# Uncertainties help

1. Jun 18, 2010

### spoony33

I have this question for homework and i'm well stuck!

The specific heat capacity of a liquid was found by heating a measures quantity of the liquid for a certain length of time. The following results were obtained.
Power of heater: ( 50.0 ± 0.5)W
Mass of liquid: (200 ± 10)g
Time of heating: (600 ± 1)s
Temperature rise: (50.0 ± 0.5)°C

I managed to get the percentage error in each reading but i just don't know how to work out the approximate percentage error in the value of specific heat capacity!

A) What will the approximate percentage error in the value of specific heat capacity?
B) Suggest one way in which to reduce the percentage error obtained for the specific heat capacity?

I managed to work out this but i dont know how to work out the approximate percentage error in the value of specific heat capacity?
The percentage of uncertainty of each is:
Power of heater:
(±0.5)/50.0 = ± 1%

Mass of liquid
(±10)/200 = ± 5%

Time of heating:
(±1)/600 = ±0.167 %

Temperature Rise:
(±.5)/50.0 = ± 1%

Thanks

2. Jun 18, 2010

### Mindscrape

Nah, you have to get the total uncertainty first.

So Q=mcT

I don't remember how the uncertainties propagate for products, but it's something you can look up, or if you know partial derivatives you can use the "master formula."

3. Jun 18, 2010

### Jolsa

For products, the relative error on the measured quantity is equal to the sum of the relative errors. So if

$$z=x\cdot y$$

then

$$\frac{\Delta z}{z}=\frac{\Delta x}{x}+\frac{\Delta y}{y}$$

A more general approach is the formula for uncertainty on a measured quatity f(x,y). The absolute error in f is given by

$$\Delta f=\frac{\partial f}{\partial x} \Delta x+\frac{\partial f}{\partial y} \Delta y$$

and the standard deviation is given by

$$\sigma _f=\sqrt{(\frac{\partial f}{\partial x} \Delta x)^2+(\frac{\partial f}{\partial y} \Delta y)^2}$$

If you plug $$f(x,y)=x \cdot y$$ into the formula for absolute uncertainty and rearrange a little, you'll notice that it reduces to the first formule for absolute error for products.