Numerical Problems

Step 1: Identify the operation and rule

The volume V is calculated by multiplication: V = l × w × h. The rule for multiplication is that the result must have the same number of significant figures as the measurement with the fewest significant figures.

Step 2: Count significant figures

• l = 15.12 cm has 4 significant figures.
• w = 3.45 cm has 3 significant figures.
• h = 1.78 cm has 3 significant figures.

The result must be rounded to 3 significant figures.

Step 3: Calculate the volume

V = 15.12 cm × 3.45 cm × 1.78 cm = 92.78592 cm³

Step 4: Round to the correct number of significant figures

Rounding 92.78592 to 3 significant figures gives 92.8.

Step 1: Set up the conversion factors

• Meters to kilometers: 1 km = 1000 m
• Seconds to hours: 1 hour = 3600 s

Step 2: Perform the conversion

Speed = (3.00 × 10⁸ m/s) × (1 km / 1000 m) × (3600 s / 1 hour)

The units of meters and seconds cancel out, leaving km/hour.

Speed = (3.00 × 10⁸ × 3600) / 1000 km/h

Speed = (3.00 × 10⁸ × 3.6 × 10³) / 10³ km/h

Speed = 3.00 × 3.6 × 10⁸ km/h = 10.8 × 10⁸ km/h

Step 3: Express in proper scientific notation

Move the decimal to get 1.08 × 10⁹ km/h.

Step 1: Analyze the Left-Hand Side (LHS)

The dimension of the LHS is the dimension of the period T, which is [T].

Step 2: Analyze the Right-Hand Side (RHS)

The term 2π is a dimensionless constant, so we ignore it for dimensional analysis.

The dimensions of the RHS are those of √(m/k).

Dimension of m/k = [m]/[k] = [M] / [MT⁻²]

Simplifying the dimensions: [M] / [MT⁻²] = [M¹⁻¹ T²] = [T²]

Now, take the square root: √([T²]) = [T].

Step 3: Compare LHS and RHS

Dimension of LHS = [T]

Dimension of RHS = [T]

Step 1: Calculate the mean resistance (a)

R_mean = (4.10 + 4.14 + 4.08 + 4.12) / 4 = 16.44 / 4 = 4.11 Ω

Step 2: Calculate the absolute errors for each measurement

• |4.11 - 4.10| = 0.01 Ω
• |4.11 - 4.14| = 0.03 Ω
• |4.11 - 4.08| = 0.03 Ω
• |4.11 - 4.12| = 0.01 Ω

Step 3: Calculate the mean absolute error (b)

ΔR_mean = (0.01 + 0.03 + 0.03 + 0.01) / 4 = 0.08 / 4 = 0.02 Ω

Step 4: Calculate the percentage error (c)

Relative Error = ΔR_mean / R_mean = 0.02 / 4.11 ≈ 0.004866

Percentage Error = 0.004866 × 100% ≈ 0.49%

Step 5: Express the final result (d)

The result should be written as R ± ΔR, where the uncertainty is rounded to one significant figure, and the main value is rounded to the same decimal place.

ΔR = 0.02 Ω. This is in the hundredths place.

R_mean = 4.11 Ω. This is also in the hundredths place.

Step 1: Calculate the mean resistance

R_mean = V_mean / I_mean = 10.0 V / 2.50 A = 4.00 Ω

Step 2: Calculate the relative errors of V and I

Relative error in Voltage: ΔV/V = 0.2 / 10.0 = 0.02

Relative error in Current: ΔI/I = 0.05 / 2.50 = 0.02

Step 3: Add the relative errors to find the relative error in R

For division, the relative errors add: ΔR/R = ΔV/V + ΔI/I

ΔR/R = 0.02 + 0.02 = 0.04

Step 4: Calculate the percentage error in R

Percentage Error = (ΔR/R) × 100% = 0.04 × 100% = 4%

Step 5 (Optional): Find the absolute error and final result

Absolute error ΔR = R_mean × (ΔR/R) = 4.00 Ω × 0.04 = 0.16 Ω

Final Result: R = (4.00 ± 0.16) Ω. Rounded properly: R = (4.0 ± 0.2) Ω.

