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.

7. 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.

Explanation / Answer

for vertical path , the ean for normal force is

N cos theta = mg+ Ff sin theta

N * cos 20 + 2 * 9.81 + 0.4* 1 * 9.81 * sin 20

N = 24.4 N

Noe using it for Horizonta force

T + FF cos20 +N sin20 = mr W^2

where W is angular velocity = V/r

so

Wmax = 3.7 rad/s

For vertical F sin 20 + N cos 20 = 2* 9.81

N = 18.23 N

for vertical

T + N sin 20 - f cos 20 = mrW^2

W = 14.3 rad/s

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