You are an astronaut, living for a long time interval in the international space station (iss). during your off-duty hours, you have run out of books to read and video games to play. so, your mind wanders to your hobby of music. the last book you read discussed gauss's law, and you get an inspiration. you plan to attach two nonconducting spheres of radius r = 1.40 cm together using a light insulating string of length l and linear mass density μ = 6.00 ✕ 10−3 kg/m, with the string attached at the surface of each sphere. then, using the electrical system on the iss, you will be able to electrify each sphere to a charge of q = 55.0 µc, uniformly spread over the surface of the sphere. the combination will then be allowed to float freely in the iss. the spheres will repel, creating a tension in the string. when you pluck the string, you wish it to play a perfect middle c, at 262 hz. determine the length of the string (in cm) that you need. (assume the frequency of 262 hz is the fundamental frequency. round your answer to at least one decimal place.)

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?