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Newton's second law and centripetal force

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A small, frictionless ball, mass M= 2.5 kg, rests on the surface of a rotating solid cone. The surface is at angle b= 30 degrees. The ball is held in place at distance L= 1.85 m from the apex by a cord. The cord exerts force T on the ball, parallel to the cone's surface. SEE ATTACHMENT #1 for a diagram showing parameters.
Part a. Find maximum angular speed in rad/sec for which the ball remains in contact.
Part b. If the cone slows to half its maximum speed, find the tension in the cord.

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Solution Summary

Newton's second law and centripetal force are demonstrated.

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Part a. Step 1.
SEE ATTACHMENT #2 for vector arrows for all forces (with components shown as dotted line arrows) on the ball as the cone rotates.
Step 2.
Note that for the maximum angular speed, the ball barely remains in contact and the normal force N is zero. The only forces on the ball are then T and M g.
Step 3.
Note that the ball rotates in a circle about a point on the cone's axis. Since the y coordinate of the ball is constant, Newton's second law requires that;
(1) 'net ...

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