The period T of a simple pendulum is given by where L is the length of the pendulum and g is the acceleration due to gravity.

a. Assume g = 9.80 m/s2 exactly, and that L= 0.742 ± 0.005 m. Estimate T, and find the uncertainty in the estimate.

b. Assume L = 0.742 m exactly, and that T = 1.73 ± 0.01 s. Estimate g, and find the uncertainty in the estimate.

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

=

=

Substitute “u” value in above equation we get

=

=

=

=

=

= 1.1645

Hence, = 1.1645.

Now,

The estimate of uncertainty is given by

= (1.1645)(0.005)

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