MATH NERD QUESTION OF THE DAY:
O.K. since I'm a married man with children, I spend the better part of my existence in the park playing with my eldest daughter. While she was on the swings yesterday, I question that has plagued me since early childhood reared its head. I have often wondered if you can go fast and high enough on a swing set to actually flip over the crossbar. Is this possible? Is my fear realistic or a physical impossibility? Sixteen tons of invisible ice cream to whoever can demonstrate the answer to me (assume a ten-foot high swingset crossbar).
how long is the chain connecting the swing to the seat? That is most important. As far as the mass of the individual child is concerned, that doesn't matter as it is cancelled out in the calculations. Assuming a chain length of 2 meters, the velocity required to go from a low of 0 meters (ground level) to 4 meters (completely above the swing) is roughly 17 mph (I converted that for all us proud Americans).
ReplyDeleteNow the predicament faced is that once an individual on the swing reaches an angle greater than 90 degrees to the ground, that individual better hope to be going fast enough to go completely around the top, or not go any higher than 90 degrees, because when that individual reaches 0 velocity (like a ball reaches 0 velocity on the top or the arc as it starts to fall) that individual will not fall in the direction of the chain, but continue in the arc back toward the swing which really would hurt.
To maintain 90 degree maximum height, velocity must be 8.6 mph. To increase this to 17.2 mph in one pumping motion (assuming you have the entire downward motion from 90 degrees to zero degrees to increase velocity from 0 to 17.2 mph) the force required would be over three times the body weight and that's assuming it is applied at the seat and not somewhere along the chain, which would reduce the applied force. And that's not including any wind resistance. So, unless you can expel a force greater than 3 times your body weight (probably more like 9 times taking into account all variables) then, no, I don't think you can swing yourself completely around the top of a swing... of course, I welcome corrections, since I threw this stuff together in approximately a half hour
just a reminder, conservation of energy requres potential energy to equal kinetic energy and while potential energy equals mgh, kinetic energy isn't just 1/2mv^2 but also I*omega^2 (omega = angular velocity of an object moving in a circular pattern) so the total kinetic energy (I=mr^2 of a point particle as the individual on a swing would be considered) equals 3/2mv^2 not 1/2 mv^2 again, I'm hurrying this along and may have missed something
ReplyDeleted'oh I just read the post again and saw the "assume a 10-foot crossbar" - I assumed 2 meters or 6.6 feet... just makes it that much harder...
ReplyDeleteand because I can I will add... I could've been playing Hexic
ReplyDeleteMythBusters addressed that question in one of their episodes. It's possible, but I wouldn't want to try it.
ReplyDeleteThe chain and the swing set had better be pretty strong. Most backyard swing sets would probably topple over if you got swinging that hard.