Propagating Error in a Pendulum

[nubey1]

Homework Statement

The length of a string attached to a pendulum is measured with a precision of (+or-)0.2. The time of the oscillation is measured to a precision of (+or-)0.1. How many periods must you measure so that the contribution of the uncertainty in time is smaller than the uncertainty in length, when calculating g?

2. Homework Equations

T=2pi(l/g)^(1/2)

3. The Attempt at a Solution

g=((2pi)^2(l+delta:l))/(T+delta:T)^2
I don't know where to go from here.
Delta l and T are the error in those measurements

 

[Jhero]

it is important to always consider the uncertainties and errors in our measurements and calculations. In this case, we need to determine how many periods we must measure in order for the uncertainty in time to be smaller than the uncertainty in length when calculating g.

To do this, we can use the equation for calculating g using the period and length of a pendulum, which is:

g = (4pi^2 * l) / T^2

Where l is the length of the string and T is the period of oscillation.

We also need to consider the uncertainties in these measurements, which are given as ±0.2 for length and ±0.1 for time.

To simplify our calculations, we can use the maximum possible uncertainties for both length and time, which would be +0.2 for length and -0.1 for time. This will give us the largest possible difference between the uncertainties in length and time.

Substituting these values into the equation for g, we get:

g = (4pi^2 * (l + 0.2)) / (T - 0.1)^2

Now, we need to determine the number of periods we need to measure in order for the uncertainty in time to be smaller than the uncertainty in length. To do this, we can set up an inequality where the uncertainty in time is less than the uncertainty in length:

0.1 < 0.2

Substituting our equation for g into this inequality, we get:

(4pi^2 * (l + 0.2)) / (T - 0.1)^2 < 0.2

We can then solve for T by multiplying both sides by (T - 0.1)^2 and rearranging the terms:

(T - 0.1)^2 > (4pi^2 * (l + 0.2)) / 0.2

T > ((4pi^2 * (l + 0.2)) / 0.2)^(1/2) + 0.1

This means that in order for the uncertainty in time to be smaller than the uncertainty in length, we need to measure at least T periods, where T is given by the above equation. Note that this is the minimum number of periods we need to measure, and we can always measure more if we want to reduce the uncertainty even further.

In conclusion, as a scientist