Will a round-headed rod topple if it slides down a frictionless slope?

In summary: Welcome to PF. In summary, the problem is that, someone told me that a ball won't roll when sliding down a frictionless slope because the resultant force mgsinx is parallel to the slope which means that the ball will slide down the slope. Now, replace the ball with a round headed rod, does this means that the rod won't topple no matter the rod is perpendicular to the slope or not? Actually, I think the torque is not "around the point of contact" but rather around the center of mass of the rod.
  • #1
James Brown
21
11
Well, the problem is that, someone told me that a ball won't roll when sliding down a frictionless slope because the resultant force mgsinx is parallel to the slope which means that the ball will slide down the slope. Now, replace the ball with a round headed rod, does this means that the rod won't topple no matter the rod is perpendicular to the slope or not?
 
Last edited:
  • Wow
Likes Delta2
Physics news on Phys.org
  • #2
James Brown said:
...a ball won't roll when sliding downslope...
Not true. A simple test will bear that out.

And maybe don't trust that person to tell you things.
 
  • Like
Likes vanhees71
  • #3
DaveC426913 said:
Not true. A simple test will bear that out.

And maybe don't trust that person to tell you things.
Well it is a frictioness slope, u mean the ball with also roll when sliding downa frictioness slope? Well if I trust him I won't be asking this question
 
  • Like
Likes vanhees71
  • #4
I missed the "frictionless" reference. mea culpa.
 
  • Like
Likes berkeman, vanhees71 and hutchphd
  • #5
Welcome to PF.
James Brown said:
Well it is a frictioness slope, u mean the ball with also roll when sliding downa frictioness slope?
Without friction there will be no rotation, so the ball is in effect falling.
But there is no such thing as a real frictionless surface.

The rod will not topple in the frictionless model, but it will topple in practice.
 
  • Like
Likes vanhees71 and Delta2
  • #6
Baluncore said:
Without friction there will be no rotation, so the ball is in effect falling.
So, does that hold for the round-ended rod as well?
 
  • #7
James Brown said:
Now, replace the ball with a round headed rod, ...
Does it have a flat foot?
It matters not, head or foot, flat or round, the rod would slide down a virtual frictionless surface.
 
  • Like
Likes Delta2
  • #8
DaveC426913 said:
So, does that hold for the round-ended rod as well?
I am doubtful that the rod will stay upright. The momentum of the center of mass will, it seems to me, not allow the middle of the rod to keep up w/ the rounded bottom as it slides down so the rounded bottom will slide out from under the center of mass the the rod will topple.

@Baluncore I take it you do not agree w/ this.
 
  • #9
phinds said:
... so the rounded bottom will slide out from under the center of mass ...
Then in your analysis, that must also be true of the ball.

Do you assume the rod is vertical or perpendicular to the surface?
James Brown said:
... , does this means that the rod won't topple no matter the rod is perpendicular to the slope or not?
 
  • #10
Baluncore said:
Then in your analysis, that must also be true of the ball.
No, because the normal force of the ball pressing against the slope runs THROUGH the center of mass so absent any impediment (friction) to the point of contract, the ball has no reason to roll. With the vertical rod, the normal force against the point of contact with the slope does NOT go through the center of mass, so there is an imbalance and the rod topples.
 
  • Like
Likes Steve4Physics
  • #11
phinds said:
With the vertical rod, ...
Ah. You assume the rod is vertical.
The OP assumed the rod could be perpendicular to the surface.
 
  • #12
@Baluncore what do you mean rod vertical and rod perpendicular, vertical and perpendicular are synonyms depending on the context. A scheme would help here.

If the rod is going to topple there must be a net torque around some point, I don't see any net torques here.
 
  • #13
Actually I think I am wrong, the weight has a net torque around the point of contact
 
  • #14
Delta2 said:
Actually I think I am wrong, the weight has a net torque around the point of contact
Exactly. The rod does, the ball doesn't. See post #10

Actually, I think the torque is not "around the point of contact" but rather around the center of mass of the rod.
 
