# 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/s

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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. Find the percentage error.
1. 3.2%
2. 4.11%
3. 2.24%
4. 1.22%

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Correct Answer - Option 4 : 1.22%

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.
• Relative error: The ratio of the mean absolute error (or final absolute error) to the mean value of the experimental value measured.

$Percentage ~error = {Δa_{mean} \over a_{mean}}\times100$

EXPLANATION:

• 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
• The relative error is the ratio of the mean absolute error (or final absolute error) Δgmean to the mean value gmean of the experimental value measured.

$Relative~ error = {Δg_{mean} \over g_{mean}}$

The relative error is = ${0.12 \over 9.81}=0.0122324$

• The relative error is ≈ 0.01223

percentage error = relative error × 100

Percentage error = 0.01223 × 100 ≈ 1.22%

• So the correct answer is option 4.