Propagating Error in a Pendulum

In order for the uncertainty in time to be smaller than the uncertainty in length when calculating g, we need to measure at least T periods, where T is given by the equation above. In summary, we must measure a minimum of T periods, as determined by the equation (T > ((4pi^2 * (l + 0.2)) / 0.2)^(1/2) + 0.1), to ensure that the uncertainty in time is smaller than the uncertainty in length when calculating g.
  • #1
nubey1
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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
 
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  • #2


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
 

1. How do you calculate the error in a pendulum's period?

The error in a pendulum's period can be calculated using the formula: ΔT = 2π√(L/g) * ΔL, where ΔT is the error in period, L is the length of the pendulum, g is the acceleration due to gravity, and ΔL is the error in length.

2. What factors contribute to the error in a pendulum's period?

The main factors that contribute to the error in a pendulum's period are the length of the pendulum, the accuracy of the measuring instrument used to measure the length, and the accuracy of the measurement of the acceleration due to gravity.

3. How does the error in a pendulum's period affect its accuracy?

The error in a pendulum's period directly affects its accuracy. A larger error in period will result in a larger overall error in the time measurement of the pendulum's oscillations, making it less accurate.

4. Can the error in a pendulum's period be reduced?

Yes, the error in a pendulum's period can be reduced by using more accurate measuring instruments, taking multiple measurements and averaging them, and reducing any sources of external interference that may affect the pendulum's motion.

5. How does the error in a pendulum's period change with different lengths?

The error in a pendulum's period increases as the length of the pendulum increases. This is because a longer pendulum will have a larger period, making any error in measurement of length more significant. Therefore, it is important to use accurate measuring instruments when working with longer pendulums.

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