Welcome to March edition of Monthly Challenge Question from Specs & Techs by GlobalSpec:
Three identical ladders standing squarely on a plane are simultaneously released, starting thirty degrees to vertical. One falls in an open frictionless area. The second falls, without friction, away from a wall that the base contacts initially. The third falls without slipping. In what order do they hit the ground?
Thanks to Physicist? for submitting this question!
(Update: March 18, 8:52 AM EST) And the Answer is...
The first is the earliest to hit the ground. The second takes the longest to fall.
For simplicity, consider the situation in the absence of air damping. As all situations are then lossless (at least until the ladders hit the ground), the total Kinetic Energy depends only on the height of the Centre of Gravity. The KE is the sum of three components: vertical, lateral, and inertial. Because the relation between the height of the CofG and the angle of the ladders is the same for all three ladders, they all display the same relation between vertical and inertial KEs, so the only effective difference is the amount of KE that is required for lateral movement. The CofG of the first ladder falls vertically, so all the available energy is available for fallingandrotation. The second and third arrangements are identical while the CofG is accelerating horizontally away from the base of the ladder. However, once the angle becomes flat enough, the horizontal component of the CofG's velocity will need to reduce if the base is to remain stationary; for the nonslipping ladder, that is indeed what happens, but there is nothing to restrain movement of the second ladder away from the wall, so it departs with the CofG maintaining constant lateral velocity. Thus, once the lateral velocity of the second and third ladders reaches their peaks, the vertical KE of the nonslipping ladder can outstrip that of the second ladder, so the third ladder goes ahead of the second. Neither viscous damping nor turbulence would change this order.
Can't wait til the end of the month for a new Challenge Question? Check out these weekly challenges from CR4:
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