Support Notes for Test
Physics Practice – Learn by Doing
Learn Concepts Faster
Q1. Error in Calculation
Rule: When multiplying/dividing quantities with powers, the relative error adds with the power as a multiplier: \[ \frac{\Delta P}{P} = n\frac{\Delta a}{a} + m\frac{\Delta b}{b} + \cdots \] Example: \[ P = \frac{a^3 b^2}{cd}, \quad \frac{\Delta P}{P} = 3\% + 2(2\%) + 3\% + 4\% = 14\%. \]
Q2. Fundamental Units
Force is a fundamental unit:
Q3. Units vs Dimensions
- Displacement: has both unit & dimension. - Speed: has both. - Angle: has units (radians) but is dimensionless: \[ \theta = \frac{\text{arc length}}{\text{radius}}, \quad \frac{L}{L} = 1. \]
Q4. Dimensions of Pascal-second
Pascal = Pressure = \(\frac{F}{A} = ML^{-1}T^{-2}\). Multiply by time → \[ \text{Pa·s} = ML^{-1}T^{-1}, \] which is the dimension of viscosity.
Q5. Different Dimensional Quantity
- Spring constant: \(F/x \sim MT^{-2}\). - Surface tension: Force/length \(\sim MT^{-2}\). - Surface energy: Energy/area \(\sim MT^{-2}\). - Acceleration due to gravity: \(LT^{-2}\). ⇒ Different dimensions → \(g\).
Q6. Pressure Gradient
Pressure gradient = \(\frac{\Delta P}{\Delta x}\). Dimensions: \[ [P] = ML^{-1}T^{-2}, \quad \frac{P}{L} = ML^{-2}T^{-2}. \] Not the same as velocity, potential, or energy gradients.
Q7. Dimensions of Resistance
Ohm’s law: \(V = IR\). Resistance: \[ [R] = \frac{[V]}{[I]}. \] Volt = Joule/Coulomb: \[ [V] = \frac{ML^2T^{-2}}{AT}. \] So: \[ [R] = ML^2T^{-3}A^{-2}. \]
Q8. Dimensions of \((\mu₀ε₀)^{-\tfrac{1}{2}}\)
From Maxwell’s equations: \[ c = \frac{1}{\sqrt{\mu_0 \epsilon_0}}. \] Dimensions of speed: \[ [c] = LT^{-1}. \]
Q9. Ratio of L/R and R·C
Inductance/resistance: \[ \frac{L}{R} = \frac{\text{Henry}}{\Omega} = \text{seconds}. \] Resistance × capacitance: \[ RC = \Omega \cdot \text{Farad} = \text{seconds}. \] ⇒ Both represent time constants.
Q10. Planck’s Constant / Moment of Inertia
Planck’s constant: \[ [h] = ML^2T^{-1}. \] Moment of inertia: \[ [I] = ML^2. \] Ratio: \[ \frac{h}{I} = T^{-1}, \] which is the dimension of frequency.