Walking by a pond, you find a rope attached to a stout tree limb that is above ground level. you decide to use the rope to swing out over the pond. the rope is a bit frayed, but supports your weight. you estimate that the rope might break if the tension is greater than your weight. you grab the rope at a point from the limb and move back to swing out over the pond. (model yourself as a point particle attached to the rope from the limb.)

(a) what is the maximum safe initial angle between the rope and the vertical at which it will not break during the swing?

(b) if you begin at this maximum angle, and the surface of the pond is below the level of the ground, with what speed will you enter the water if you let go of the rope when the rope is vertical?

Amass m = 74 kg slides on a frictionless track that has a drop, followed by a loop-the-loop with radius r = 18.4 m and finally a flat straight section at the same height as the center of the loop (18.4 m off the ground). since the mass would not make it around the loop if released from the height of the top of the loop (do you know why? ) it must be released above the top of the loop-the-loop height. (assume the mass never leaves the smooth track at any point on its path.) 1. what is the minimum speed the block must have at the top of the loop to make it around the loop-the-loop without leaving the track? 2. what height above the ground must the mass begin to make it around the loop-the-loop? 3. if the mass has just enough speed to make it around the loop without leaving the track, what will its speed be at the bottom of the loop? 4. if the mass has just enough speed to make it around the loop without leaving the track, what is its speed at the final flat level (18.4 m off the ground)? 5. now a spring with spring constant k = 15600 n/m is used on the final flat surface to stop the mass. how far does the spring compress?