Question 1181818: A rectangular brass bar of mass M has dimensions a, b, c as shown in the
figure below. The moment of inertia I about the axis at the centre of the ab
face and perpendicular to it, is given by:
I=(M/12)(a^2 + b^2)
The following measurements are made:
𝑀=135.0±0.1𝑔
𝑎 =80±1 𝑚𝑚
𝑏 =10±1 𝑚𝑚
𝑐 =20.00±0.01 𝑚𝑚
Calculate the standard error in:
a) the density ρ of the material
b) the moment of inertia I
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! Here's how to calculate the standard error in density (ρ) and moment of inertia (I):
**a) Standard Error in Density (ρ)**
1. **Calculate the nominal density:**
ρ = M / (abc) = 135.0 g / (80 mm * 10 mm * 20.00 mm) = 0.084375 g/mm³
2. **Calculate the fractional uncertainties:**
* ΔM/M = 0.1 g / 135.0 g = 0.000741
* Δa/a = 1 mm / 80 mm = 0.0125
* Δb/b = 1 mm / 10 mm = 0.1
* Δc/c = 0.01 mm / 20.00 mm = 0.0005
3. **Calculate the fractional uncertainty in density:**
(Δρ/ρ)² = (ΔM/M)² + (Δa/a)² + (Δb/b)² + (Δc/c)²
(Δρ/ρ)² = (0.000741)² + (0.0125)² + (0.1)² + (0.0005)²
(Δρ/ρ)² = 0.00000055 + 0.00015625 + 0.01 + 0.00000025 = 0.010157
Δρ/ρ = √0.010157 ≈ 0.1008
4. **Calculate the absolute uncertainty in density:**
Δρ = (Δρ/ρ) * ρ = 0.1008 * 0.084375 g/mm³ ≈ 0.0085 g/mm³
5. **Express the density with its uncertainty:**
ρ = 0.0844 ± 0.0085 g/mm³ (rounded to appropriate significant figures)
**b) Standard Error in Moment of Inertia (I)**
1. **Calculate the nominal moment of inertia:**
I = (M/12)(a² + b²) = (135.0 g / 12) * ((80 mm)² + (10 mm)²) = 11.25 g * (6400 mm² + 100 mm²) = 11.25 g * 6500 mm² = 73125 g mm²
2. **Calculate the fractional uncertainties:** (Same as in part a)
* ΔM/M = 0.000741
* Δa/a = 0.0125
* Δb/b = 0.1
3. **Calculate the fractional uncertainty in I:**
(ΔI/I)² = (ΔM/M)² + [2*(Δa/a)]² + [2*(Δb/b)]²
(ΔI/I)² = (0.000741)² + [2*(0.0125)]² + [2*(0.1)]²
(ΔI/I)² = 0.00000055 + 0.000625 + 0.04 = 0.04062555
ΔI/I = √0.04062555 ≈ 0.2016
4. **Calculate the absolute uncertainty in I:**
ΔI = (ΔI/I) * I = 0.2016 * 73125 g mm² ≈ 14750 g mm²
5. **Express the moment of inertia with its uncertainty:**
I = 73100 ± 14800 g mm² (rounded to appropriate significant figures)
**Key Points:**
* The uncertainty in *b* has the largest impact on both ρ and I because it has the largest fractional uncertainty.
* When calculating the uncertainty in I, the uncertainties in *a* and *b* are multiplied by 2 because they are squared in the formula for I.
* Always round your final answers to the appropriate number of significant figures. The uncertainties should usually have only one significant figure (sometimes two if the leading digit is a 1 or 2), and the measured values should be rounded to the same decimal place as the uncertainty.
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