Students `I_(1),J_(1), J_(3)` and `I_(2)` perform an experiment for measuring the acceleration due to gravity (g) using a simple predulum, they use different lengths of the pendulum and record time for different number of oscillations. The observations are shown in the table. Least count for length `= 0.1 cm`, least count for time = 1s
`{:("Students",underset("pendulum (cm)")("Length of the"),underset("oscillations(n)")("No.of"),underset("of pendulum (s)")("Time period")),(I_(1),100.0,20,20),(J_(1),400.0,10,40),(J_(1),100.0,10,20),(I_(1),400.0,20,40):}`
If `P_(1),P_(2),P_(3)` and `P_(4)` are the % error in g for students `I_(1),J_(1),J_(3)` and `I_(2)` respectively then -

Options

A. `P_(1) = P_(3)`
B. `P_(3)` is maximum
C. `P_(4)` is minimum
D. `P_(a) = P_(4)`

Answer

B, C

Explanation

`T=2 pi sqrt((l)/(g))rArr g prop lT^(-2) = (Deltag)/(g)=(Delta l)/(l)+(2 Delta T)/(T)`
Therefore `P=((Delta l)/(l)+(2 Delta T)/(T))100 rArr P_(1)=((0.1)/(100)+(2(1))/(400))xx100 =0.6 % -P_(2)=((0.1)/(400)+(2(1))/(400))xx100 =0.42 %`
`P_(3)=((0.1)/(100)+(2(1))/(200))xx100=1.1 %,P_(4)=((0.1)/(400)+(2(1))/(800))xx100=0.28 %`.