How Many Pendulum Periods Minimize Time Uncertainty in Gravity Calculations?

In summary, the conversation discusses the measurement of a pendulum's length and time of oscillation with certain precisions. The goal is to determine how many periods must be measured in order to have an uncertainty in time that is smaller than the uncertainty in length when calculating the value of g. The attempt at a solution involves an equation for g that takes into account the errors in length and time measurements. It is suggested to assume that the errors are small relative to the actual values and an error equation for g is provided using calculus. Without calculus, it is important to know what uncertainty relations are given to guide the next steps in the solution.
  • #1
nubey1
5
0

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?

Homework Equations



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

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
nubey1 said:

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?

Homework Equations



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

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.

The next thing to do is assume that the error is really small relative to the actual value.
[tex]\delta l << l[/tex] and [tex]\delta T << T[/tex]. I think you might have an error in your equation for g:
[tex] g +\delta g= (2 \pi)^2 \frac{l+\delta l }{(T+\delta T)^2}[/tex] where [tex] \delta g[/tex] is the error in g.
If you are familiar with calculus then this comes out to:
[tex]\delta g = |\frac{\partial g}{\partial l}| \delta l + |\frac{\partial g}{\partial T}| \delta T[/tex]
If you don't have the luxury of calculus we might need to know what relations for uncertainty you are given to clue you in
 
  • #3


I understand the importance of accurately measuring and minimizing error in experiments. In this case, we are looking at the measurement of a pendulum's length and time of oscillation and how it affects the calculation of g, the acceleration due to gravity. The length of the string is measured with a precision of (+/-) 0.2 and the time of oscillation is measured with a precision of (+/-) 0.1.

To determine the number of periods needed to minimize the contribution of uncertainty in time, we can use the equation T=2pi(l/g)^(1/2). This equation shows that g is directly proportional to the square of the time (T) and inversely proportional to the square root of the length (l). Therefore, to minimize the contribution of uncertainty in time, we want to increase the number of periods measured, which will result in a larger T value.

To determine the minimum number of periods needed, we can use the equation for error propagation: delta g = ((2pi)^2(l+delta:l))/(T+delta:T)^2. Here, delta l and delta T represent the error in the measurements of length and time, respectively.

To make the contribution of uncertainty in time smaller than the uncertainty in length, we want delta g to be smaller than delta l. This means we need to increase the value of T until delta g is smaller than delta l. This can be achieved by increasing the number of periods measured until the value of delta g is smaller than delta l.

In conclusion, to minimize the contribution of uncertainty in time when calculating g, we need to measure enough periods so that the value of delta g is smaller than delta l. This can be achieved by increasing the number of periods measured until the value of delta g is smaller than delta l.
 

Related to How Many Pendulum Periods Minimize Time Uncertainty in Gravity Calculations?

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

To calculate the error in a pendulum's period, you need to know the length of the pendulum, the acceleration due to gravity, and the uncertainty in these measurements. The error in the period can be calculated using the formula: ∆T = 2π∆L/g, where ∆T is the error in the period, ∆L is the uncertainty in the length, and g is the acceleration due to gravity.

2. What is the effect of air resistance on the error in a pendulum's period?

Air resistance can cause the pendulum to lose energy, resulting in a slightly shorter period. This can lead to a larger error in the period, as the pendulum's motion may not be as consistent due to the varying effects of air resistance. It is important to minimize air resistance when conducting experiments with pendulums to reduce the error.

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

Yes, the error in a pendulum's period can be reduced by taking multiple measurements and finding the average, using more precise measurement tools, and minimizing external factors such as air resistance. Additionally, using a larger pendulum with a longer period can also decrease the relative error.

4. What is the difference between random and systematic errors in a pendulum's period?

Random errors in a pendulum's period are caused by factors that are unpredictable and can vary from measurement to measurement, such as human error or environmental conditions. Systematic errors, on the other hand, are caused by consistent factors that affect all measurements in the same way, such as a misaligned timer or a pendulum with a slightly uneven weight distribution.

5. How does the length of a pendulum affect the error in its period?

The length of a pendulum can affect the error in its period in two ways. First, a longer pendulum will have a longer period, which can lead to a smaller relative error. Second, if the length of the pendulum is not measured accurately, it can introduce a larger error in the period. Therefore, it is important to measure the length of the pendulum as precisely as possible to minimize error.

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