Three Blocks Sliding
Randy was exploring the top of a glacier when it suddenly cracked into two pieces, which then started sliding without friction.
How will Randy move, relative to his initial position?
Note : This ice is very smooth and slides without friction with other objects (i.e., the ground and Randy).
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4 solutions
I agree with the answer, but only for the first moments of the movement. After a certain amount of time, that depend on the size of the pieces, Randy will probably hit the bottom piece and then move to the left.
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This is also true. However, I believe the problem is talking about the moments immediately after the break.
Quite right, I was a little confused by this. The problem didn’t explicitly state that the ice was horizontal, although this was clear from the diagram. I also thought that eventually he would have to move down and left.
Of course as Randy is "exploring" he will have some initial horizontal velocity which would be retained. How he is moving given the surface is frictionless is another matter though - back mounted fan maybe.
Wouldn't the normal force exerted by the top glacier and the normal force exerted by the bottom glacier be equal and opposite to one another, according to Newton's third law? There is no mention of the masses of both glacier so how would we know if the displacement of the top glacier to the left is equal in magnitude to the displacement of the bottom glacier to the right? Or are we assuming that both top and bottom glaceirs have equal mass?
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The ice block falls gravity. It doesn't matter which block is more massive; either the top block would push the bottom one away, or vice versa and Randy would slide relative to the top ice block.
My thoughts exactly. Well said.
The same force which is pushing the bottom glacier to the right is pushing the upper glacier to the left. Isn't it?
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Yes. How would this affect the solution?
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The same force which is pushing the bottom glacier to the right is pushing the upper glacier to the left. Randy stays in place (left/right) until the top block hits the ground and tilts. At that moment Randy will be either on the top block or on the bottom block (not yet on the ground?) and will start moving to the right or to the left ... But that is not in scope of the question I assume, so the answer is indeed: Straight down
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@A Former Brilliant Member – Yes, you are correct in every way. Although, it would depend on the masses of the ice blocks.
He will move down, then hit the bottom block and start moving down and left. Since it didn't specifically restrict it to just his initial movement (I did double check for that, since it wasn't at all clear what it was asking), I still say he moves down and left overall.
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I agree. The top block will move to the left as it slides down the lower block. Randy, being on the upper block, will move to the left with it.
Thank you!!! I’m glad someone else noticed this.
This is the correct answer.
It was not specified when to calculate his position. I too understood the block would move but the hiker wouldn't as there was no friction applied to the hiker. And eventually the hiker would follow the same path as the smaller piece of glacier moving down the larger piece.
Why would he move to the left?
Hmm, I'm not so sure about this. Suppose the lower block were extremely massive, and the upper block were very light, say a sled. It seems in the limit he would just slide down the incline and the massive block hardly move at all, certainly not enough to keep up with his speed. Will have to think some more about it, but it seems that his motion in the rest frame needs to take the mass of the blocks into account.
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I can't agree more. Masses are very relevant to answer the question. EDIT: But altough the smaller block moves left, Randy doesn't because ice is frictionless.
From the center of mass point of view, he would slide downwards iff the masses of the blocks are the same. If, for example, the bottom block was larger, a small displacement of it to the right would require a big displacement of the smaller block to the left. Otherwise, the center of mass would move! EDIT: I was wrong; Randy isn't moving along with the smaller block because of zero friction. Very tricky question!
I think the question wasnt asked correctly .. shouldn't it be how will he move relative to the piece of ice he is on. It just asked how will he move to his initial position which can be taken to mean otherwise
This was a really misleading question. Too bad I can't unanswer it.
Even without friction don't the two blocks have different mass and will accelerate differently?
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Yes, it would. However, that aspect does not matter in this question because we are considering the motion of Randy and not the glaciers. At first glance, I thought that Randy will tag along with the top glaciers and I got it wrong.
If there is friction present, from the point of view of the glacier at the instant where it starts moving, it would seem like Randy is moving to the right relative to the glacier. Therefore the friction would act in the opposite direction - in the direction that the glacier is moving - in order to resist the relative motion of Randy.
However, when there is no friction. Randy and the glacier are two bodies detached from one another. There would be no force acting to resist the relative motion of Randy. Hence, Randy would remain where he is in the horizontal direction and simply fall straight down.
Definitely a head scratcher!
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But if Randy is detached to top glacier, then he would surely strike the bottom glacier and hence gets a normal force acted upon him. The figure is misleading a d should have mentioned that Randy won't strike the bottom glacier during the motion.
Path of motion of each block
The above picture represents the motion of each block. Treat each part as a block. The
red
line the different possible paths of motion of each block and the
green
line represents the resultant motion of each block
as seen from ground frame
. The lower most block will only move forward direction as it has no other else option as the center of mass should remain at rest. The middle block and the upper most block are in contact with each other with
no inclination
. So, we can
treat them as one body
. And hence, the resultant path of motion of both the upper and middle blocks are same. According to the diagram as the middle block moves straight down the upper block to follows the same path. Hence, the
upper block moves straight down
.
The question is not clear enough, it stated that the pieces started sliding without friction, which I assumed was between them only.
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I’m still a bit confused. Is he moving straight down because the wedges are both sliding? If so, I agree that the question is very unclear. If there exists friction between the bottom wedge and the ground then there would exist a normal force pushing the top wedge to the left.
Yes. The question should mention that all surfaces are friction-less. Wait ! I will post a report on this question.
Correct Saso, the question is phrased in a way that suggests there is no friction between the two blocks of ice - not between the climber and the ice. Who goes climbing ice without a good pair of ice shoes anyway?
I disagree since the problem disregards the difference in inertia between the two ice blocks.
