Question

Suppose that after a number of measurements, you were to find an average time ∆t = (0.0031±0.0002) s using a gate separation ∆d = (1.0±.1) cm when dropping a ball from the height h = (0.510±0.002) m. Calculate g. Which fractional uncertainty (∆t, ∆d, or h) is greater? Using this fractional uncertainty, find the uncertainty δg.

Answer 1

Measured values of t, d & h are:

(t±∆t) = (0.00310.0002)s

(d±∆d)=(1.0±0.1)cm

(h±∆h)=(0.510±0.002)m

Fractional error or uncertainty of t,d,h are as follows:

(∆t / t ) = (0.0002 / 0.0031) = 0.0645

(∆d / d) = (0.1 /1.0) = 0.1

(∆h / h) = (0.002 / 0.510) = 0.039

From above equations , it is concluded that fractional uncertainty in the measurement of 'd' is greater than 't' which is greater than that of 'h'.

Formula for time of fall of a freely falling body released from a height 'h' is : t =√(2h/g)

On squaring and rearranging the terms we get the .acceleration due to gravity at that place is g = 2h/t^{​​​​​​2}

^{​​​​​By applying logarithm on both sides,we get :}

=

On differentiation, we get

±(∆g /g) =±(2∆h /2h) - (±2∆t / t)

Maximum possible fractional error is :

(∆g / g) = 0.039 +2(0.0645) = 0.04 + 0.13 = 0.17