Advertisements
Advertisements
प्रश्न
Answer the following question.
Show that if Z = `"A"/"B"`, `(triangle"Z")/"Z" = (triangle"A")/"A" + (triangle "B")/"B"`
Advertisements
उत्तर
Errors in divisions:
Suppose, Z = `"A"/"B"` and measured values of A and B are (A ± ΔA) and (B ± ΔB) then,
Z ± ΔZ = `("A" +- triangle"A")/("B"+-triangle"B")`
∴ Z`(1 +- (triangle"Z")/"Z") = ("A"[1 +-(Δ"A"//"A")])/("B"[1 +-(Δ"B"//"B")])`
`= "A"/"B" xx [1 +-(Δ"A"//"A")]/[1 +-(Δ"B"//"B")]`
As, `(triangle"B")/"B" ≪ 1,` expanding using Binomial theorem,
`"Z"(1 +- (triangle"Z")/"Z") = "Z" (1 +- (triangle"A")/"A") xx (1 bar+ (Δ"B")/"B")` ....`(because "A"/"B" = "Z")`
∴ `1 +- (Δ"Z")/"Z" = 1 +- (Δ"A")/"A" bar+ (Δ"B")/"B" +- (Δ"A")/"A" xx (Δ"B")/"B"`
Ignoring term `(Δ"A")/"A" xx (Δ"B")/"B", (Δ"Z")/"Z" = +- (Δ"A")/"A" bar+ (Δ"B")/"B"`
This gives four possible values of `(Δ"Z")/"Z"` as
`(+ (Δ"A")/"A" - (Δ"B")/"B"), (+(Δ"A")/"A" + (Δ"B")/"B"), (-(Δ"A")/"A" - (Δ"B")/"B") and (-(Δ"A")/"A" + (Δ"B")/"B")`
∴ Maximum relative error of `(Δ"Z")/"Z" +- ((Δ"A")/"A" + (Δ"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
संबंधित प्रश्न
Define Mean absolute error.
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"`
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?
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 length of a cylinder is l = (4.00 ± 0.001) cm, radius r = (0.0250 ± 0.001) cm and mass m = (6.25 ± 0.01) g. Calculate the percentage error in the determination of density.
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 relative density of a metal may be found by hanging a block of the metal from a spring balance and nothing that in air, the balance reads (5 .00 ± 0.05)N while in water, it reads (4.00 ± 0.05)N. The relative density would be quoted as ______.
Time for 20 oscillations of a pendulum is measured as t1 = 39.6 s; t2 = 39.9 s; t3 = 39.5 s. What is the precision in the measurements? What is the accuracy of the measurement?
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.
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.
If E, L, M, and G denote the quantities as energy, angular momentum, mass, and constant of gravitation respectively, then the dimensions of P in the formula P = EL2M−5G−2 are ______.
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 is used to determine the refractive index of a glass slab. If 40 divisions are there in 1 cm on main scale and 50 Vernier scale divisions are equal to 49 main scale divisions, then least count of the travelling microscope is ______ × 10-6 m.
If the zero of the vernier lies on the right-hand side and the fourth division coincides with the main scale division when the jaws are in contact so the correction will be ______.
A thermometer always reads 2°C higher than the actual temperature because its zero mark is incorrect. What type of error is this?
What are random errors?
A weighing scale that always reads a constant +0.2 kg because its zero mark is misaligned is a clear example of a(n):
Which one of the following is an effective way to reduce systematic errors in an experiment?
If the pointer of the voltmeter is not exactly at the zero of the scale, then the error is called ______.
Zero error of an instrument introduces ______.
Constant error can be caused due to ______.
