Question 239: Units , Dimensions and Error Analysis

Question

Three students S1, S2 and S3 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.

StudentNo.
Length ofPendulum (cm)
No. ofoscillations (n)
Total time forn oscillations
Timeperiod (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 E1, E2 and E3 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

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