Question

The dimensions of a physical quantity X is [M^a L^−b T^−c]. If the maximum percentage errors in the measurement of M, L and T respectively are x%, y% and z%, the maximum percentage error in the measurement of X is -

1. (ax + by + cz)%
2. (ay + bx - cz)%
3. (az - bz + cy)%
4. (ax - by - cz)%

Answer 1

Correct Answer - Option 4 : (ax - by - cz)%

The correct answer is option 4) i.e. (ax - by - cz)%

CONCEPT:

• Combination of errors: When physical quantities involving a mathematical operation have errors associated with them, there will be an error in the result arising from its combination.
• This is identified based on the following rules:
  • The error of a sum or a difference: If ​two physical quantities A and B have measured values A ± ΔA, B ± ΔB respectively where ΔA and ΔB are their absolute errors, the possible error ΔZ in the operation Z = A ± B is given by:

± ΔZ = ± ΔA ± ΔB

• The error of a product or a quotient: If ​two physical quantities A and B have measured values A ± ΔA, B ± ΔB respectively where ΔA and ΔB are their absolute errors, the possible error ΔZ in the operation Z = AB or Z = A ÷ B is given by:

\( \frac{\Delta Z}{ Z} = \frac{∆A}{A} + \frac{∆B}{B}\)

• The error in case of a measured quantity raised to a power: 

​If Z = Ax then 

\(\frac{\Delta Z}{Z} = x\frac{\Delta A}{A}\)

EXPLANATION:

Given that:

X = [Ma L−b T−c]

\(\frac{∆M}{M} = x\%\), \(\frac{∆L}{L} = y\%\) and \(\frac{∆T}{T} = z\%\)

Using the rule of the quantity raised to a power and product, \( \frac{\Delta X}{ X} =a \frac{∆M}{M} +(- b)\frac{∆L}{L} + (-c)\frac{\Delta T}{T}\)

\(\Rightarrow \frac{\Delta X}{ X} =a x + (-b)y + (-c)z\)

\(\Rightarrow \frac{\Delta X}{ X} =(a x-by -cz)\%\)