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Dustgil
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Homework Statement
A cockroach crawls with constant speed in a circular path of radius b on a phonograph turntable rotating with constant angular speed omega. The circle path is concentric with the center of the turntable. If the mass of the insect is m and the coefficient of static friction with the surface is the table is mu sub s, how fast, relative to the turntable, can the cockroach crawl before it starts to slip if it goes (a) in the direction of rotation and (b) opposite to the direction of rotation?
Homework Equations
The Attempt at a Solution
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Best I saw was to pick the reference frame of the turntable. In this fram, the cockroach walks in a circle of radius b.
[tex]
r^{'} = bcos\theta i+bsin\theta j[/tex]
[tex]v^{'} = -bsin\theta i+bcos\theta j[/tex]
[tex]a^{'} = -bcos\theta i - bsin\theta j[/tex]
The transverse force is zero (the reference frame is rotating with constant angular velocity), so the equation of motion reads as
[tex]F - 2m\omega \times v^{'}-m\omega \times (\omega \times r^{'}) = ma^{'}[/tex]
Omega is entirely in the k direction, and the only force acting on the cockroach in the reference frame is the frictional force, so by evaluating the cross products the equation becomes
[tex]F_{f}= -2m\omega b(cos\theta i+ sin\theta j)-m\omega ^{2}b(cos\theta i -sin\theta j) -mbcos\theta i - mbsin\theta j[/tex]
We can then separate this into components and solve for the max forces in either direction. Since the normal force and the graviational force oppose each other, the frictional force must be less than [tex] \mu_{s}mg[/tex]
so, in the x direction for example,
[tex]-2m\omega b cos\theta - m\omega^{2} b cos\theta-mbcos\theta < \mu_{s}mgcos\theta[/tex]
[tex]-2\omega b - \omega^{2} b-b < \mu_{s}g[/tex]
So, if we solve this for b we can put that into our equation for velocity to find the max velocity the cockroach can travel in either direction. But it doesn't seem totally correct to me. It seems messy...plus, when i used to same process for the cockroach traveling opposing the direction of rotation, it was possible to make b undefined for some values of omega. So, is my approach correct?