(Solved): Problem 1: Using air pressure, the \( 0.25 \mathrm{~kg} \) ball is forced to move through the tube ...

Problem 1: Using air pressure, the \( 0.25 \mathrm{~kg} \) ball is forced to move through the tube lying in the horizontal plane and having the shape of a logarithmic spiral. Assume that the tangential force exerted on the ball due to the air is \( F=4 \mathrm{~N} \). Determine: (a) the rate of increase in the ball's speed at the instant \( \theta=\frac{\pi}{3} \), (b) the acceleration of the ball in the radial direction at the same instant, and \( (\mathrm{c}) \) the direction of the the tangential force exerted on the ball at the same instant (this is the angle that the tangential force \( F \) makes with positive direction of the \( x \)-axis counted counterclockwise). Note: To get full credit, please, respond correctly and show all the details of the solution process, including free-body diagram, labels, and reference systems.