Errors

 ERRORS

The result of every measurement by any measuring instrument contains some uncertainty this uncertainty is called errors.  The errors in a measurement is equal to the difference between the true value and the measured value of any quantity.

Errors = True value – Measured value

Absolute Errors:- the several values obtained in an experiment measured are a₁, a₂, a₃….an the arthmetic mean of these values will be

Now error in first measurement = Da₁ = a_mean –a₁

Similarly in second measurement = Da₂  = a_mean –a₂   and so on

 Relative error:Relative error=∆ameanamean

 

Combination of errors (maximum possible error) Addition or subtraction

Product or multiplication 

Let

 

Division

 

     

 

(4) Powers

Let C = aⁿb^m

CONCEPT WITH EXAMPLES

Role of constants

The radius of sphere is measured to be (2.1 ± 0.5) cm and is surface area with error limits.

Solution:

Surface area = A = 4pr²

 

 

Now% error

 

= 47.62 %

Other way

= 4 p r²

 

= 55.4 cm²

 

 

 

= 26.4 cm²

Now (A ± DA)

 = (55.4±26.4)  

FINDING QUANTITY 

Percentage error in determining of acceleration of gravity with the help of simple Pendulum time period .  Given error in length of pendulum 4% while in The time period it is 2% then percentage error in g = ?

 

Solution

LEAST COUNT AND DIFFERENT UNITS

If a particle of mass m = 25.0 Kg is moving in a circular path of radius r = 50.00 cm with constant speed of v = 10.0 Km/hr them find error in force required by particle.

Now Given

m = 25.0 Kg

    = L.c. = 0.1

V = 10.0 Km/h

    = L.c = 0.1

DV = 0.1, V = 10

R = 50.00 cm

Dr = 0.001, r = 50cm

NOTE: never change units in error.

Least count

25.00 – L.c = 0.01

16.50 – 50 –L.c = 0.01

76.03 – 03 –L.c = 0.01

15.0002 –L.c = 0.0001

17.030 – L.c = 0.01

 

Solution:

 

 

 

 

 

= 1.21 x 2= 2.42% Ans.

UNIQUE CONCEPT

Calculate focal length of a spherical Mirror from the following observations Object distance u = (50.1 ± 0.5) cm. and image distance v = (20.1 ± 0.2) cm.

 

 

 

   = 14.3 cm

Now

 

     

 

    

 

      ± 0.4 cm

Note:  Similar case in resistance connected in parallel combinations

    

SIGNIFICANT FIGURES

The number of significant figures of a numerical quantity is the number of reliably known digits it contains:

Rules:

1. Zeros at the beginning of a number are not significant.

Ex. 0.0523 – there S.f. (5, 2, 3)

2. Zeros within a number are significant.

Ex: 2056 –Four S.f.    (2, 0, 5, 6)

3. Zeros at the end of a number after decimal points are significant.

Ex. 3702.0 –five S.f. (3, 7, 0, 2, 0)