of main scale wincide with "line" and zero of cs coincides with reference line of ms simultaneously, no zero error. * **Circular scale reading:** * Coinciding circular scale division with reference line of MS * **least conut:** * LC = pitch/no of CSD * pitch = distance moved by thimble on main scale in one complete rotation. * Generally → P = IMSD * **+ve zero error : →** * **tve zero error:** * OMS * "MS reference line" * O(CS) * = CC&D * LC * = (2) * 1mm → IMSD * 50 * **-ve zero error:** * O(MS) * O(CS) * O of MS is visible. * -ve error: * = (total no of CSD - CCSD) * LC * = (50 - 47) x 1mm/50 = Pitch → IMSD * **+ ve zero error:** * O of MS is not visible. ## Thank you 🙂