Question

If the length and time period of an oscillating pendulum have error in estimate of $ 1\% $ and $ 2\% $ respectively, then the percentage error in estimate of $ g $ is:
(A) $ 1\% $
(B) $ 2\% $
(C) $ 3\% $
(D) $ 5\% $

Answer 1

Hint :Here, the length of the pendulum has the error of $ 1\% $ and the time period has the error of $ 2\% $ and we have to calculate the error in $ g $ . As the time period is dependent on the acceleration due to gravity. First we have to place these errors in the formula $ \dfrac{{\Delta x}}{x} = \% error $ .

Complete Step By Step Answer:
Let $ L $ be the length of the pendulum and $ \Delta L $ be the difference between actual length and calculated length. Similarly, $ T $ be the time period and $ \Delta T $ be the difference, $ g $ be the acceleration due to gravity and $ \Delta g $ be the difference.
Now, let us put the values in the formula of percentage error as:
 $ \dfrac{{\Delta L}}{L} = 1\% $ And $ \dfrac{{\Delta T}}{T} = 2\% $ …… (Given)
No, we use the formula for time period to calculate percentage error in $ g $ .
 $ T = 2\pi \sqrt {\dfrac{L}{g}} $ $ \Rightarrow g = 4\pi \dfrac{L}{{{T^2}}} $
 $ \Rightarrow \dfrac{{\Delta g}}{g} \times 100 = \left[ {\dfrac{{\Delta L}}{L} \times 100} \right] + 2\left[ {\dfrac{{\Delta T}}{T} \times 100} \right] $
 $ \Rightarrow \dfrac{{\Delta g}}{g} = 1\% + 2(2\% ) = 5\% $
The percentage error in the acceleration due to gravity is $ 5\% $ after calculating above equations.
Thus, the correct answer is option D.

Note :
Here, we have used the formula of the percentage error and put all the given values in the formula $ \dfrac{{\Delta x}}{x} = \% error $ . Thus, we observed the acceleration due to gravity and its percentage error. So, we have to calculate the above equations carefully and relate all the values given in the problem. This type of error problems are observed when there is calculative or observational error in the values of some observations while experimenting.