  • Informative
Likes Delta2
  • #15
I think both the ball and the rod have zero weight torque around their CM.
 
  • #16
Baluncore said:
Does it have a flat foot?
It matters not, head or foot, flat or round, the rod would slide down a virtual frictionless surface.
Omg this is incredible! Now do it in practice, place the rod on the slope, the rod will rotate without sliding at first right? Now, assume the friction is gone, if it is from your assumption then there is net torque so u think the rod will slide down without topple?
 
  • #17
We need an expert in rigid body dynamics here. I am a mathematician, my physics/mechanical intuition says that the rod will topple or not depending on the height of the rod , the diameter of the rounded head and the position of its CM.
 
  • Like
Likes James Brown
  • #18
Delta2 said:
We need an expert in rigid body dynamics here. I am a mathematician, my physics/mechanical intuition says that the rod will topple or not depending on the height of the rod , the diameter of the rounded head and the position of its CM.
Yeah you are right
 
  • Like
Likes Delta2
  • #19
Actually I keep switching between yes and no the more I keep thinking about this problem. My thoughts on this are a pure confusio so I just can't post them because even if I did no one, maybe not even me would understand what I said.
 
  • Like
Likes James Brown
  • #20
James Brown said:
Now, replace the ball with a round headed rod, does this means that the rod won't topple no matter the rod is perpendicular to the slope or not?

I think that even if there is friction between the object and the slope, as long as the friction between all parts of the object in contact with the slope is the same, the object will not necessarily tip over in this equilibrium state.
 
  • Like
Likes James Brown
  • #21
Delta2 said:
what do you mean rod vertical and rod perpendicular, vertical and perpendicular are synonyms depending on the context.
The two special cases are with (1) the rod perpendicular to the slope, and (2) the rod vertical.
1. The rod would remain perpendicular to the slope as it slides, with all parts accelerating at the same rate. A sphere is an example of a short rod, with CofM on the perpendicular to the contact point on the slope.
2. A vertical rod would rotate about it's CofG, and would fall as the foot slid down-slope.
 
  • Like
Likes Steve4Physics, hutchphd and Ibix
  • #22
alan123hk said:
I think that even if there is friction between the object and the slope, as long as the friction between all parts of the object in contact with the slope is the same, the object will not necessarily tip over in this equilibrium state.
I think we mean a perfect rounded head that has only one point of contact with the slope.
 
  • Informative
Likes alan123hk
  • #23
Delta2 said:
I think we mean a perfect rounded head that has only one point of contact with the slope.
I think you mean a perfect rounded foot.
 
  • Haha
  • Informative
Likes hutchphd and alan123hk
  • #24
Delta2 said:
Actually I keep switching between yes and no the more I keep thinking about this problem. My thoughts on this are a pure confusio so I just can't post them because even if I did no one, maybe not even me would understand what I said.
Same here, I think if the rod will topple or bot depends on how you put the rod
 
  • Like
Likes Delta2
  • #25
Delta2 said:
I think we mean a perfect rounded head that has only one point of contact with the slope.
Indeed
 
  • #26
I think this is the point @Baluncore made in #21 in diagrammatic form.
1654937500580.png

There are only two forces here in the absence of friction - the green weight and the red normal reaction force. The weight has no moment about the center of mass of the rod so is irrelevant to whether or not it rotates. The normal reaction force, however, can cause it to rotate except in the special case that it points at the center of mass - i.e., when the rod is perpendicular to the slope. In the case shown, the rod will rotate clockwise as it slides down to the right, so "fall backwards".

In reality, a frictional force pointing up the slope will tend to cause the rod to rotate anticlockwise. Which way the rod actually falls depends on the balance between the torques from the normal and frictional forces.
 
