A student measures three lengths a, b and c in cm and a time t in seconds:
a = 40 ± 5
b = 30 ± 3
c = 20 ± 1
t = 1.2 ± 0.1
What is a+b
a+b+c
a/t
(a+c)/t
Calculate (1.23 ± 0.03) + pi
Question 9.4. Calculate (1.23 ± 0.03) × pi
Can you do a sample calculation of each of those please? This is for error analysis.
Sure! To calculate these expressions, we need to consider both the values and uncertainties of the given quantities. The uncertainties are represented by the ± symbol.
Here's how to calculate each expression step by step:
1. a + b:
- Add the values of a and b: 40 + 30 = 70
- Add the uncertainties: 5 + 3 = 8
- The result is (70 ± 8)
2. a + b + c:
- Add the values of a, b, and c: 40 + 30 + 20 = 90
- Add the uncertainties: 5 + 3 + 1 = 9
- The result is (90 ± 9)
3. a / t:
- Divide the value of a by the value of t: 40 / 1.2 = 33.33...
- Calculate the fractional uncertainty: (uncertainty of a / value of a) + (uncertainty of t / value of t) = (5/40) + (0.1/1.2) = 0.125 + 0.083... = 0.208...
- Multiply the value by the fractional uncertainty: 33.33... * 0.208... = 6.94...
- The result is (33.33... ± 6.94...)
4. (a + c) / t:
- Add the values of a and c: 40 + 20 = 60
- Divide the result by the value of t: 60 / 1.2 = 50
- Calculate the fractional uncertainty: (uncertainty of a / value of a) + (uncertainty of c / value of c) + (uncertainty of t / value of t) = (5/40) + (1/20) + (0.1/1.2) = 0.125 + 0.05 + 0.083... = 0.258...
- Multiply the value by the fractional uncertainty: 50 * 0.258... = 12.92...
- The result is (50 ± 12.92...)
5. (1.23 ± 0.03) + pi:
- Add the value of 1.23 and the value of pi: 1.23 + π
- Add the uncertainties: 0.03
- The result is (1.23 + π ± 0.03)
6. (1.23 ± 0.03) × pi:
- Multiply the value of 1.23 by pi: 1.23 × π
- Calculate the fractional uncertainty: (uncertainty / value) = 0.03 / 1.23 ≈ 0.024...
- Multiply the value by the fractional uncertainty: (1.23 × π) * 0.024... ≈ 0.076...
- The result is (1.23 × π ± 0.076...)
I hope this helps! Let me know if you have any further questions.