Support Notes for Test I
Physics Practice – Learn by Doing
Learn Concepts Faster
Q1. Unit of Reduction Factor
Reduction factor is related to **resistance** in electrical circuits. Resistance is measured in Ohms (Ω). \[ R = \frac{V}{I}, \quad [R] = \Omega \]
Q2. Lumen as a Unit
Lumen (lm) is the SI unit of **luminous flux** – the total visible light emitted per second by a source. \[ 1\ \text{lumen} = 1\ \text{candela} \cdot \text{steradian}. \]
Detailed Notes
Lumen, Candela, Steradian — How They Relate
\[ \boxed{1\ \mathrm{lumen}\ (1\,\mathrm{lm}) \;=\; 1\ \mathrm{candela}\ (1\,\mathrm{cd}) \times 1\ \mathrm{steradian}\ (1\,\mathrm{sr})} \]
What each term means
- \(\mathrm{candela}\) (\(\mathrm{cd}\)): luminous intensity — brightness in a particular direction (human-eye weighted).
- \(\mathrm{steradian}\) (\(\mathrm{sr}\)): solid angle — the 3D “size” of a cone (beam spread).
- \(\mathrm{lumen}\) (\(\mathrm{lm}\)): luminous flux — total visible light passing through that solid angle.
Therefore, \[ \Phi \;=\; I\,\Omega, \] where \(\Phi\) is luminous flux (lm), \(I\) is intensity (cd), and \(\Omega\) is solid angle (sr).
So, candela × steradian = lumens:
If a source has intensity \(I = 1\,\mathrm{cd}\) in some direction, and you collect the light over a cone of solid angle \(\Omega = 1\,\mathrm{sr}\), then \[ \Phi \;=\; I\,\Omega \;=\; 1 \times 1 \;=\; 1\,\mathrm{lm}. \]
Two quick examples
- Isotropic point source (same intensity in all directions) with \(I = 1\,\mathrm{cd}\). The full sphere has solid angle \(4\pi\,\mathrm{sr}\). Hence \[ \Phi_{\text{total}} \;=\; I \times 4\pi \;=\; 4\pi\ \mathrm{lm} \;\approx\; 12.57\ \mathrm{lm}. \]
- Flashlight beam: Suppose a lamp has \(I = 50\,\mathrm{cd}\) within a narrow beam of \(\Omega = 0.2\,\mathrm{sr}\). Then \[ \Phi_{\text{beam}} \;=\; I\,\Omega \;=\; 50 \times 0.2 \;=\; 10\ \mathrm{lm}. \] (Widening the beam \(\Rightarrow\) larger \(\Omega\) \(\Rightarrow\) more lumens in that beam for the same cd.)
Related (often confused) unit
Illuminance (lux, \(\mathrm{lx}\)) is flux per area: \[ 1\ \mathrm{lx} \;=\; 1\ \frac{\mathrm{lm}}{\mathrm{m}^2}. \] It tells how much luminous flux lands on each square meter of a surface.
Q3. Young’s Modulus – Units
Young’s modulus is stress/strain. Stress has units of pressure: \[ Y = \frac{\text{Stress}}{\text{Strain}}, \quad [Y] = \frac{F/A}{\Delta L/L} = \frac{N}{m^2} = \text{Pa}. \] So valid units are N/m², dyne/cm², MPa — but not N/m.
Q4. Conversion of Young’s Modulus
Given: \[ Y = 18 \times 10^{11}\ \text{dyne cm}^{-2} \] Convert 1 dyne/cm²: \[ 1\ \frac{\text{dyne}}{\text{cm}^2} = \frac{10^{-5}\ \text{N}}{(10^{-2}\ \text{m})^2} = 0.1\ \text{Pa}. \] So: \[ Y = 18 \times 10^{11} \times 0.1 = 1.8 \times 10^{11}\ \text{Pa}. \]
Q5. Precision of Measurement
Precision = smallest count of the measuring instrument + number of decimal places. Example: \[ 5.00\ \text{mm} \quad (\text{smallest unit, most decimals}) \implies \text{highest precision}. \]
Q6. Significant Figures
Rules: - All non-zero digits are significant. - Zeros between non-zero digits are significant. - Leading zeros are not significant. - Trailing zeros (after decimal) are significant. Examples: \[ 23.023 \ (5 \text{ s.f.}), \quad 0.0003 \ (1 \text{ s.f.}), \quad 2.1 \times 10^{-3} \ (2 \text{ s.f.}). \]
Q7. Screw Gauge Measurement
Formula: \[ \text{Reading} = \text{MSR} + (\text{CSR} \times \text{LC}) \pm \text{Zero Error}. \] Example: \[ (0.5\ \text{cm} + 0.025\ \text{cm}) - (-0.004\ \text{cm}) = 0.521\ \text{cm}. \]
Q8. Percentage Error in Volume
Volume of sphere: \[ V = \frac{4}{3}\pi r^3. \] Error rule: \[ \frac{\Delta V}{V} = 3 \frac{\Delta r}{r}. \] So if error in radius = 2%, \[ \Delta V = 3 \times 2\% = 6\%. \]
Q9. Dimensional Formula of Energy
Work/Energy = Force × Distance: \[ E = F \cdot d = (MLT^{-2}) \cdot L = ML^2T^{-2}. \] So the dimensional formula of energy is: \[ [E] = M^1L^2T^{-2}. \]
Q10. Dimensional Formula of Power
Power = Energy / Time: \[ P = \frac{E}{T} = \frac{ML^2T^{-2}}{T} = ML^2T^{-3}. \]
Q11. Pressure and Its Dimensions
Pressure = Force / Area: \[ P = \frac{F}{A} = \frac{MLT^{-2}}{L^2} = ML^{-1}T^{-2}. \] Unit: Pascal (Pa).
Q12. Velocity and Acceleration
Velocity: \[ v = \frac{\Delta x}{\Delta t}, \quad [v] = LT^{-1}. \] Acceleration: \[ a = \frac{\Delta v}{\Delta t}, \quad [a] = LT^{-2}. \]
Q13. Hooke’s Law and Elasticity
Hooke’s Law: \[ F = k \Delta x. \] Here, \(k\) = spring constant: \[ [k] = \frac{F}{x} = \frac{MLT^{-2}}{L} = MT^{-2}. \] Related to Young’s modulus: \[ Y = \frac{\text{Stress}}{\text{Strain}}. \]