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Gravitational acceleration at a place using a second pendulum is calculated. The reading comes out to be 9.86 m/s2, 9.72 m/s2, 9.64 m/s2, and 9.98 m/s2. Calculate the range where gravitational acceleration lies in that place.
1. between 9.69 m/s2 and 9.93 m/s2
2. between 9.69 m/s2 and 9.81 m/s2
3. between 9.81 m/s2 and 9.93 m/s2
4. between 9.86 m/s2 and 9.98 m/s2

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Correct Answer - Option 1 : between 9.69 m/s2 and 9.93 m/s2

CONCEPT:

  • Error: The result of every measurement of experiments by any measuring instrument contains some uncertainty. This uncertainty is called error. 
  • Absolute error: The magnitude of the difference between the measurement of the experimental readings and the true value of the quantity is called the absolute error.
    • This is denoted by |Δa |. 
    • Absolute error is always positive.
    • If there is no true value, we can take the mean of all measured values as a true value.

CALCULATION:​

  • The mean value of gravitational acceleration

\(g= {9.86\; + \;9.72\; + \;9.68\; + \;9.98 \over 4}=9.81\)

  • Absolute error is the magnitude of the difference between the measured value while doing the experiment measurement and the true value.
  • When there is no true value, we take mean value of measurements as a true value. 

So true value of g = 9.81

  • Now, The errors in the measurements are

Absolute error = |measured value - true value|

Now, The errors in the measurements are

|9.86 - 9.81| = |0.05| = 0.05

|9.72 - 9.81| = |-0.09| = 0.09 

|9.64 - 9.81| = |-0.17| = 0.17

|9.98 - 9.81| = |0.17| = 0.17

  • The arithmetic mean of all the absolute errors (for the arithmetic mean, we take only the magnitudes i.e. positive value) is

\(\Delta g_{mean} = {0.05\;+\;0.09\;+\;0.17\;+\;0.17 \over 4}\)

\(\Delta g_{mean} = 0.12\)

  • So mean absolute error is 0.12 m/s2
  • That means the gravitational acceleration at that place is (9.81 ± 0.12) m/s2 i.e.

It lies between (9.81 + 0.12) m/s2 and (9.81 - 0.12) m/s2 or between 9.69 m/s2 and 9.93 m/s2

  • Hence the correct answer is option 1.

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