The length of a pendulum is (1.5 ± 0.01) m and the acceleration due to gravity is taken into account as (9.8 + 0.1) ms⁻². Calculate the time period of the pendulum with uncertainty in it. (2.5 ± 0.8 %) 


Given:

Length of the pendulum = l = (1.5 ± 0.01) m 
Acceleration due to gravity = g = (9.8 ± 0.1) ms⁻² 


To Find :

The time period of the pendulum = T = ?


Solution:

For division, percentage uncertainties are added. So, converting the fractional uncertainty to percentage uncertainty
l = (1.5 ± 0.01) m = (1.5 ± 0.011.5ｘ100) m = (1.5 ± 0.667% ) m
g = (9.8 ± 0.1) m s⁻² = (9.8 ± 0.19.8ｘ100) m s⁻² = (9.8 ± 1.02%) m s⁻²

Now by using the Time Period formula 

T = 2𝝿 √lg

T = 2𝝿 √(1.5±0.687%)m(9.8±1.02%)ms⁻²

The fraction/percentage uncertainty is added in addition, subtraction, multiplication, and division. So,

T = 2𝝿 √1.59.8s⁻²±(0.667%+1.02%)

T = 2𝝿 √1.59.8s⁻²± √0.667%+1.02%

T = 2𝝿 (0.39) s ±  (1.687%)12

the power percentage uncertainty is multiply with the power

T = 2.458 s ±  12(1.69% )

T = 2.458 s ±  0.843 %

or

T = (2.458  ±  0.8 %) s   ------------Ans

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