A cockroach crawls with a constant speed on a phonograph turntable that is rotat

Question

A cockroach crawls with a constant speed on a phonograph turntable that is rotating with angular velocity ?. If the bug moves at constant speed away from the center of the turntable draw all forces acting on the bug (in the bug's frame).(b) Now say the bug tries to move along the circuference of the turntable at a constant distance r. Draw all the forces that the bug feels.(c ) Now the bug wants to travel at a constant speed v' along the circumference of the turntable, but the coefficient of static friction between the bug and table is ?_s. What is the maximum speed v' that the bug can move before it starts to slip. Assume that the bug is moving in the direction of rotation.

Explanation / Answer

For: b = path radius wt = angular speed of turntable wi = angular speed of the insect, relative to turntable Well, the centripet force acting on the insect ONLY due to the turntable is: Ft = m a = (m)( w^2 )(b) And what force holds him into his circular path? just friction, which is: Ff = (Ms)(N) = (Ms)(m)(g) So, the insect crawls, lets say at an relative angular to the turntable speed: wi case (a): Wtot = wt + wi case (b): Wtot = wt - wi The maximum TOTAL angular speed is when friction force = centripet force, so (m)( w^2 )(b) = (m)(Ms)(g) (*Note that the mass is no longer relevant) Wmax = maximum TOTAL permited angular speed Wmax = sqrt( (Ms)(g)/b ) Then, is easily seen (from above) that the relative speeds are: (a) wi = Wmax - wt (b) wi = wt - Wmax answer(a): wi = sqrt( (Ms)(g)/b ) - wt answer(b): wi = wt - sqrt( (Ms)(g)/b ) Hope it was clear. Interesting also to note, that wi(b) is 2 times wi(a). As expected.

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