I disagree. As long as the bottom block is much larger, the inertia will make the top block move faster, forcing it to go sideways.
What about inertia? If he hits the big block, it should not move to the right immediately because it has to accelerate. Randy would slide to the left.
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When he hit the big block relative motion between him and smaller block will stop and he may move as the corner of small wedge moves
The top purple block and the middle blue block cannot be treated as one body , and they do not have the same motion.
The question asks about Randy (the top block), and the correct answer is that he will move straight down. But if the question had been about the wedge of ice (middle block), the answer would be different. That wedge will slide down and left following the incline.
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Thank you for clearing this up.
Yeah, but as soon as the blue block slides out from Randy's legs, he will end up on the inclined side of the big block of ice. And will slide to the right (unless we assume Randy has the same mass as the big block of ice).
Let mg be weight of blue wedge it's normal reaction component will b mgcos@ acting perpendicular to diagonal of wedge. It's horizontal component will push blue wedge horizontally leftward and it's reaction push green wedge horizontally rightward. But there is no friction between blue wedge and person only forces acting on him is his weight and normal reaction by blue wedge. So only vertical force is Der and he will come down
How can u assume that the larger part of the glacier supporting the small part can move freely? I mean it's only the small glacier that broke
Since there is nothing directly pushing Randy, the only horizontal force could be friction with the top piece of ice. However, the question states that there is no friction. Therefore, as the two pieces of ice move down Randy will slide along the top piece so as to remain in the same horizontal position, as there is no sidewards force acting on him. Therefore he will move straight down .
From a energy point of view, the smaller block displacement will be greater than for the bigger block to transfer the same energy. So the smaller block get a bigger acceleration to the left that the bigger block to the right. At some point the smaller block will escape Randy feet and after that Randy will follow the smaller block to the left too.
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If an ant ice skated and crashed into a giant block that is infinitely heavier, would the ant catapult itself in the opposite direction at the speed of light because the block is heavier, or would he simply transfer his momentum to the block.
Also, if the small block were to move to the left, Randy would not follow as there is no friction and hence no leftward force applying to Randy. (without friction, he'd just watch the block slide from under him without being able to do anything about it.)
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The super heavy block will take a insignificant fraction of the collision energy proportional to the ratio of the masses, and so the ant will take virtually all the collision energy. If none of them can dissipate energy by deformation, then the only way how the ant can keep his energy is by traveling in an other direction. To simplify: the block will stay in place and the ant bounce.
Thinking in terms of F=ma, wouldn't the right-directed force of the smaller block acting on the larger block be equal to the left-directed force of the larger on the smaller? If so, wouldn't the smaller block accelerate faster because of less mass? Which would imply that the net motion of the smaller block would be to the left? But then... if there were no friction, the smaller block would move out from under Randy, as his feet would not grip the surface, so he would fall straight down.
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A world such as this would be a very peculiar place.
In short, since the right and left directed forces are equal, they cancel each other out and a total force of 0 applies for both blocks. When zero force is applied, mass is irrelevant as 0 times any value is always 0.
I get how your thinking goes, but this is not like a lighter cannon ball accelerating faster than the cannon does in the opposite direction - the blocks are "pushing" against one another and there is no third force at the center that wishes to separate them.
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It's exactly like if a lighter cannon ball hit a heavy cannon. If none of them can lost energy by deformation, all the energy collision will be shared by the masses ratio. So the lighter cannon ball will bounce and the heavy cannon will stay mostly in place. Likewise in the initial question the force F will result in different acceleration depending of the mass (a = F/m) and so the larger block will always move slower than the small block. So the light block will go left from the Randy point of view, escaping him.
I agree but since the little block will be accelerating faster than the larger block at some point in Randy's downward trajectory he will make contact with the sloped surface of the larger block and Randy will start traveling down the slop or the smaller block will make contact with the ground and then the horizontal surface will start to slope downward to the right as the blocks travel away from each other and Randy will be going right. Either way Randy will only go straight down for the initial movement. This question should be reworded as such.
One criterion not mentioned was the sizes/masses of the broken pieces. As drawn, the upper ice shard is much smaller then the base piece. Therefore, the upper piece should accelerate faster to the left than the larger bottom piece would to the right, giving a net left direction, due to inertia(imagine if the upper blue broken piece was barely larger than the person above it and the bottom piece, to scale filled the page, you could conclude logically, that the upper piece, and the man on top would slide down and to the left.
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But the man on top has no friction with the upper piece. Even if the upper piece slides to the left, he will still slide along it to remain horizontally stationary.
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True only until the upper piece will be no longer under the man (because the upper piece slide to the left more quickly than the lower piece go to the right due to masses ratio). At some point the man will contact the larger piece and because his mass is lower than the lower block, he will start sliding to the left (like the upper piece before him).
I agree with Mitchell Day below, and it also assumes that the block of ice was in a stationary state prior to Randy jumping on the block causing it to crack
Put a little common sense and u can get the answer
Why make such a meaningless comment? If it was all just 'common sense' it wouldn't be a problem on this website.
There are also airodynamics staff, why don't we consider that? Usually cyclers which competes in teams and the ones that rides at the back spend less energy because of air resistance.
Relevant wiki: Newton's First Law
According to Isaac Newton's first law of motion,
Let's assume the ice is frictionless. Once the glacier breaks, Randy will start to go down due to gravity, the outside force. Note that, since there is no friction, Randy will not move to the side. If Randy was on a block of wood instead of ice, he would move down and left because he would be acted on by friction , an outside force.
It is important to note the technically ice has some amount of friction because there is no such thing as a frictionless surface. Therefore, Randy will move slightly to the left as well.