Solved: The period T of a simple pendulum is given by

Chapter 3, Problem 5E

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QUESTION:

The period T of a simple pendulum is given by \(T=2 \pi \sqrt{L / g}\) where L is the length of the pendulum and g is the acceleration due to gravity.

a. Assume \(g=9.80 \mathrm{\ m} / \mathrm{s}^{2}\) exactly, and that \(L=0.742 \pm 0.005 \mathrm{\ m}\). Estimate T, and find the uncertainty in the estimate.

b. Assume \(L=0.742 \mathrm{\ m}\) exactly, and that \(T=1.73 \pm 0.01 \mathrm{\ s}\). Estimate g, and find the uncertainty in the estimate.

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QUESTION:

The period T of a simple pendulum is given by \(T=2 \pi \sqrt{L / g}\) where L is the length of the pendulum and g is the acceleration due to gravity.

a. Assume \(g=9.80 \mathrm{\ m} / \mathrm{s}^{2}\) exactly, and that \(L=0.742 \pm 0.005 \mathrm{\ m}\). Estimate T, and find the uncertainty in the estimate.

b. Assume \(L=0.742 \mathrm{\ m}\) exactly, and that \(T=1.73 \pm 0.01 \mathrm{\ s}\). Estimate g, and find the uncertainty in the estimate.

ANSWER:

Solution 5E

Step1 of 3:

Let T be the period of of a simple pendulum and it is given by

Where,

L = length of the pendulum

g = acceleration due to gravity

Here our goal is:

a).We need to Estimate T, and find the uncertainty in the estimate by Assuming g = 9.80 exactly, and that L = 0.7420.005m

b).We need to Estimate g, and find the uncertainty in the estimate by Assuming L = 0.742 m exactly, and that T = 1.730.01 s.


Step2 of 3:

a).

From the given information we have g = 9.80, L = 0.742,

Here  

mathematical constant and its value is 3.14

         

                                   =  

                       = 6.28

                     = 6.28(0.2751)

           = 1.7280

Hence,  1.7280.

Consider,

Differentiate above equation with respect to “L” we get

Let u = L/g

     du = 1/g

Apply chain rule to above equation to get a differential equation and it is given by

     

     =

  =

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