Question 276: Units , Dimensions and Error Analysis

Question

Student I, II and III perform an experiment for measuring the acceleration due to gravity (g) using a
simple pendulum. They use different length of the pendulum and/or record time for different number
of oscillations. The observations area shown in the table.
Least count for length = 0.1 cm
Least count for time = 0.1 s
Student
Length of the
pendulum
(cm)
No. of
oscillations
(n)
Total time
for(n)
oscillations
(s)
Time
periods
(s)
I
64.0
8
128.0
16.0
II
64.0
4
64.0
16.0
III
20.0
4
36.0
9.0
If E
I
, E
II
and E
III
are the percentage errors in g, i.e., $$\left(\frac{	riangle g}g	imes100ight)$$ for students I, II and III, respectively,then

Accepted Answer

Time period (T) $$ = 2\pi \sqrt {{l \over g}} $$
or $${t \over n} = 2\pi \sqrt {{l \over g}} $$
$$	herefore$$ $$g = {{(4{\pi ^2})({n^2})l} \over {{t^2}}}$$
% error in $$g = {{\Delta g} \over g} 	imes 100$$
$$ = \left( {{{\Delta l} \over l} 	imes {{2\Delta l} \over l}} ight) 	imes 100$$
$${E_I} = \left( {{{0.1} \over {64}} + {{2 	imes 0.1} \over {128}}} ight) 	imes 100 = 0.3125\% $$
$${E_{II}} = \left( {{{0.1} \over {64}} + {{2 	imes 0.1} \over {64}}} ight) 	imes 100 = 0.46875\% $$
$${E_{III}} = \left( {{{0.1} \over {20}} + {{2 	imes 0.1} \over {36}}} ight) 	imes 100 = 1.055\% $$
Hence, $$\mathrm{E_I}$$ is minimum.

Answer

E
I
= 0

Answer

E
I
is minimum

Answer

E
I
= E
II

Answer

E
II
is maximum

Units , Dimensions and Error Analysis

Choose the correct answer: