A block is tied to a string of length l [meters] which is pinned down to a smooth horizontal floor. The block is pushed so that it slides in uniform circular motion (where the radius of the circular trajectory is l [meters]. The magnitude of the acceleration of the block is a [m/s²] as it goes around the circle. Suddenly the string is cut! The block then slides straight up a nearby smooth ramp, inclined at an angle θ degrees above the horizontal. (The block's speed at the bottom of the ramp is the same speed it had when the string was cut, but the velocity vector now points up-ramp (along the ramp). The block slides up a vertical distance h [m] and reaches a smooth horizontal surface that is h [m] above the surface where it started. It then slides L [m] on this higher horizontal surface and slides off the edge! It then lands on the surface below, falling a vertical distance h [m], and landing a horizontal distance d [m] from the edge of the higher surface. Answer the following in terms of the variables given above and use g for the magnitude of free-fall acceleration.
a) Find an expression for the speed of the block before the string is cut. (We are assuming a "perfect cut" where the speed of the block does NOT change due to the cutting of the string.)
b) Find an expression for the time it takes the block to make one complete circle before the string is cut.
c) Find an expression for the speed of the block when it reaches the higher horizontal surface.
d) Find an expression for d (the horizontal component of the displacement of the block while it is in projectile motion) in terms of the variables given.