A 1.0 kg weight, hanging vertically from the bottom of an inverted cone, is held

Question

A 1.0 kg weight, hanging vertically from the bottom of an inverted cone, is held in place by a 2.0-kg-wheel that is rolling (without slipping) with a constant speed in a horizontal circle on the inside of the cone, as shown in the figure. They are attached by a string that passes through a pulley and a hole in the middle of the cone. If the cone makes an angle of 20 degrees with re-spect to the horizontal and the wheel is always 1.0 m from the center line of the cone, what is the a) maximum and b) minimum speed with which the wheel can go to keep the hanging weight in place, given the coefficient of static friction between the cone and the wheel is 0.4.

Please answer the question thoroughly with full work and correct answers as best as possible. My TA will be directly checking the answer to be sure. Also, here is an image. The first best answer will get a full rating.

Explanation / Answer

1. the weight of the hanging mass (mg = 1x9.8 = 9.8 N)
2. the friction force between the wheel and the table (Ff = mu*N = 0.4x1x9.8) = 3.92 N

The maximum centripetal force = 9.8 + 3.92 = 13.72N
The minimum centripetal force = 9.8 - 3.92 = 5.88N

The maximum speed is given by mv2 / r = 13.72.

assume r= 0.33; Solving for v we have: V(max) = 2.13 m/s

The minimum speed: mv2 / r = 5.88, therefore V(min) = 1.3 m/s

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