The moment of inertial I of a metre rule of rectengular cross-section is $$ I=\frac{b d^{2}}{12} $$ alculate I and its standard error (se) if $$ b=2.66 \pm 0.01 \mathrm{~cm}\langle $$ (s.e). $$ d=0.65 \pm 004 $$ .e.).

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We are given that the moment of inertia is

I = (b d²)/12

with

b = 2.66 ± 0.01 cm  and  d = 0.65 ± 0.04 cm.

Step 1. Calculate I using the central values.

d² = (0.65)² = 0.4225
  b·d² = 2.66 × 0.4225 ≈ 1.12385
  I = (1.12385)/12 ≈ 0.0937

Thus, the best‐estimate is I ≈ 0.0937 (in units of cm³).

Step 2. Find the standard error (se) in I.

When a quantity is a function of measured variables, the uncertainty is given (to first order) by

(δI)² = (∂I/∂b · δb)² + (∂I/∂d · δd)²

Calculate the partial derivatives:
  ∂I/∂b = d²/12  and  ∂I/∂d = (2b d)/12 = (b d)/6.

Now substitute the numbers:

• For b:
  d²/12 = 0.4225/12 ≈ 0.03521
  Contribution from b: (0.03521 × 0.01)² = (0.0003521)² ≈ 1.239×10⁻⁷

• For d:
  (b d)/6 = (2.66 × 0.65)/6.
  2.66 × 0.65 ≈ 1.729, so (b d)/6 ≈ 1.729/6 ≈ 0.28817.
  Contribution from d: (0.28817 × 0.04)² = (0.011527)² ≈ 0.0001329

Now sum the squares:
  (δI)² ≈ 1.239×10⁻⁷ + 0.0001329 ≈ 0.0001330

Taking the square root:
  δI ≈ √(0.0001330) ≈ 0.01153

Thus, the standard error is approximately 0.01153.

Step 3. Write the final answer

Rounding appropriately, we have

I = 0.094 ± 0.012 (in cm³).

An alternative method is to use relative errors. Since
  I = (b d²)/12,
the fractional uncertainty is
  (δI/I) = √[(δb/b)² + (2 δd/d)²].

Here,
  δb/b = 0.01/2.66 ≈ 0.00376,  δd/d = 0.04/0.65 ≈ 0.06154.
Then,
  (δI/I) = √[(0.00376)² + (2×0.06154)²]
      = √[1.414×10⁻⁵ + (0.12308)²]
      = √[1.414×10⁻⁵ + 0.01516] ≈ √(0.015174) ≈ 0.1233.
Multiplying by I:
  δI ≈ 0.1233 × 0.0937 ≈ 0.01155,
which is consistent with our earlier calculation.

Final Answer:
  I = 0.094 ± 0.012 cm³.

Note: The units come out as cm³ because our formula is I = (b d²)/12 (using b and d in cm). In some contexts (such as when calculating the area moment of inertia) one might expect different exponents or units, but here we follow the given formula.
The moment of inertia $$ I $$ is approximately 0.094 cm³ with a standard error of 0.012 cm³.

The moment of inertial I of a metre rule of rectengular cross-section is \[ I=\frac{b d^{2}}{12} \] alculate I and its standard error (se) if \( b=2.66 \pm 0.01 \mathrm{~cm}\langle \) (s.e). \( d=0.65 \pm 004 \) .e.).

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