Step 1: Assume the relationship

Let f = k × Lᵃ × Fᵇ × μᶜ, where k is a dimensionless constant.

Step 2: Write the dimensional equation

[T⁻¹] = [L]ᵃ × [MLT⁻²]ᵇ × [ML⁻¹]ᶜ

Combine the powers for each dimension:

[M⁰L⁰T⁻¹] = [Lᵃ] × [MᵇLᵇT⁻²ᵇ] × [MᶜL⁻ᶜ]

[M⁰L⁰T⁻¹] = [Mᵇ⁺ᶜ Lᵃ⁺ᵇ⁻ᶜ T⁻²ᵇ]

Step 3: Equate the powers for each dimension

1. For Mass [M]: 0 = b + c => c = -b
2. For Time [T]: -1 = -2b => b = 1/2
3. For Length [L]: 0 = a + b - c

Step 4: Solve for the exponents

From (2), we have b = 1/2.

Using (1), c = -b = -1/2.

Using (3), 0 = a + (1/2) - (-1/2) = a + 1 => a = -1.

Step 5: Substitute the exponents back into the assumed relationship

f = k × L⁻¹ × F¹/² × μ⁻¹/²

This can be rewritten as: f = (k/L) × √(F/μ)

Step 1: Write the formula for relative error propagation

Given the relationship P = a³b² / (c¹/² d¹), the relative error in P is the sum of the relative errors of each variable multiplied by the magnitude of its power.

ΔP/P = 3(Δa/a) + 2(Δb/b) + (1/2)(Δc/c) + 1(Δd/d)

Note: Errors are always added, regardless of whether the variable is in the numerator or denominator.

Step 2: Convert percentage errors to relative errors (or work directly with percentages)

We are given the percentage errors, which is (Δx/x) × 100%. We can use these values directly.

• % error in a = 3 × 1% = 3%
• % error in b = 2 × 3% = 6%
• % error in c = (1/2) × 4% = 2%
• % error in d = 1 × 2% = 2%

Step 3: Sum the resulting percentage errors

Total % error in P = (Contribution from a) + (Contribution from b) + (Contribution from c) + (Contribution from d)

Total % error = 3% + 6% + 2% + 2% = 13%

Part 1: Surface Area (A = 4πr²)

1. Calculate the mean area:

A_mean = 4π(2.15 cm)² = 4π(4.6225 cm²) ≈ 58.088 cm²

2. Calculate the relative error in area:

Using the power rule, ΔA/A = 2(Δr/r).

Δr/r = 0.05 / 2.15 ≈ 0.02326

ΔA/A = 2 × 0.02326 = 0.04652

3. Calculate the absolute error in area:

ΔA = A_mean × (ΔA/A) = 58.088 × 0.04652 ≈ 2.702 cm²

4. Express the final result for area:

Round error to one sig fig: ΔA ≈ 3 cm². Round mean value to the same decimal place (the ones place): A_mean ≈ 58 cm².

Surface Area = (58 ± 3) cm²

Part 2: Volume (V = (4/3)πr³)

1. Calculate the mean volume:

V_mean = (4/3)π(2.15 cm)³ = (4/3)π(9.938375 cm³) ≈ 41.629 cm³

2. Calculate the relative error in volume:

Using the power rule, ΔV/V = 3(Δr/r).

ΔV/V = 3 × 0.02326 = 0.06978

3. Calculate the absolute error in volume:

ΔV = V_mean × (ΔV/V) = 41.629 × 0.06978 ≈ 2.905 cm³

4. Express the final result for volume:

Round error to one sig fig: ΔV ≈ 3 cm³. Round mean value to the same decimal place (the ones place): V_mean ≈ 42 cm³.

Volume = (42 ± 3) cm³