How do you handle uncertainties in derived quantities?

Uncertainties in derived quantities are handled by using the rules of propagation of uncertainties.

When you're dealing with derived quantities in physics, it's important to understand that uncertainties can propagate through your calculations. This means that the uncertainty in your final result is influenced by the uncertainties in the quantities you used to calculate it. The rules of propagation of uncertainties provide a systematic way to calculate this.

If you're adding or subtracting quantities, the absolute uncertainties add up. For example, if you're measuring the length of a table using a ruler with an uncertainty of ±1 cm, and you make two measurements, the total uncertainty is ±2 cm. This is because the uncertainties could add up in the worst-case scenario.

When you're multiplying or dividing quantities, or raising them to a power, you need to add the relative (or percentage) uncertainties. For example, if you're calculating the area of a rectangle, you multiply the length by the width. If the length has a relative uncertainty of 2% and the width has a relative uncertainty of 3%, the area has a relative uncertainty of 2% + 3% = 5%. This is because the uncertainties could multiply in the worst-case scenario.

If you're dealing with a more complex equation, you might need to use calculus to calculate the propagation of uncertainties. This involves taking the derivative of the equation with respect to each variable, multiplying by the uncertainty in that variable, and then adding these up in quadrature (i.e., squaring each one, adding them up, and taking the square root). This gives you the total uncertainty in your derived quantity.

Remember, handling uncertainties correctly is crucial in physics. It allows you to quantify how confident you can be in your results, and it's a key part of any experimental investigation. So, make sure you understand the rules of propagation of uncertainties and apply them correctly in your work.