From: Sue 21/05/99 17:34:43 Subject: Falling into a hole in a large sphere post id: 13203 Sorry to repeat this, but now this question has everyone here intrigued, and we'd like to add some additional questions too! If you had an earth-sized sphere, with the same mass as the earth, and hypothetically it was solid throughout and all at room temperature, and there was an elevator-shaft sized hole going centrally from one side of this sphere to the other, and the hole had air in it, and a person standing beside this hole jumped into it, where would the person end up? The first part of the journey would be similar to freefall, wouldn't it...but what would happen then? And would the person necessarily be able to stay clear of the walls? New related questions: 1) What would be the differences between the above scenario, and one where there was a vacuum in the hole? (assume the person is wearing a spacesuit and will survive - I'm not asking a biology question here :-) 2) Could you actually get air to be at STP in such a hole, assuming the whole sphere was at room temperature, or would differences in gravitational forces make that impossible? 3) And could somebody explain to me what happens to the gravitational force as you get closer to the centre of the sphere, and why? 4) If the frictionless scenario enabled you to come to the surface at the other side, would you be able to hold onto the edges of the hole and pull yourself out of it to stop yourself falling back down again? Or couldn't you come to the surface? 4) What would be the maximum velocity you would reach in both scenarios (with friction, without friction)? 5) How would your velocity change as you are travelling through this hole, again comparing both scenarios? Anyone got time for this? My mind is boggling. From: Pete Mousley 21/05/99 18:31:56 Subject: re: Falling into a hole in a large sphere post id: 13224 . Anyway, I haven't really thought about it but here's my response: 1) If it were a perfect vacuum and there were no friction from other sources then yes, in a simple world you would just make it to the other side. However, if your planet was a perfect sphere with perfect mass distribution, then there would still be other influences, such as the gravitational pull from the moon and other planets... (I'll leave that to someone else) 2) Depends on your room temperature - think about it :) 3) Neglecting the outside influences (the moon and other planets for example) the gravitational pull at the centre of the tunnel would be equal in all directions. If you were stopped at the centre of the tunnel you would stay there - gravitational acceleration would be equal in all directions. Though it might pull you apart... 4) You'd have to grab on pretty quick otherwise you might spend a lot of time going back and forth! 4) (again! Is this you way of making it seem as though you didn't have so many questions???) I'll let someone else do the calculation for the frictionless senario (it is a Friday night after all!) (though don't forget that acceleration due to gravity is not constant!) As for the case with friction, with STP conditions it would be similar to that experienced sky diving - the terminal velocity, which depending on body size and position could be between 150km/h and 250km/h. 5) In both cases you'd accelerate at a decreasing rate. With friction, the increasing drag would cause your rate of acceleration to decrease with speed, until your acceleration was zero at your terminal velocity. Without friction, the reduction of your acceleration would be caused by the reduction in acceleration due to gravity as you moved through the tunnel, reaching zero acceleration at the centre (where your velocity is maximum) and then your acceleration would be an increasing negative (back towards the centre) as you moved from the centre to the other side. Hope that makes some sort of sense. Your final comment - I'm not surprised your mind is boggling! Go out and enjoy your evening before they lock you up :) From: Chris W (Plebeian) 21/05/99 19:33:36 Subject: re: Falling into a hole in a large sphere post id: 13249 The Avatars have been a bit thin on the ground today, but then they have a life too! There are some miscellaneous ramblings on this very topic here, including JR and Chris. From: Terry Frankcombe 21/05/99 23:11:17 Subject: re: Falling into a hole in a large sphere post id: 13284 OK sue. Though Pete had a good go at it, here's my attempt to reinforce my reputation of someoone who's interested in working out mindless details. If you had an earth-sized sphere, with the same mass as the earth, and hypothetically it was solid throughout and all at room temperature, and there was an elevator-shaft sized hole going centrally from one side of this sphere to the other, and the hole had air in it, and a person standing beside this hole jumped into it, where would the person end up? The first part of the journey would be similar to freefall, wouldn't it...but what would happen then? And would the person necessarily be able to stay clear of the walls? The first part is almost similar to freefall. The thing to remember is that as you drop below the surface of your sphere the gravitational force decreases linearly with r rather than increasing with the inverse of r squared, because suddenly you start to be attracted graviatiationally upwards with all the mass that is 'above' you. So you will pass the centre with less than your normal terminal velocity. You'll probably pass through the centre, and then stop somewhere between the centre and the surface and start heading back to the middle. If your air resistance was high enough (low enough 'normal' terminal velocity) then you would never make it past the middel. I can't work out the required integrals at the moment because