Fraternity Physics. The brothers of Iota Eta Pi fraternity build a platform, supported at all four corners by vertical springs, in the basement of their frat house. A brave fraternity brother wearing a football helmet stands in the middle of the platform; his weight compresses the springs by 0.18 m. Then four of his fraternity brothers, pushing down at the corners of the platform, compress the springs another 0.53 m until the top of the brave brother’s helmet is 0.90 m below the basement ceiling. They then simultaneously release the platform. You can ignore the masses of the springs and platform. (a) When the dust clears, the fraternity asks you to calculate their fraternity brother’s speed just before his helmet hit the flimsy ceiling. (b) Without the ceiling how high would he have gone? (c) In discussing their probation the dean of students suggests that the next time they try this, they do it outdoors on another planet. Would the answer to part (b) be the same if this stunt were performed on a planet with a different value of ?g?? Assume that the fraternity brothers push the platform down 0.53 m as before. Explain your reasoning.
Solution 76P Introduction We have to first calculate the velocity of the person at 0.9 m height. Then we have to calculate the maximum possible height of the person if there was no roof and finally we have discuss if this height will be same in other planet having different g. Step 1 Suppose the mass of the person is m. Hence the weight of the person is The spring compresses by x 1 0.18 m when the person stands on the platform, hence the spring constant is Now the spring was compressed by 0.53 m more by pussing it. So the total compression is x2= 0.18 + 0.53 = 0.71 m. Hence the potential energy at that point, because of the compression of the spring is Now at x 3 0.9 m height, the potential energy of the person due to gravity will be Now the kinetic energy of the person at that point is Now if v is the speed at that point, then we have Hence the speed just before the the helmet hit the ceiling is 3.1 m/s. Step 2 Suppose the height of the person without ceiling is h. Then, at the maximum height h, the gravitational potential energy of the person will be equal to the potential energy of the spring. So the potential energy at h is Equating with the potential energy of the spring we have Hence the maximum height of the person will be 1.4 m.