Question

Explain

(a) Error of a sum or a difference

(b) Error of a product or a quotient

(c) Error in case of a measured quantity raised to a power​

Answer 1

Answer:

(a) Error of a sum or a difference:

When two quantities are added or subtracted, the absolute error in the final result is the sum of individual absolute errors.

For ex. Let A and B be the two physical quantities which are to be added or subtracted and Z be their resultant physical quantity. Let ΔA, ΔB and ΔZ be their respective absolute errors.

Here, A±ΔA, B±ΔB, Z±ΔZ

For sum, Z=A+B

        Z±ΔZ=(A±ΔA) + (B±ΔB)

Maximum possible error in Z= ΔZ=ΔA+ΔB

For difference, Z=A-B

                 Z±ΔZ= (A±ΔA) - (B±ΔB)

Maximum possible error= ΔZ=ΔA+ΔB

(b) Error of a product or a quotient:

When two physical quantities are multiplied or divided , the relative error in the resultant physical quantity is the sum of relative error of individual quantities.

For ex. Suppose the measured value of physical quantities A and B be A±ΔA and B±ΔB. Let Z±ΔZ be the resultant physical quantity.

Now, Z=AB

Z±ΔZ= (A±ΔA)(B±ΔB)

Z±ΔZ= AB±ΔAB±AΔB±ΔAΔB

ΔAΔB being very, small can be neglected.

Dividing LHS by Z and RHS by AB

Z/Z±ΔZ/Z= AB/AB ± ΔAB/AB ± AΔB/AB

1 + ΔZ/Z = 1 + ΔA/A + ΔB/B

So, ΔZ/Z = ΔA/A + ΔB/B

Above relation is used for division also.

(c) Error in case of a measured quantity raised to a power:

When a physical quantity is raised to a power, the resultant quantity is multiplied by the power on the quantity.

For ex. Z=A^n

So, ΔZ/Z = nΔA/A

Hope it will help you..