When the 1-kg ball starts to rotate around the horizontal circular path A with a

Question

When the 1-kg ball starts to rotate around the horizontal circular path A with a constant speed of 1 m/s. the rope connecting the ball passes through a pulley (in point C, not shown in the figure), which is fixed to the ceiling. A force F is applied as shown in the figure. If the force F on the cord increases, the ball rises and eventually reaches a new position and rotates around the horizontal circular path B. The distances between the pulley and the point-mass in the two conditions are shown in the figure. a. Determine the analytical relationship relating the applied force F and the angle theta_1 b. Determine the speed of the ball and the angle between the cord and vertical when the ball is rotating around the path B (a different force F is applied in this condition)

Explanation / Answer

a ) Using

F = [ ( m g )^2 + ( m v^2 / L sin )^2 ]^1/2   

F = [ 1 * 9.8^2 + ( 1 * 1 / 0.6^2*sin )^2 ]^1/2   

F = ( 9604 + 2.78 / sin^2(1) )^1/2

b ) Here

cos1 / sin1 = ( L2 sin2 ) g / V1^2

V1L1sin1 = V2L2sin2   

V2 = 2 sin1 / sin2   

V2 cos2 / 2 sin1 = 0.3*2*sin1*g / V2^3

V2^4 = 0.3*4*g*sin^2(1) / cos2

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