Two intervals of time are measured as `Delta t_(1) = (2.00 pm 0.02) s and Delta t_(2) (4.00pm 0.02)s`. The value of `sqrt( ( Deltat_(1)) ( Deltat_(2)))` with correct significant figures and error is

Options

A. `(2.828 pm 0.01)s`
B. `(2.83 pm 0.01)s`
C. `(2.828 pm 0.0075)s`
D. `(2.83 pm 0.0075)s`

Answer

B

Explanation

Here, `Delta t_(1) = (2.00 pm 0.02) s and Delta t_(2) = ( 4.00 pm 0,02)s `
Given , `T = sqrt((Deltat_(1)) ( Deltat_(2)) ) = sqrt((2.00) (4.00)) = 2.828427 s`
Now, according to the relative error of product
`pm (Delta T)/T = pm ( 1/2 (Delta t_(1))/t_(1) + 1/2 (Delta t_(2))/t_(2))`
since, `Deltat_(1) and Delta_(2)` have 3 significant figures, then T also has 3 significant figures. Hence, after rounding off,
`therefore T= 2.83 s`
Now `Delta T = pm 1/2 ((0.02)/(2.00)+(0.02)/(4.00))xx 2.8284`
`DeltaT = 0.02121 s`
so, after rounding off,
`Delta T = 0.02`
Hence, `T = (2.83 pm 0.02)s`
so, most matched answer is (b).