Prelab_Centripetal Force

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Indiana University, Bloomington *

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Aerospace Engineering

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Oct 30, 2023

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Pre-Lab: Centripetal Force Introduction This pre-lab focuses on preparing you for the uncertainty calculations in the lab. First, it includes a general discussion on how to calculate the uncertainties for the centripetal force equation. Next, sample data is used to calculate relative uncertainties. Finally, these relative uncertainties are combined in quadrature to obtain the uncertainty for the centripetal force. Submit a clear image or scan of your work, including both calculations and final numbers. Discussion on How to Calculate Uncertainties In the Introduction to the lab, we provide the centripetal force equation: 𝐹𝐹 𝐶𝐶 = 4 𝜋𝜋 2 𝑚𝑚𝑚𝑚 𝑇𝑇 2 In this equation, the 4 and π are constants. These will have no effect on the final uncertainty since they are exact numbers. This leaves us with the variables m, r and T . Recalling the Rules A ddition begins with “A”, as does a bsolute. Combine a bsolute uncertainties when doing a dditions or subtractions. Multiplication and divisions are r atios. R atios begins with “R”, as does r elative. Use r elative uncertainties for multiplications or divisions. Applying the Rules The centripetal force equation does not have any additions or subtractions, only multiplications and divisions. Therefore, we will use relative uncertainties for the entire calculation.

Sample Data m = 170.8 + 0.1 g r = 16.4 + 0.1 cm W clicker = 0.4 cm Period of Rotation Data Trial Time for 10 rotations (s) 1 8.62 2 8.63 3 8.56 4 8.66 5 8.54 6 8.71 7 8.56 8 8.65 Uncertainty in the Period T We have multiple measurements of the rotational period, T . Therefore, we can find its mean and standard deviation. The uncertainty will be the standard deviation of the mean, the standard deviation divided by the square root of the number of trials. One thing to keep in mind is that we are not timing just one rotation at a time in this lab. If we did, we could do the calculation and be done! However, this value would have a large uncertainty because of the reaction time of whoever makes the measurement. For example, if the period of rotation is 1 s and the person timing has a reaction time of 0.5 s (which is not unusual), the time for one period would be 1.0 + 0.5 s, which has a 50% uncertainty! Measuring ten rotations will not affect the person’s reaction time, but it does give us a much longer time to measure. Our time for one period is now 10.0 + 0.5 s, which is a 5% uncertainty . How does this affect your results? You have the time for 10 rotations, so to get the time for 1 rotation, you need to divide your mean value and standard deviation by 10. Keep in mind that this is different than dividing by the square root of the number of trials to get the standard deviation of the mean! You will need to divide the standard deviation twice: once by the number of rotations, and once by the square root of the number of trials. Exercise 1 Calculate the best value and uncertainty for the period T using the sample data. Convert the absolute uncertainty for the period T to a relative uncertainty.

Uncertainty in the Mass m Fortunately, finding the uncertainty in the mass of the bob is easy! The absolute uncertainty was determined from the scale used to measure it. Exercise 2 Convert the absolute uncertainty in the mass of the bob to a relative uncertainty. Uncertainty in the Radius r We just had to convert δ m from an absolute to relative uncertainty. Can we do the same for δ r? Unfortunately not... The radius of rotation actually has two sources of uncertainty: the uncertainty in our ruler, and an experimental uncertainty due to the width of the clicker. The value of r given in the sample data was measured with a ruler, so it has an uncertainty around δ r ruler = 0.1 cm. The width of the clicker is typically about W clicker = 0.4 cm. Hearing the click is how we maintain a constant radius. However, since the clicker will click if the bob passes anywhere along the radius, this acts as another source of uncertainty. In our uncertainty equation, we use half the total width as the uncertainty. Ignoring this would cause us to greatly underestimate our total uncertainty. There may be even more sources of uncertainty. In the lab, you will be asked to assess whether there is a wobble to the entire system that may contribute. You should always be looking for these additional factors and including them in your calculations whenever possible. For the pre-lab, however, we will stick with these two. Recall that to combine two uncertainties, we need to propagate them in quadrature. Since we want to add uncertainties, we will do this with absolute uncertainties . Combining these together yields: 𝛿𝛿𝑚𝑚 = � ( 𝛿𝛿𝑚𝑚 𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 ) 2 + � 1 2 𝑊𝑊 𝑐𝑐𝑟𝑟𝑐𝑐𝑐𝑐𝑐𝑐𝑟𝑟𝑟𝑟 � 2 Exercise 3 Showing your work, combine the absolute uncertainties in quadrature to get the absolute uncertainty δ r. Co nvert the absolute uncertainty δ r to a relative uncertainty.

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Uncertainty in the Centripetal Force F C Now we have the uncertainty for each of our variables: m, r and T . We are ready to combine these relative uncertainties in quadrature. Recall that since the equation contains entirely multiplications and divisions, we only need relative uncertainties for this. The quadrature equation for the uncertainty in F C is: 𝛿𝛿𝐹𝐹 𝐶𝐶 𝐹𝐹 𝐶𝐶 = � � 𝛿𝛿𝑚𝑚 𝑚𝑚 � 2 + � 𝛿𝛿𝑚𝑚 𝑚𝑚 � 2 + � 𝛿𝛿𝑇𝑇 2 𝑇𝑇 2 � 2 We have all of our relative uncertainties, so we should be ready to go! ...Right? If you’re paying attention, you may have noticed that we have the relative uncertainty 𝛿𝛿𝛿𝛿 𝛿𝛿 , but the equation calls for 𝛿𝛿𝛿𝛿 2 𝛿𝛿 2 . Fortunately, making this transition is not as difficult as you may think. In Appendix B, there is a section on “Power Rules for Uncertainties in Calculations”. From this, we can use the rule: When squaring a number, double the relative uncertainty. Taking this into account, we can rewrite the quadrature equation to use the relative uncertainty 𝛿𝛿𝛿𝛿 𝛿𝛿 : 𝛿𝛿𝐹𝐹 𝐶𝐶 𝐹𝐹 𝐶𝐶 = � � 𝛿𝛿𝑚𝑚 𝑚𝑚 � 2 + � 𝛿𝛿𝑚𝑚 𝑚𝑚 � 2 + 2 � 𝛿𝛿𝑇𝑇 𝑇𝑇 � 2 Exercise 4 Showing your work, calculate the relative uncertainty in centripetal force using the relative uncertainties for m, r and T .

Work to Submit for this Pre-Lab Exercise 1 Calculate the best value and uncertainty for the period T using the sample data. Convert the absolute uncertainty for the period T to a relative uncertainty. Exercise 2 Convert the absolute uncertainty in the mass of the bob to a relative uncertainty. Exercise 3 Showing your work, combine the absolute uncertainties in quadrature to get the absolute uncertainty δ r. Convert the absolute uncertainty δ r to a relative uncertainty. Exercise 4 Showing your work, calculate the relative uncertainty in centripetal force using the relative uncertainties for m, r and T .