Class 11 Physics ⚡

Options: A. Velocity   B. Acceleration   C. Force   D. Work Answer: C. Force Reason: Force = mass × acceleration = kg × m/s² = [L¹M¹T⁻²]. Velocity=[LT⁻¹], Acceleration=[LT⁻²], Work=[L²MT⁻²].

Options: A. 1%   B. ½%   C. 2%   D. None Answer: C. 2% Reason: Area = L × B. Error in product → ΔA/A = ΔL/L + ΔB/B = 1% + 1% = 2%.

Options: A. Time   B. Mass   C. Distance   D. Luminosity Answer: C. Distance 1 light year = 9.467 × 10¹⁵ m = distance travelled by light in one year.

Options: A. Force   B. Acceleration   C. Work   D. Pressure Answer: C. Work KE = ½mv² = [L²M¹T⁻²]. Work = F.d = [L²M¹T⁻²]. Both same.

Options: A. cm   B. kg   C. centigrade   D. volt Answer: D. Volt Volt is a derived unit (V = kg m² s⁻³ A⁻¹). cm and kg are units of fundamental quantities. Centigrade (°C) is a unit of temperature (fundamental).

Unit: The standard measure of any quantity is called the unit of that quantity. A measured quantity is expressed as a number followed by corresponding unit. Example: length of wire = 5 m. Need: Measurement always involves comparison with a standard. Without a universally accepted standard, measurements made in different places or by different people would be meaningless and cannot be compared. Hence, internationally accepted standard units are essential.

Fundamental quantities: Physical quantities which do NOT depend on any other physical quantities. There are 7 fundamental quantities. Examples: length (m), mass (kg), time (s), temperature (K). Derived quantities: Physical quantities which depend on fundamental quantities and are expressed in terms of them. Examples: velocity (m s⁻¹), force (kg m s⁻²), pressure (kg m⁻¹ s⁻²).

Besides the 7 fundamental units, there are 2 more units called supplementary units: (i) Plane angle (dθ): Ratio of arc length to radius of circle. dθ = ds/r. Measured in radian (rad). (ii) Solid angle (dΩ): Ratio of area of surface of sphere to square of its radius. dΩ = dA/r². Measured in steradian (sr). A full sphere subtends 4π sr at its centre.

Parallax: The apparent change in position of an object due to a change in the position of the observer is called parallax. Formula: If b = baseline (distance between two observation points) and θ = parallax angle in radian, then: D = b/θ where D = distance of the object. For planet size: d = α × D, where α = angular diameter.

Accuracy: How close a measurement is to the actual (true) value of that quantity. A measurement is accurate if the measured value matches the true value. Precision: Multiple measurements of the same quantity give nearly identical values (i.e., reproducible results). Precision means consistency. Note: A measurement can be precise but not accurate (if there is systematic error). The goal is to achieve both accuracy and precision.

Systematic errors: Errors that are NOT determined by chance but are introduced by inaccuracy inherent to the system. They tend to be in one direction (either + or −). Sources: (i) Instrumental error: Due to defective calibration, incorrect zero reading. Example: ammeter showing 0.5A even when not connected. (ii) Imperfect experimental technique: Defective setting of instrument. Example: graduated tube not held vertical gives wrong volume. (iii) Personal error: Due to observer's bias or carelessness. Example: parallax error while reading a ruler.

Significant figures: The number of digits in a measurement about which we are certain, plus one additional digit about which we are not certain. Rules: 1. All non-zero digits are significant. (178.43 → 5 s.f.) 2. Zeros between non-zero digits are significant. (165.02 → 5 s.f.) 3. Zeros to left of first non-zero digit are NOT significant. (0.0014 → 2 s.f.) 4. Trailing zeros with decimal point are significant. (1.500 → 4 s.f.)

