Question 226: Units , Dimensions and Error Analysis

Question

Three students S
1
, S
2
and S
3
perform an experiment for determining the acceleration due to gravity (g) using a simple pendulum. They use different lengths of pendulum and record time for different number of oscillations. The observations are as shown in the table.
Student
No.
Length of
Pendulum (cm)
No. of
oscillations (n)
Total time for
n oscillations
Time
period (s)
1
64.0
8
128.0
16.0
2
64.0
4
64.0
16.0
3
20.0
4
36.0
9.0
(Least count of length = 0.1 cm and Least count for time = 0.1 s)
If E
1
, E
2
and E
3
are the percentage errors in 'g' for students 1, 2 and 3 respectively, then the minimum percentage error is obtained by student no. ______________.

Accepted Answer

$$T = {t \over n} = 2\pi \sqrt {{l \over g}} $$
$$ \Rightarrow g = {{4{\pi ^2}l} \over {{T^2}}}$$
$$ \Rightarrow {{\Delta g} \over g} 	imes 100 = {{\Delta l} \over l} 	imes 100 + 2{{\Delta T} \over T} 	imes 100$$
$$ = \left( {{{\Delta l} \over l} + {{2\Delta T} \over {T}}} ight)100\% $$
$${E_1} = {{20} \over {64}}\% $$
$${E_2} = {{30} \over {64}}\% $$
$${E_3} = {{19} \over {18}}\% $$

Units , Dimensions and Error Analysis

Integer / Numeric Answer Type