How Do Measurement Uncertainties Impact the Calculation of Gravity?

[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 ...