Dimensional formula: When a derived quantity is represented with appropriate powers of symbols of fundamental quantities, such an expression is called dimensional formula. It is expressed in square brackets. Examples: Velocity = displacement/time → [L¹M⁰T⁻¹] Force = mass × acceleration → [L¹M¹T⁻²] Work = force × displacement → [L²M¹T⁻²]

Absolute error: The magnitude of the difference between mean value and each individual observed value. Δaᵢ = |a_mean − aᵢ| Mean absolute error: Arithmetic mean of all the absolute errors. Δa_mean = (Δa₁+Δa₂+...+Δaₙ)/n Relative error: Ratio of mean absolute error to arithmetic mean value. Relative error = Δa_mean / a_mean Percentage error: = (Δa_mean / a_mean) × 100%

Principle of Homogeneity: In any correct physical equation, the dimensions of all the terms on both sides of the equation must be the same (equal). Application: Used to check the correctness of physical equations. Example: v = u + at LHS: [v] = [LT⁻¹] RHS: [u] = [LT⁻¹], [at] = [LT⁻²][T] = [LT⁻¹] LHS = RHS → equation is dimensionally correct ✓

(i) To check correctness of physical equations: Using Principle of Homogeneity — dimensions on both sides of equation must be equal. If equal → dimensionally correct. Example: v = u + at → Both sides have dimension [LT⁻¹] ✓ (ii) To establish relationship between physical quantities: If T of pendulum depends on l and g: T = k lᵃ gᵇ Equating dimensions: [T¹] = [L]ᵃ [LT⁻²]ᵇ → a=½, b=−½ ∴ T = k √(l/g) → T = 2π√(l/g) (k found experimentally) (iii) To find conversion factor between units: 1 J = x erg. Using [L²M¹T⁻²]: x = (m/cm)²(kg/g)(s/s)² = (100)²(1000)(1) = 10⁷ ∴ 1 Joule = 10⁷ erg

1. The value of dimensionless constants cannot be found using dimensional analysis. They must be determined experimentally. Example: k = 2π in T = 2π√(l/g). 2. Dimensional analysis cannot be used to derive relations involving trigonometric, exponential and logarithmic functions as these are dimensionless. 3. This method is not useful if the constant of proportionality is NOT dimensionless. Example: Gravitational constant G in F = Gm₁m₂/r² has dimensions. 4. If the correct equation contains more terms of the same dimension, this method cannot reveal them. Example: S = ½at² is dimensionally correct but complete equation is S = ut + ½at².

Random errors: These are the errors which are introduced even after following all procedures to minimize systematic errors. Nature: These may be positive or negative (both directions). They occur due to variation in conditions during the experiment. Causes: Temperature change during experiment, pressure change of gas used, random fluctuation in voltage of power supply, etc. Minimization: Random errors cannot be eliminated completely but can be minimized by: 1. Taking repeated observations of the same quantity. 2. Calculating the arithmetic mean (average) of all readings. 3. The mean value is taken as the most probable value of the quantity.

Given: Time period T depends on length l and acceleration due to gravity g. Step 1: T ∝ lᵃ gᵇ → T = k lᵃ gᵇ (k = dimensionless constant) Step 2: Write dimensions: [L⁰M⁰T¹] = k [L]ᵃ [LT⁻²]ᵇ = k [L^(a+b) T⁻²ᵇ] Step 3: Compare powers: L: a + b = 0 → a = −b T: −2b = 1 → b = −1/2 → a = +1/2 Step 4: T = k l^(1/2) g^(−1/2) = k √(l/g) Step 5: k is found experimentally = 2π ∴ T = 2π √(l/g)

(i) Error in sum and difference (Z = A ± B): If A ± ΔA and B ± ΔB are measured values: Z ± ΔZ = (A ± ΔA) ± (B ± ΔB) Maximum absolute error: ΔZ = ΔA + ΔB Rule: When two quantities are added or subtracted, the maximum absolute error in result = sum of absolute errors in individual quantities. (ii) Error in product and division (Z = AB or A/B): Z ± ΔZ = (A ± ΔA)(B ± ΔB) = AB ± AΔB ± BΔA ± ΔAΔB Neglecting ΔAΔB (very small): ΔZ/Z = ΔA/A + ΔB/B Maximum relative error: ΔZ/Z = ΔA/A + ΔB/B Rule: When two quantities are multiplied or divided, the maximum relative error in result = sum of relative errors in each quantity.