  • Like
Likes Delta2
  • #27
Ibix said:
I think this is the point @Baluncore made in #21 in diagrammatic form.
View attachment 302676
There are only two forces here in the absence of friction - the green weight and the red normal reaction force. The weight has no moment about the center of mass of the rod so is irrelevant to whether or not it rotates. The normal reaction force, however, can cause it to rotate except in the special case that it points at the center of mass - i.e., when the rod is perpendicular to the slope. In the case shown, the rod will rotate clockwise as it slides down to the right, so "fall backwards".

In reality, a frictional force pointing up the slope will tend to cause the rod to rotate anticlockwise. Which way the rod actually falls depends on the balance between the torques from the normal and frictional forces.
Well so if the Reaction force does not point to the center of weight the rod will topple?
 
  • #28
James Brown said:
Well so if the Reaction force does not point to the center of weight the rod will topple?
Looks like it to me. If it feels slightly counter intuitive, I think it's the lack of friction messing with your notion of how stuff behaves.
 
  • #29
Ibix said:
I think this is the point @Baluncore made in #21 in diagrammatic form.
View attachment 302676
There are only two forces here in the absence of friction - the green weight and the red normal reaction force. The weight has no moment about the center of mass of the rod so is irrelevant to whether or not it rotates. The normal reaction force, however, can cause it to rotate except in the special case that it points at the center of mass - i.e., when the rod is perpendicular to the slope. In the case shown, the rod will rotate clockwise as it slides down to the right, so "fall backwards".

In reality, a frictional force pointing up the slope will tend to cause the rod to rotate anticlockwise. Which way the rod actually falls depends on the balance between the torques from the normal and frictional forces.
Wait… the resultant force that act on a rod will becomes mgsinx right?
 
  • #30
@Ibix suppose the starting position for the rod is such that the normal force passes through the center of mass. What about the torque of mgsinθ (that is the component of weight parallel to the slope), wouldn't that make the rod topple? Why do we consider torque only around center of mass?
 
Last edited:
  • #31
Cancel all friction, then place one coin on the slope, and watch it accelerate away down-slope.

Place one coin on top of another, then place the two on the slope. The two coins, with one resting on the other, will accelerate together down-slope, with identical velocities.

A stack of coins will not tumble, it will move as one cylinder down-slope. The stack of coins is a virtual rod, that is standing perpendicular to the slope.

The rod need not have a rounded foot. Like a coin, it could have a flat foot.
 
  • Informative
Likes DaveC426913
  • #32
Delta2 said:
Why do we consider torque only around center of mass?
You can consider moments around the end of the rod if you like, but then you need to include the ##ma## of the rod's center of mass accelerating downslope, which I didn't draw (it has no moment about the center of mass so it doesn't matter to my earlier argument). Since ##ma=mg\sin\theta##, the "rotation" of the rod around its end is just the rate of change of its angular coordinate as it slides down the slope.
 
  • Informative
Likes Delta2
  • #33
I agree that a rod, perpendicular to slope, with a rounded tip will slide without tipping. Depending on the curvature of the rod's rounded tip, this orientation may be stable or unstable.

A vertical rod with a rounded tip can slide without tipping if the center of curvature coincides with the rod's center of gravity and the slope is gentle enough that the contact point stays somewhere in the interior of the rounded tip.

A vertical rod with a rounded tip can slide and tip toward a stable perpendicular orientation if the center of curvature is beyond the rod's center of gravity and if the slope is sufficiently gentle to keep the contact point in bounds. The rod will rock back and forth about the perpendicular in this case.

Like one of these.
1654950951927.png

Edit to add one more case...

If center of curvature of the tip of a non-perpendicular rod is on the near side of the center of gravity then instead of rocking toward the perpendicular, the rod will topple away from the perpendicular. This is likely the case that most of us have in mind when we think about a rod with a rounded tip on an slippery inclined slope.
 
Last edited:
  • Like
Likes Steve4Physics, alan123hk and Ibix
  • #34
jbriggs444 said:
I agree that a rod, perpendicular to slope, with a rounded tip will slide without tipping. Depending on the curvature of the rod's rounded tip, this orientation may be stable or unstable.
You seem to be assuming there is some friction present at the contact patch.
Without friction, if the contact patch, contains the perpendicular through the centre of gravity, then the shape of the rod-end, and of the contact patch is quite irrelevant.
 
