Question

The time period of oscillation of a simple pendulum is \[T=2\pi \sqrt{\dfrac{L}{g}}\]. Measured value of length is 20cm known to have 1mm accuracy and the time for 100 oscillation of the pendulum is found to be 90s using wrist watch of 1s resolution.what is the accuracy in determination of g?
A. 2.5%
B. 2.7%
C. 2%
D. 3%

Answer 1

Hint: We must have a basic idea of the combination of errors. We will firstly find g in terms of T. Then we will use the formula to find percentage error and by doing suitable calculations after substituting the given values in the formula, we will obtain our desired result.

Formula used:
\[\begin{align}
  & z=\dfrac{{{A}^{p}}{{B}^{q}}}{{{C}^{r}}} \\
 & \dfrac{\Delta Z}{Z}=p\dfrac{\Delta Z}{Z}+q\dfrac{\Delta Z}{Z}+r\dfrac{\Delta Z}{Z} \\
\end{align}\]

Complete step by step answer:
As given in the quention
\[T=2\pi \sqrt{\dfrac{L}{g}}\]
Firstly we will change the equation in terms of g.
We will square on both the sides .
\[\begin{align}
  & {{T}^{2}}=4{{\pi }^{2}}\dfrac{L}{g} \\
 & g=4{{\pi }^{2}}\dfrac{L}{{{T}^{2}}} \\
\end{align}\]
Now we will use the formula to find percentage error in g we know the formula is
\[\dfrac{\Delta Z}{Z}\times 100=p\dfrac{\Delta Z}{Z}\times 100+q\dfrac{\Delta Z}{Z}\times 100+r\dfrac{\Delta Z}{Z}\times 100\]
Now L=20.0 cm , \[\Delta \]L=1mm =0.1 cm ,T for 100 oscillations =90 s , \[\Delta \]T=1s.
\[\therefore \dfrac{\Delta g}{g}\times 100=\dfrac{\Delta L}{L}\times 100+2\times \dfrac{\Delta T}{T}\times 100\]
Now putting values we get
\[\begin{align}
  & \therefore \dfrac{\Delta g}{g}\times 100=\dfrac{0.1}{20}\times 100+2\times \dfrac{1}{90}\times 100 \\
 & =0.5+2.22=2.72\%=3\% \\
\end{align}\]
Accuracy in determination of g is 3%.

Hence, the correct option is (D).

Additional Information:
All measurements of physical quantities are uncertain and imprecise to some limit. There are three sources of errors. Negligence or inexperience of a person. Faulty apparatus. Inappropriate method or technique. Difference between error and uncertainties.
The basic difference between errors and uncertainties is that error is the difference between the calculated value and actual value, while uncertainty is usually described as an error in measurement.

Note:
We know in all mathematical operations errors are of additive nature. If a quantity appears with a powerless than one in an expression, then its error contribution in the final result is reduced. Calculations must be performed carefully. There are two different formulas for calculating errors and a percentage so one should be careful while using it.