Advertisements
Advertisements
Question
If the measured values of the two quantities are A ± ΔA and B ± ΔB, ΔA and ΔB being the mean absolute errors. What is the maximum possible error in A ± B? Show that if Z = `"A"/"B"`
`(Delta "Z")/"Z" = (Delta "A")/"A" + (Delta "B")/"B"`
Advertisements
Solution
Suppose, Z = `"A"/"B"` and measured values of A and B are (A ± ΔA) and (B ± ΔB) then,
`"Z" +- Delta"Z" = ("A" +- Delta "A")/("B" +- Delta"B")`
∴ Z`(1 +- (Delta"Z")/"Z") = "A"/"B"([1 +- (Delta "A"//"A")]/[1 +- (Delta "B"// "B")])`
`= "A"/"B" xx [1 +- (Delta "A"//"A")]/[1 +- (Delta "B"// "B")]`
As, `(Delta "B")/"B"`<< 1, expanding using Binomial theorem,
`"Z"(1 +- (Delta"Z")/"Z") = "Z" xx (1 +- (Delta "A")/"A") xx (1 -+ (Delta"B")/"B") ...(because "A"/"B" = "Z")`
∴ `1 +- (Delta"Z")/"Z" = 1 +- (Delta "A")/"A"-+ (Delta"B")/"B" +- (Delta"A")/"A" xx (Delta"B")/"B"`
Ignoring term `(Delta "A")/"A" xx (Delta "B")/"B", (Delta "Z")/"A" = +- (Delta "A")/"A" = (Delta "B")/"B"`
This gives four possible values of `(Delta "Z")/"Z"` as `(+ (Delta "A")/"A" - (Delta "B")/"B"), (+ (Delta "A")/"A" + (Delta "B")/"B"), (- (Delta "A")/"A" - (Delta"B")/"B")` and `(- (Delta "A")/"A" + (Delta "B")/"B")`
∴ Maximum relative error of `(Delta "Z")/"Z" = +- ((Delta "A")/"A" + (Delta "B")/"B")`
Thus, when two quantities are divided, the maximum relative error in the result is the sum of relative errors in each quantity.
APPEARS IN
RELATED QUESTIONS
Define Mean absolute error.
Define relative error.
Define percentage error.
Answer the following question.
Show that if Z = `"A"/"B"`, `(triangle"Z")/"Z" = (triangle"A")/"A" + (triangle "B")/"B"`
The masses of two bodies are measured to be 15.7 ± 0.2 kg and 27.3 ± 0.3 kg. What is the total mass of the two and the error in it?
The distance travelled by an object in time (100 ± 1)s is (5.2 ± 0.1) m. What are the speed and its maximum relative error?
Find the percentage error in kinetic energy of a body having mass 60.0 ± 0.3 g moving with a velocity of 25.0 ± 0.1 cm/s.
In Ohm’s experiments, the values of the unknown resistances were found to be 6.12 Ω, 6.09 Ω, 6.22 Ω, 6.15 Ω. Calculate the (mean) absolute error, relative error, and percentage error in these measurements.
Solve the numerical example.
If the formula for a physical quantity is X = `("a"^4"b"^3)/("c"^(1//3)"d"^(1//2))` and if the percentage error in the measurements of a, b, c and d are 2%, 3%, 3% and 4% respectively. Calculate percentage error in X.
In an experiment, the percentage of error occurred in the measurement of physical quantities A, B, C and D are 1%, 2%, 3% and 4% respectively. Then the maximum percentage of error in the measurement X, where X = `(A^2B^(1/2))/(C^(1/3)D^3)`, will be:
If Q = `"X"^"n"/"Y"^"m"` and Δx is absolute error in the measurement of X, Δy is absolute error in the measurement of Y, then absolute error ΔQ in Q is ______.
The mean length of an object is 5 cm. Which of the following measurements is most accurate?
The vernier scale of a travelling microscope has 50 divisions which coincide with 49 main scale divisions. If each main scale division is 0.5 mm, calculate the minimum inaccuracy in the measurement of distance.
A physical quantity X is related to four measurable quantities a, b, c and d as follows: X = a2 b3 c5/2d–2. The percentage error in the measurement of a, b, c and d are 1%, 2%, 3% and 4%, respectively. What is the percentage error in quantity X? If the value of X calculated on the basis of the above relation is 2.763, to what value should you round off the result.
The resistance R = `"V"/"I"`, where V = (50 ± 2) V and I = (20 ± 0.2)A. The percentage error in R is 'x' %.
The value of x to the nearest integer is ______.
In a Screw Gauge, the fifth division of the circular scale coincides with the reference line when the ratchet is closed. There are 50 divisions on the circular scale, and the main scale moves by 0.5 mm on a complete rotation. For a particular observation, the reading on the main scale is 5 mm and the 20th division of the circular scale coincides with the reference line. Calculate the true reading.
The dependence of g on geographical latitude at sea level is given by `g = g_0(1 + betasin^2Phi)` where Φ is the latitude angle and β is a dimensionless constant. If Δg is the error in the measurement of g then the error in measurement of latitude angle is ______.
A travelling microscope has 20 divisions per cm on the main scale while its Vernier scale has total 50 divisions and 25 Vernier scale divisions are equal to 24 main scale divisions, what is the least count of the travelling microscope?
The least count of the main scale of a screw gauge is 1 mm. The minimun number of divisions on its circular scale required to measure 5 µm diameter of a wire is ______.
What is an error in measurement?
Reading a graduated cylinder at an angle (parallax error) is a specific example of which type of systematic error?
Systematic errors cause all measurements to shift in the same direction. What does this primarily affect?
Which of the following is an environmental error?
