Question

Part 1: (Theory) Simple Pendulum 1. Consider a mass m hanging from a string of length L that makes an angle with the vertical (shown below). Assume the string is massless and that the hanging object is a point mass. Use Newton's Second Law directly to show that the equation of motion for this simple pendulum can be written: (LO) = -mgsin(o), (1) dia where is the angular displacement of the pendulum from its vertical equilibrium position (and is a function of time). Notice that for small angles, sin(0) 0 (in radians), and so Eq. 1 simplifies to: m f(0) 20. (2) d12 2. Show by direct substitution that 0(0) - Oscos(wl ++) is a solution to B 2 if w=1, and thus - 2 Notice that the period T does not depend on m here, unlike in the case of the mass-spring system you studied last week. (Think about why this is!)

Answer 1

Dusplace the bob by small angle from its vertical position. Draw the free body duagram and apply Newtons second law of motion. Make assumption for the small angular displacement.

2 do ar 0 (t) = 0 cos (Wt to - Woo sinut to -w²0 cos (eet +O) d20 -w2 olt If war L d20 do? of this is same as so, ean is a solution 0(4) = 0o (o) (C++0) to eane does not depend Ta 20 29 or mass m