Question

The standard gravity acceleration is 9.81 m/s^2 In the laboratory conducting some experiments using a pendulum we obtain four values for g = 9.76 m/s^2, 9.6 m/s^2 and 9.89 m/s^2 . The variation in values is the product of errors. Find the uncertainty range in this measurement.

Answer 1

Answer: The uncertainty range in this measurement is approximately ±0.122 m/s². This means the true value of gravity (g) is likely to be within 0.122 m/s² of the measured values (9.76 m/s², 9.6 m/s², 9.89 m/s², etc.).

Step-by-step explanation:

To find the uncertainty range in this measurement, we first need to calculate the error for each measurement and then determine the overall uncertainty.

Step 1: Calculate the errors for each measurement:

Error for each measurement = |Measured value - Standard value|

For g = 9.76 m/s²:

Error = |9.76 m/s² - 9.81 m/s²| = 0.05 m/s²

For g = 9.6 m/s²:

Error = |9.6 m/s² - 9.81 m/s²| = 0.21 m/s²

For g = 9.89 m/s²:

Error = |9.89 m/s² - 9.81 m/s²| = 0.08 m/s²

Step 2: Calculate the product of errors (variation in values):

Product of errors = Error1 * Error2 * Error3 * Error4

Product of errors = 0.05 m/s² * 0.21 m/s² * 0.08 m/s² = 0.00084 m²/s⁴

Step 3: Find the overall uncertainty range:

The overall uncertainty range is typically given as the standard deviation of the measurements. Since we have only four values, we can use the sample standard deviation formula:

Sample standard deviation = sqrt((sum of (measurement - mean)²) / (n - 1))

Where "n" is the number of measurements (in this case, n = 4).

Let's calculate the mean of the measurements first:

Mean = (9.76 m/s² + 9.6 m/s² + 9.89 m/s² + 9.81 m/s²) / 4 = 9.765 m/s²

Now, calculate the sum of (measurement - mean)² for each measurement:

(9.76 m/s² - 9.765 m/s²)² = 0.000025 m²/s⁴

(9.6 m/s² - 9.765 m/s²)² = 0.027025 m²/s⁴

(9.89 m/s² - 9.765 m/s²)² = 0.015625 m²/s⁴

(9.81 m/s² - 9.765 m/s²)² = 0.002025 m²/s⁴

Now, calculate the sum:

Sum of (measurement - mean)² = 0.000025 m²/s⁴ + 0.027025 m²/s⁴ + 0.015625 m²/s⁴ + 0.002025 m²/s⁴ = 0.0447 m²/s⁴

Now, calculate the sample standard deviation:

Sample standard deviation = sqrt(0.0447 m²/s⁴ / (4 - 1)) = sqrt(0.0149 m²/s⁴) ≈ 0.122 m/s²

Therefore, the uncertainty range in this measurement is approximately ±0.122 m/s². This means the true value of gravity (g) is likely to be within 0.122 m/s² of the measured values (9.76 m/s², 9.6 m/s², 9.89 m/s², etc.).