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)

Answer 1

a )

using equation

F = [ ( m g )² + ( m v² / L sinϴ )² ]^1/2   

F = [ 1 X 9.8² + ( 1 X 1 / 0.6²sinϴ )² ]^1/2   

F = ( 9604 + 2.78 / sin²ϴ₁ )^1/2  

b )

cosϴ₁ / sinϴ₁ = ( L₂ sinϴ₂ ) g / V₁²

V₁ X L₁ X sinϴ₁ = V₂ X L₂ X sinϴ₂   

V₂ = 2 sinϴ₁ / sinϴ₂   

V₂ cosϴ₂ / 2 sinϴ₁ = 0.3 X 2 X sinϴ₁ X g / V₂³

V₂⁴ = 0.3 X 4 X g X sin²ϴ₁ / cosϴ₂