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(II) Derive a formula for the maximum speed $v_{\max }$ of a simple pendulum bob in terms of $g$ , the length $\ell$ , and the maximum angle of swing $\theta_{\max }$ .

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$$\sqrt{2 g L\left(1-\cos \theta_{\max }\right)}$$

03:19

Shital Rijal

Physics 101 Mechanics

Chapter 14

Oscillators

Motion Along a Straight Line

Motion in 2d or 3d

Periodic Motion

University of Michigan - Ann Arbor

Hope College

McMaster University

Lectures

04:01

2D kinematics is the study of the movement of an object in two dimensions, usually in a Cartesian coordinate system. The study of the movement of an object in only one dimension is called 1D kinematics. The study of the movement of an object in three dimensions is called 3D kinematics.

02:18

In physics, an oscillation is the repetitive variation, typically in time, of some measure about a central value or between two or more different states. The oscillation may be periodic or aperiodic.

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for this problem on the topic of oscillators, we are asked to imagine a simple pendulum bob and derive a formula for the maximum speed of this bob in terms of acceleration due to gravity, G the length of the pendulum, L. And the maximum angle of swing. Betamax. So if we can use energy conservation to relate the potential energy at the maximum height of the pendulum to the kinetic energy at the lowest point swing. If we take the lowest point to be the zero location for gravitational potential energy and using the diagram, we have, the energy at the top of the pendulum swing is equal to the mechanical energy at the bottom of it swing and so mm kinetic energy at the top plus the potential energy at the top must equal the kinetic energy at the bottom, thus the potential energy at the bottom. And so if we write the expressions for each one of these, the kinetic energy, the top is zero. Since the pendulum is moment momentarily addressed last, its gravitational potential energy MGH must equal to its kinetic energy at the bottom, at which point it's gravitational potential energy zero, so that's a half and times the maximum velocity squared. Which means if we rearrange we can find an expression for this. Maximum velocity is too quiet, so max Is equal to the square root of two G. H. To get this in terms of the length of the pendulum, using geometry becomes two G oh Into 1- Callsign Theatre.

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