The SI unit of time is second. Earlier, 1 second was defined as 1/86400 of a mean Solar day (24 × 60 × 60 = 86400 s). Problem: The length of a Solar day varies gradually due to the gradual slowing down of Earth's rotation. Hence this standard is not constant. Cesium Atomic Clock: To get a more standard and non-varying (constant) unit for time, a cesium atomic clock is used. It is based on periodic vibrations produced in a cesium atom. Definition: In cesium atomic clock, 1 second = time needed for 9,192,631,770 vibrations of radiation (wave) emitted during a transition between two hyperfine states of Cs¹³³ atom. Advantage: It gives an extremely accurate and constant (non-varying) measurement of time.

Step 1: Total mass = 15.7 + 27.3 = 43.0 kg Step 2: Error in sum = ΔA + ΔB = 0.2 + 0.3 = 0.5 kg Total mass = (43.0 ± 0.5) kg

Step 1: Speed = distance/time = 5.2/100 = 0.052 m s⁻¹ Step 2: Relative error = Δd/d + Δt/t = 0.1/5.2 + 1/100         = 0.01923 + 0.01 = 0.02923 Step 3: Absolute error in speed = 0.02923 × 0.052 = 0.00152 ≈ 0.002 m s⁻¹ Speed = 0.052 m s⁻¹, Relative error = ±0.029

Step 1: Mean = (5.63+5.54+5.44+5.40+5.35)/5 = 27.36/5 = 5.472 m Step 2: Absolute errors: |5.63−5.472| = 0.158 | |5.54−5.472| = 0.068 |5.44−5.472| = 0.032 | |5.40−5.472| = 0.072 |5.35−5.472| = 0.122 Step 3: Mean absolute error = (0.158+0.068+0.032+0.072+0.122)/5 = 0.452/5 = 0.0904 m Step 4: Relative error = 0.0904/5.472 = 0.0165 ≈ 0.017 Step 5: % error = 0.017 × 100 = 1.7%

Step 1: Volume = mass/density = M/ρ Step 2: Error in division: ΔV/V = ΔM/M + Δρ/ρ = 0.15/5 + 0.2/5 = 0.03 + 0.04 = 0.07 Step 3: % error = 0.07 × 100 % error in Volume = 7%

Step 1: T = 2π√(l/g) → T² = 4π²l/g → g = 4π²l/T² Step 2: % error in g = % error in l + 2 × (% error in T)         = (Δl/l × 100) + 2(ΔT/T × 100) Step 3: = (0.1/100 × 100) + 2(0.01/2 × 100) = 0.1 + 2(0.5) = 0.1 + 1.0 = 1.1 Maximum % error in g = 1.1%

Step 1: Mean = (3.11+3.13+3.14+3.14)/4 = 12.52/4 = 3.13 cm Step 2: Absolute errors: |3.11−3.13|=0.02 | |3.13−3.13|=0.00 | |3.14−3.13|=0.01 | |3.14−3.13|=0.01 Step 3: Mean absolute error = (0.02+0.00+0.01+0.01)/4 = 0.04/4 = 0.01 cm Step 4: % error = (0.01/3.13) × 100 = 0.32%

Step 1: KE = ½mv² → % error in KE = % error in m + 2 × % error in v Step 2: % error in m = (0.3/60.0) × 100 = 0.5% Step 3: % error in v = (0.1/25.0) × 100 = 0.4% Step 4: % error in KE = 0.5 + 2(0.4) = 0.5 + 0.8 % error in KE = 1.3%

Step 1: Mean = (6.12+6.09+6.22+6.15)/4 = 24.58/4 = 6.145 Ω Step 2: Absolute errors: |6.12−6.145|=0.025 | |6.09−6.145|=0.055 | |6.22−6.145|=0.075 | |6.15−6.145|=0.005 Step 3: Mean absolute error = (0.025+0.055+0.075+0.005)/4 = 0.16/4 = 0.04 Ω Step 4: Relative error = 0.04/6.145 = 0.0065 Step 5: % error = 0.65%

(a) 0.003 m²: Zeros left of non-zero are not significant. 1 s.f. (b) 0.1250 g cm⁻²: 1, 2, 5, 0 are significant (trailing zero with decimal). 4 s.f. (c) 6.4×10⁶ m: 6 and 4. 2 s.f. (d) 1.6×10⁻¹⁹ C: 1 and 6. 2 s.f. (e) 9.1×10⁻³¹ kg: 9 and 1. 2 s.f.