When finding uncertainties in measurements, I know that if you are adding or subtracting, you must add the uncertainties of the two values. But what do I do if I multiply/divide?
If X=A*B or X=A/B, then the uncertainty in X is:
X*sqrt( ((uncertainty in A)/A)^2 + ((uncertainty in B/B))^2 )
I'm not entirely sure what you mean, but after a thorough search through notes I think I might get it... correct me if I'm wrong?
I need to use relative uncertainties, and then use the relative uncertainty to find the uncertainty?
(I know how to do both of those, I just want to make sure that that is what I'm supposed to be doing.)
Thanks
Yes, since the uncertainty of a product or quotient of two values depends on the uncertainty of each of those two values. (The more uncertain the values, the more uncertain the product or quotient of those values.)
When dealing with multiplication or division, you need to consider the percentage uncertainties rather than absolute uncertainties. Here's what you need to do:
1. For multiplication: First, find the percentage uncertainty for each value by dividing its uncertainty by the measured value, and then multiply it by 100 to express it as a percentage. Add the percentage uncertainties for both values. Lastly, multiply this combined percentage uncertainty by the measured value obtained through multiplication.
Example:
Let's say you have measured a length of 10 cm with an uncertainty of 0.2 cm. You also have a width of 5 cm with an uncertainty of 0.1 cm. To find the area (length multiplied by width), follow these steps:
- Percentage uncertainty for length = (0.2 cm / 10 cm) x 100% = 2%
- Percentage uncertainty for width = (0.1 cm / 5 cm) x 100% = 2%
- Combined percentage uncertainty = 2% + 2% = 4%
- Area = 10 cm x 5 cm = 50 cm²
- Uncertainty in the area = 4% of 50 cm² = 2 cm²
Therefore, the area is 50 cm² with an uncertainty of ±2 cm².
2. For division: Similar to multiplication, calculate the percentage uncertainty for each value. Then, add the percentage uncertainties of both values and multiply the combined percentage uncertainty by the division result.
Example:
Let's say you measured a distance of 100 m with an uncertainty of 1 m. You also measured the time of 10 s with an uncertainty of 0.1 s. To find the average speed (distance divided by time), follow these steps:
- Percentage uncertainty for distance = (1 m / 100 m) x 100% = 1%
- Percentage uncertainty for time = (0.1 s / 10 s) x 100% = 1%
- Combined percentage uncertainty = 1% + 1% = 2%
- Average speed = 100 m / 10 s = 10 m/s
- Uncertainty in the average speed = 2% of 10 m/s = 0.2 m/s
Therefore, the average speed is 10 m/s with an uncertainty of ±0.2 m/s.