  • #35
Guys (@jbriggs444 and @Baluncore) when you say vertical rod you mean with its axis parallel to gravity, while when you say perpendicular to the slope you mean with its axis perpendicular to the slope right?
 
<h2>1. Will a round-headed rod topple if it slides down a frictionless slope?</h2><p>The answer to this question depends on the shape and dimensions of the rod, as well as the angle of the slope. Generally, if the center of mass of the rod is above the point where it touches the slope, it will topple. However, if the center of mass is below this point, it will not topple.</p><h2>2. What factors affect whether a round-headed rod will topple when sliding down a frictionless slope?</h2><p>The main factors that affect whether a round-headed rod will topple are the shape and dimensions of the rod, the angle of the slope, and the location of the center of mass of the rod. Other factors such as the material of the rod and the surface of the slope may also play a role.</p><h2>3. Can a round-headed rod with a low center of mass topple when sliding down a frictionless slope?</h2><p>No, a round-headed rod with a low center of mass will not topple when sliding down a frictionless slope. This is because the center of mass is below the point where the rod touches the slope, making it stable and less likely to topple.</p><h2>4. How does friction affect the stability of a round-headed rod sliding down a slope?</h2><p>Friction can play a significant role in the stability of a round-headed rod sliding down a slope. If there is friction between the rod and the slope, it can provide a stabilizing force and prevent the rod from toppling. However, if the slope is completely frictionless, the rod may be more likely to topple.</p><h2>5. Is there a specific angle at which a round-headed rod will always topple when sliding down a frictionless slope?</h2><p>No, there is not a specific angle at which a round-headed rod will always topple when sliding down a frictionless slope. The angle at which a rod will topple depends on its shape, dimensions, and center of mass. In general, the steeper the slope, the more likely the rod is to topple.</p>

1. Will a round-headed rod topple if it slides down a frictionless slope?

The answer to this question depends on the shape and dimensions of the rod, as well as the angle of the slope. Generally, if the center of mass of the rod is above the point where it touches the slope, it will topple. However, if the center of mass is below this point, it will not topple.

2. What factors affect whether a round-headed rod will topple when sliding down a frictionless slope?

The main factors that affect whether a round-headed rod will topple are the shape and dimensions of the rod, the angle of the slope, and the location of the center of mass of the rod. Other factors such as the material of the rod and the surface of the slope may also play a role.

3. Can a round-headed rod with a low center of mass topple when sliding down a frictionless slope?

No, a round-headed rod with a low center of mass will not topple when sliding down a frictionless slope. This is because the center of mass is below the point where the rod touches the slope, making it stable and less likely to topple.

4. How does friction affect the stability of a round-headed rod sliding down a slope?

Friction can play a significant role in the stability of a round-headed rod sliding down a slope. If there is friction between the rod and the slope, it can provide a stabilizing force and prevent the rod from toppling. However, if the slope is completely frictionless, the rod may be more likely to topple.

5. Is there a specific angle at which a round-headed rod will always topple when sliding down a frictionless slope?

No, there is not a specific angle at which a round-headed rod will always topple when sliding down a frictionless slope. The angle at which a rod will topple depends on its shape, dimensions, and center of mass. In general, the steeper the slope, the more likely the rod is to topple.

Similar threads

Replies
16
Views
16K
Replies
11
Views
8K
  • Introductory Physics Homework Help
Replies
13
Views
3K
  • Introductory Physics Homework Help
Replies
4
Views
1K
  • Introductory Physics Homework Help
Replies
6
Views
2K
  • Introductory Physics Homework Help
Replies
8
Views
2K
  • Introductory Physics Homework Help
Replies
33
Views
7K
  • Introductory Physics Homework Help
Replies
4
Views
1K
  • Introductory Physics Homework Help
Replies
6
Views
2K
Replies
10
Views
2K
Back
Top