I've just come back form a DCS Smoko. Translation for non-UQ chemistry students: I've been drinking. If the person was standing next to the hole and jumped in, then the exact details of the drag on the person would determine whether they would hit the wall or not. Lots of drag compared to the hole diameter then no, less drag/smaller hole then yes. If they were suspended over the hole then they would not hit the sides. New related questions: 1) What would be the differences between the above scenario, and one where there was a vacuum in the hole? (assume the person is wearing a spacesuit and will survive - I'm not asking a biology question here :-) In this case you would keep accelerating until you passed the middle, where you would start decelerating. You would reach zero velocity at exactly the same place (but oposite) as you jumped form on the opposite side. 2) Could you actually get air to be at STP in such a hole, assuming the whole sphere was at room temperature, or would differences in gravitational forces make that impossible? My gut feeling is that something close to STP throughout the hole would be possible, though I'll have to think about it some more. 3) And could somebody explain to me what happens to the gravitational force as you get closer to the centre of the sphere, and why? As I said, as you get closer to the centre the gravitational force drops linearly with the distance from the centre. Essentially what is happening is that the mass above you (less, but closer) is balancing with the mass on the other side of the sphere (more, but further away) so that the force you feel is the same as what you would feel for a sphere of the same density and of radius equal to the distance between you and the centre. If you do the intgrals it works out. 4) If the frictionless scenario enabled you to come to the surface at the other side, would you be able to hold onto the edges of the hole and pull yourself out of it to stop yourself falling back down again? Or couldn't you come to the surface? You would come to exactly the same place as you left. If you jumped into the hole, then you would end up with your feet as far out of the hole as they were when you started dropping on the other side. Whether you could grab hold or not depends on whether you could have grabbed hold or not on the other side when you were entering the hole. If you were hanging by your fingertips on the other side before letting go then you would be able to just get a grip with your fingertips on the fallen-to side. 4) What would be the maximum velocity you would reach in both scenarios (with friction, without friction)? Um, ask me again tomorrow. (see the original question). 5) How would your velocity change as you are travelling through this hole, again comparing both scenarios? Again, get back to me tomorrow! Anyone got time for this? My mind is boggling. At this point I'd like to say that I hope you have appreciuated my recent use of colour. I have endeavoured to not use the same colour twice since you pointed out I always used red! From: bob s 22/05/99 6:18:43 Subject: re: Falling into a hole in a large sphere post id: 13390 I posted an answer to the sphere with a hole and no air case elswhere. If there was air then the density of the air would increase to a maximum at the center of the sphere The actual density of the air would depend on the size of the sphere and depending on the temperature could actually become liquid before the center was reached. The drag in that case would possibly be high enough to cause your motion to become almost zero as the center was approached. In that case you would not pass the center by much. In the vacuum case the motion conditions would be very similar to that of a pendulum. From: Sue 24/05/99 11:08:49 Subject: re: Falling into a hole in a large sphere post id: 13512 Hi Terry! Thanks for the effort, looking forward to your additional thoughts. For when you do those, actually, let me rephrase the STP question: I think I said 20 degrees C in my original question, should be 0 degrees C, I suspect. So if you assume the sphere has a temperature of 0 degrees throughout, would it necessarily follow (as that rogue seemed to suggest) that the atmosphere inside the hole would therefore be at STP? I can't even get my mind around a "gravity map" in such a sphere, let alone work out how that would affect the characteristics of the atmosphere inside the hole. But I have this hunch that it couldn't be homogeneous. What do you think? Sue From: Terry Frankcombe 25/05/99 1:01:09 Subject: re: Falling into a hole in a large sphere post id: 13636 Sue, I think you'll find that that was not really red. The HTML gods have decreed that that colour is infact 'orangered' :-P As was pointed out by Dr Karl, you go through the hole with SHM. This means your velocity (and position) vary like a sine wave. For the air case, I can't work out a solution. For all you DE jocks out there, what's the general solution to damped motion described by d2x/dt2 + a (dx/dt)2 + b x =0? Or what is a transform that allows this to be transformed to a normal damped free vibration equation? I think bob was right. Maintaining constant temperature, air pressure would increase as you dropped below the surface. Probably the rate of pressure increase with depth would decrease as you got deeper. The SHM refered to above is for a non-rotating sphere. If the sphere was rotating then you could not avoid running into the wall without some external force. The wall you would run into would be the one towards which the sphere is rotating (ie the front wall of the tunnel). This forum is un-moderated. The views and opinions expressed are those of the individual poster and not the ABC. The ABC reserves the right to remove offensive or inappropriate messages.