The bob of a pendulum of length L is pulled aside so that

Chapter , Problem 106

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The bob of a pendulum of length L is pulled aside so that the string makes an angle with the vertical, and the bob is then released. In Example 7-5, the conservation of energy was used to obtain the speed of the bob at the bottom of its swing. In this problem, you are to obtain the same result using Newtons second law. (a) Show that the tangential component of Newtons second law gives where v is the speed and is the angle between the string and the vertical. (b) Show that v can be written (c) Use this result and the chain rule for derivatives to obtain (d) Combine the results of Parts (a) and (c) to obtain (e) Integrate the left side of the equation in Part (d) from to the final speed v and the right side from to and show that the result is equivalent to where h is the original height of the bob above the bottom of its swing.