Problem

A body has mass m = 22.42 g and volume V = 4.7 cc. Possible errors are Δm = 0.01 g and ΔV = 0.1 cc. Find the maximum percentage error in density. (a) 22 % (b) 2 % (c) 0.2 % (d) 0.02 %

Solution

Density: ρ = m / V

Maximum fractional error in ρ (additive for independent errors):

Δρ / ρ = (Δm / m) + (ΔV / V)
    

Compute fractional errors:

Δm / m = 0.01 / 22.42 ≈ 0.000446
ΔV / V = 0.1  / 4.7   ≈ 0.021277
Δρ / ρ ≈ 0.000446 + 0.021277 = 0.021723
Percentage error = 0.021723 × 100 ≈ 2.17 %
    

Therefore the nearest choice is: (b) 2 %.

Problem

If the error in the measurement of radius of a sphere is 2 %, then the error in the determination of volume of the sphere will be:

(a) 4 %   (b) 6 %   (c) 8 %   (d) 2 %

Solution

Volume of a sphere: V = (4/3)π r³

Relative (percentage) error in volume when radius has error is:

ΔV / V = 3 · (Δr / r)

Given Δr / r = 2%, therefore

ΔV / V = 3 × 2% = 6%

Correct option: (b) 6 %

Problem (restated)

Measured percentage errors:

• a : 1%
• b : 2%
• c : 3%
• d : 4%

Quantity:

P = (a³ b²) / (c d)

Step-by-step solution

1. Write P:

   P = a³ · b² · c⁻¹ · d⁻¹

2. Take natural log (to use differential rule):

   ln P = 3 ln a + 2 ln b − ln c − ln d

3. Differentiate (relative errors):

   dP / P = 3 (da / a) + 2 (db / b) − (dc / c) − (dd / d)

   For maximum percentage error we take absolute contributions (add magnitudes):

4. Convert to percentage form:

   %ΔP ≈ 3·%Δa + 2·%Δb + 1·%Δc + 1·%Δd

5. Substitute given percentages:

   3 × 1% = 3%
   2 × 2% = 4%
   1 × 3% = 3%
   1 × 4% = 4%
           

6. Add them to get total percentage error:

   %ΔP = 3% + 4% + 3% + 4% = 14%

Problem

The percentage errors in the measurements of mass and speed are 2% and 3% respectively. The error in kinetic energy obtained by measuring mass and speed will be:

(a) 12 %   (b) 10 %   (c) 8 %   (d) 2 %

Solution

Kinetic energy of a body is

K = (1/2) m v²

Take relative (fractional) errors. For a product/power:

ΔK / K ≈ Δm / m + 2 · (Δv / v)

Given:

Δm / m = 2%  (mass error)
Δv / v = 3%  (speed error)

Substitute:

ΔK / K ≈ 2% + 2 × 3% = 2% + 6% = 8%

Therefore the correct option is: (c) 8 %.