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Title1.General Physics
TagsLens (Optics) Electrical Resistance And Conductance Applied And Interdisciplinary Physics Physical Quantities Physics
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1 GENERAL PHYSICS

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ERRORS IN MEASUREMENT
i) Significant figures:

The number of significant figures in the measured value of a physical quantity gives the accuracy of its value.
Common rules of counting significant figure

1. All non-zero digits are significant.

2. All zeros occurring between two non-zero digits are significant.

3. In a number less than one, all zeros to the right of decimal point and to the left of a non-
zero digit are not significant.

4. All zeros on the right of the last non-zero digit in the decimal part are significant.

Errors: The difference between the true and the measured values of a quantity is the error.

Propagation of Errors:

a) Sum and difference of quantities:

x = a ± b

x = ( a + b)

b) Products and quotients of quantities:

x = a x b

x = a/b for both b
b

a
a

x
x

c) Powers of quantities:

m

n

b

a
x

lnx = nlna - mlnb

differentiating b
db

m
a

da
n

x
dx

For errors,

Maximum fractional error in x, b
b

m
a
a

n
x
x

d) When taking the mean (m) of several uncorrelated measurements of the same quantity, the error

dm is:
n

x....x n1

n
x

, for n measurements.

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From the data obtained, a graph showing extension (Dl) against the load (W) is plotted which is obtained
as a straight line passing through the origin. The slope of the line gives

tanq = W
l l

Mg

Now, stress = 2
Mg

r
and strain =

l
L

Y = Stress/strain = 2 2
Mg L L

r r tanl

With known values of initial length L, radius r of the experimental wire and tanq, Young’s modulus Y can be
calculated.

4) Specific Heat of a liquid using a calorimeter:

The principle is to take a known quantity of liquid in an insulated calorimeter and heat it by passing a
known current (i) through a heating coil immersed within the liquid for a known length of time (t). The mass of the
calorimeter (m1) and, the combined mass of the calorimeter and the liquid (m2) are measured. The potential drop
across the heating coil is V and the maximum temperature of the liquid is measured to q2.

The specific heat of the liquid (S
l
) is found by using the relation

(m2 - m1) Sl (q2 - q0) + m1Sc(q2 - q0) = i.V.t or (m2 - m1)Sl + m1Sc = i.V.t/(q2 - q0) ...... (1)

Here 0 is the room temperature, while Sc is the specific heat of the material of the calorimeter and the stirrer.
If Sc is known, then Sl can be determined.

On the other hand, if Sc is unknown : one can either repeat the experiment with water or a different mass
of the liquid and use the two equals to eliminate m1Sc.

The sources of error in this experiment are errors due to improper connection of the heating coil, radia-
tion, apart from statistical errors in measurement.

The direction of the current is reversed midway during the experiment to remove the effect of any differ-
ential contacts, radiation correction is introduced to take care of the second major source of systematic error.

Radiation correction:

The temperature of the system is recorded for half the length of time t,

i.e. t/2, where t is the time during which the current was switched on) after the current is switched off. The

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fall is temperature d, during this interval is now added to the final temperature q2 to give the correct final tempera-
ture:

q’2 = q2 + d.

This temperature is used in the calculation of the specific heat S
l
.

Error analysis:

After correcting for systematic errors, equation (i) is used to estimate the remaining errors.

5) Focal length of a concave mirror and a convex lens using the u-v method:

In this method one uses an optical bench and the convex lens (or the concave mirror) is placed on the holder.

The position of the lens is noted by reading the scale at the bottom of the holder.

A bright object (a filament lamp or some similar object) is placed at a fixed distance (u) in front of the lens (mirror).

The position of the image (v) is determined by moving a white screen behind the lens until a sharp image is
obtained (for real images).

For the concave mirror, the position of the image is determined by placing a sharp object (a pin) on the
optical bench such that the parallel between the object pin and the image is nil.

A plot of |u| versus |v| gives a rectangular hyperbola. A plot of gives a straight line.

The intercepts are equal to , where f is the focal length.

Error: The systematic error in this experiment is mostly due to improper position of the object on the
holder. This error maybe eliminated by reversing the holder (rotating the holder by 1800 about the vertical) and
then taking the readings again. Averages are then taken.

The equation for errors gives:

u vf u v
f u v u v

The errors u, v correspond to the error in the measurement of u and v..

6) Speed of sound using resonance column:

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The ratio arms are first adjusted so that they carry 100 each. The resistance in the rheostat arm is now
adjusted so that the galvanometer deflection is in one direction, if R = R0 (Ohm) and in the opposite direction.
This implies that the unknown resistance, S lies between R0 and R0 + 1 (Ohm). Now, the resistance in P and Q
are made 100W and 1000W respectively, and the process is repeated.

Equation (1) is used to compute S.

The ratio P/Q is progressively made 1 : 10, and then 1 : 100. The resistance S can be accurately measured.

Errors: The major sources of error are the connecting wires, unclear resistance plugs, change in resis-
tance due to Joule heating, and the insensitivity of the Wheatstone bridge. These may be removed by using thick
connecting wires, clean plugs, keeping the circuit on for very brief periods (to avoid Joule heating) and calculating
the sensitivity. In order that the sensitivity is maximum, the resistance in the arm P is close to the value of the
resistance S.

Illustration:

The sides of a rectangle are (10.5 ± 0.2) cm and (5.2 ± 0.1) cm calculate its perimeter with error limit.

Solution:

Here, l = (10.5 ± 0.2) cm

b = (5.2 ± 0.1) cm

P = 2(l + b) = 2(10.5 + 5.2) = 31.4 cm

P = 2 ( l + b) = ± 0.6

Hence perimeter = (31.4 ± 0.6) cm.

Consider the following data:

10 main scale division = 1 cm, 10 vernier division = 9 main scale divisions, zero of vernier scale is to the
right of the zero marking of the main scale with 6 vernier divisions coinciding with main scale divisions and the
actual reading for length measurement is 4.3 cm with 2 vernier divisions coinciding with main scale graduations.
Estimate the length.

Solution:

In this case, vernier constant =
1mm

0.1mm
10

Zero error = 6 x 0.1 = + 0.6 mm

Correction = -0.6 mm

Actual length = (4.3 + 2 x 0.01) + correction

= 4.32 - 0.06 = 4.26 cm

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