Physics Measurement Test 2 (With Solutions)

UMOJA WA WAZAZI TANZANIA
WARI SECONDARY SCHOOL

PHYSICS MEASUREMENT TEST 2
FORM FIVE

Time: 2:00hrs
November, 2022

INSTRUCTIONS

1. This paper consists of four (4) questions.
2. Answer all questions.

Question 1: Dimensional Analysis

(a) Definitions with Examples

Term	Definition	Example
Dimension	Physical nature of a quantity expressed in terms of fundamental quantities (M, L, T)	[Force] = MLT-2
Dimensional Analysis	Method to check equations or derive relations using dimensions	Verifying F = ma dimensionally
Dimensional Variable	Physical quantity with dimensions that can change value	Velocity (LT-1)
Dimensional Constant	Constant with fixed dimensions	Gravitational constant (M-1L3T-2)
Dimensionless Variable	Variable without dimensions	Strain (ΔL/L)
Dimensionless Constant	Numerical constant without dimensions	π (pi)

(b)(i) Dimensional Analysis Significance & Limitations

Significance:

1. Verifies dimensional consistency of equations

Limitations:

1. Cannot determine dimensionless constants
2. Doesn't confirm numerical correctness
3. Cannot identify logarithmic/exponential relations

(b)(ii) Dimensions of a and b

Given: (P + a/V²)(V - b) = constant
For dimensional consistency:
[a/V²] = [P] ⇒ [a] = [P][V²] = ML^-1T^-2 × L⁶ = ML⁵T^-2
[b] = [V] = L³

(b)(iii) Dimensions in Velocity Equation

v = at + b/(t + c)
[at] = [v] ⇒ [a] = [v]/[t] = LT^-1/T = LT^-2
[b/(t + c)] = [v] ⇒ [b] = [v][t] = LT^-1 × T = L
[c] = [t] = T

Question 2: Dimensions and Homogeneity

(a)(i) Quantities with Work Dimensions

Same dimensions as work (ML²T^-2):

• Energy (scalar)
• Torque (vector)

(a)(ii) Scalar/Vector with Same Dimensions

• Work (scalar) and Torque (vector)
• Speed (scalar) and Velocity (vector)

(a)(iii) Principle of Homogeneity

All terms in a physical equation must have identical dimensions on both sides.

(a)(iv) Limitations of Dimensional Analysis

No, dimensional analysis cannot confirm an equation is completely correct because:

• It doesn't verify numerical constants
• Different physical quantities may share dimensions
• It can't identify incorrect relationships between terms

(b) Deriving SHM Energy Expression

Assume E ∝ M^xf^yA^z

[E] = [M]^x[f]^y[A]^z
ML²T^-2 = M^x(T^-1)^yL^z
Comparing dimensions:
For M: x = 1
For T: -y = -2 ⇒ y = 2
For L: z = 2
Thus E ∝ Mf²A²
Actual formula: E = ½(2π)²Mf²A² = 2π²Mf²A²

Question 3: Unit Conversion and Errors

(a) Unit Conversions

(i) 5 g·cm·s^-2 to N:

1 N = 1 kg·m·s^-2 = 1000 g × 100 cm × s^-2 = 10⁵ g·cm·s^-2
5 g·cm·s^-2 = 5/10⁵ N = 5 × 10^-5 N

(ii) 19 × 10¹⁰ N/m² to g·cm^-1·s^-2:

1 N/m² = 1 kg·m^-1·s^-2 = 1000 g × (100 cm)^-1 × s^-2 = 10 g·cm^-1·s^-2
19 × 10¹⁰ N/m² = 19 × 10¹⁰ × 10 = 1.9 × 10¹² g·cm^-1·s^-2

(b)(i) Systematic Error Causes

1. Instrument calibration errors
2. Zero errors in measuring devices

(b)(ii) Precise vs Accurate

Precise Experiment	Accurate Experiment
Small random errors (consistent results)	Small systematic errors (close to true value)
May be consistently wrong	Requires both precision and correctness

(b)(iii) Dimension vs Unit

Dimension	Unit
Physical nature (M, L, T)	Standard of measurement (kg, m, s)
Independent of system	System-dependent (SI, CGS)

(b)(iv) Specific Gravity Error

Specific gravity = Weight in air / (Weight in air - Weight in water)
SG = 25.0/(25.0 - 10.0) = 25.0/15.0 ≈ 1.6667
ΔSG/SG = ΔW_air/W_air + (ΔW_air + ΔW_water)/(W_air - W_water)
= 0.1/25 + (0.1+0.02)/15 ≈ 0.004 + 0.008 = 0.012 (1.2%)

Question 4: Applications

(a)(i) Fluid Flow Relation

Assume Q ∝ Δp^xr^yη^zl^w

[Q] = [Δp]^x[r]^y[η]^z[l]^w
L³T^-1 = (ML^-1T^-2)^xL^y(ML^-1T^-1)^zL^w
Solving: x=1, z=-1, y=4, w=-1
Thus Q ∝ Δp·r⁴/(η·l)
Actual formula (Poiseuille's): Q = πΔp·r⁴/(8ηl)

(a)(ii) Volume Percentage Error

V = l × b × h = 15.12 × 10.15 × 5.29 ≈ 811.6 cm³
ΔV/V = Δl/l + Δb/b + Δh/h = 0.01/15.12 + 0.01/10.15 + 0.01/5.29
≈ 0.000661 + 0.000985 + 0.00189 ≈ 0.003536 (0.3536%)

(a)(iii) Percentage Error in P

P = a·b²·c³·d⁴
ΔP/P = Δa/a + 2(Δb/b) + 3(Δc/c) + 4(Δd/d)
= 0.5% + 2(0.5%) + 3(0.5%) + 4(0.5%)
= (1 + 2 + 3 + 4)(0.5%) = 10 × 0.5% = 5%