Uncertainty of measurements

[trollcast]

Homework Statement

The experiment set up is a steel ball bearing is held from an electromagnet and above a switch at the bottom. When the electromagnet is switched off a timer starts and stops when it hits the switch.

The height from the bearings initial position to the switch is measured.

These 2 measurements are then used to calculate g using the formula, $$g=\frac{2s}{t^2}$$

Which of your 2 variables uncertainties is the most significant in determining a value for g?

Homework Equations

The Attempt at a Solution

I said the time taken as in our equation its squared whereas the distance is only multiplied by 2 so any error in the value will be magnified more.

However the mark scheme says either is correct given a logically and scientifically correct reason.

 

[rock.freak667]

Given that equation, how would you determine the uncertainty in g?

 

[trollcast]

Given that equation, how would you determine the uncertainty in g?

The uncertainty of g would be equal to $$ s / 2t $$

Where s and t are the uncertainties of the same variables.

 

[rude man]

g = 2s/t²
dg = ∂g/∂s ds + ∂g/∂t dt

So the absolute error dg is
dg = 2/t² ds due to an absolute error ds, and
dg = -4s/t³ dt due to an absolute error dt.

However, if we're talking percentage errors,

dg/g = 2t^-2/2st^-2 ds = ds/s and
dg/g = -4st^-3/2st^-2 dt = -2dt/t.

Thus a percentage error in t is twice as bad as a percentage error in s. I don't know if that's what the problem was asking for ...

 

[Nickie]

Both the time and distance measurements have their own uncertainties, and it is difficult to determine which one is more significant in determining a value for g. However, it can be argued that the time measurement may have a larger impact on the overall calculation of g.

This is because the equation for g involves the squared value of time, which means any error in the time measurement will be magnified when calculating g. On the other hand, the distance measurement is only multiplied by a constant value of 2, so any error in this measurement will not have as significant of an impact on the final calculation of g.

Additionally, the time measurement may also be affected by external factors such as reaction time or human error, which can further increase its uncertainty. Therefore, it is important to carefully measure and minimize the uncertainty of both time and distance measurements in order to obtain a more accurate value for g.