# Centripetal acceleration problem

A small bug is sitting on the edge of a uniform disk of mass 2.5 kg and radius 20 cm initially at rest. There is a massless string wrapped around the disk and the coefficient of friction between the bug and the disk is u= 0.4. The string is then pulled with a constant tension of 4 N.

a) What is ther maximum linear speed the bug can have and stay on the disk?
b) How many seconds until the bug falls of the disk assuming the string is long enough to keep applying the force?

I know the solution, but I'm confused as to why my teacher did it. He said:

"Because there is a constant tangential acceleration and an increasing centripetal acceleration, so the acceleration is the resultant of those two perpendicular components.

For part a, He set potential energy equal to net force and solved for v (umg= ma). Why didn't he take into account centripetal force? Why is centripetal accel. increasing, but tangential acceleration constant?

For part b, he said v= a tangential * t. Why did he just use tangential acceleration to determine how long the bug stays on the disk?

Thanks in advance for any help.

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ehild
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The bug moves together with the disk along a circle. It sits at the rim of the disk, so its acceleration has a centripetal component v2/R. The rotation of the disk is accelerated by the tension of the string. You get the angular acceleration β by dividing the torque of the tension τ=TR with the moment of inertia of the disk (I): β =TR/I, and the linear acceleration at the rim is at=βR.

As the bug moves together with the disk, its acceleration also has centripetal and tangential components. The acceleration is the resultant of these two perpendicular accelerations, and the force is parallel to the resultant acceleration.

The magnitude of the resultant acceleration is

a=√(acp2+at2)

and the magnitude of the resultant force is ma.

The force on the bug is provided by the static friction, which maximum value is μgm: μgm=ma. Your teacher took the centripetal acceleration into account as 'a' is the resultant of both accelerations, centripetal and tangential.

As for b: the velocity is always tangential, and the bug stays at place while its speed is less than the maximum value. The rim moves with uniform tangential acceleration at, so the speed is proportional to the elapsed time: v=at*t .

ehild