The bob of a pendulum of length L is pulled aside so the string makes an angle θ₀ with the vertical, and the bob is then released. In Example 7-2, 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 Newton's second law. 

(a) Show that the tangential component of Newton's second law gives dv/dt = –g sin θ, where v is the speed and θ is the angle made by the string and the vertical. 

(b) Show that v can be written v = L dθ/dt. 

(c) Use this result and the chain rule for derivatives to obtain

(d) Combine the results of (a) and (c) to obtain 

vdv= -gL sin θdθ

(e) Integrate the left side of the equation in part (d) from v = 0 to the final speed v and the right side from θ = θ₀ to θ = 0, and show that the result is equivalent to v = √2gh₀, where h is the original height of the bob above the bottom.

Transcribed Image Text:

dv v dv dv d0 dt d0 dt