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Solved: The period T of a simple pendulum is given by
Chapter 3, Problem 5E(choose chapter or problem)
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.
Questions & Answers
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
=
=