# Block Toppling on an incline

1. Oct 26, 2014

### Satvik Pandey

1. The problem statement, all variables and given/known data
A block of base $10cm \times 10cm$ and height $15cm$ is kept on an incline plane. The coefficient of friction between them is $\sqrt { 3 }$. The inclination of the incline plain is $\theta$. Find the value of theta for which the block begins to topple.

2. Relevant equations
1)$\tau =I\alpha$
2)For equilibrium $\sum { \tau } =0$

3. The attempt at a solution

I
tried to make free body diagram of the block.

As the block begins to topple the I think at that moment point A can be treated as Instantaneous Axis of rotation. Can it be?

When the block begins to topple, does the Normal force on block would shift to point A?

If so then, if I take the moment of forces acting on the block then torque of force $N$ and $f$ about A will be $0$. But that's not correct.
How should I proceed. Sorry if I am missing something very common.

Last edited: Oct 26, 2014
2. Oct 26, 2014

### BvU

In your post I miss the relevant equations....

First thing that strikes me in the picture is that the block isn't resting on its base...

Your question about the action line of the normal force is interesting; intuitively I'm with you, but not quite certain. So perhaps we can try to make it irrelevant.

3. Oct 26, 2014

### Satvik Pandey

I have added two equations in my questions. Please have a look over them.

Could you please tell me how to proceed.

4. Oct 26, 2014

### voko

Very good question. To answer it, consider a more general problem: at which point should the normal force be applied for the block to be in equilibrium on an incline?

5. Oct 26, 2014

### ehild

Analyse the situation when the block is raised up a little bit from the slope, so that it touches it with the bottom edge only. Releasing, it either returns to its stable position, or topples over.
In that raised --up position, the normal force acts et the edge.

6. Oct 26, 2014

### BvU

Time to write down some of the torques around point A ....

7. Oct 26, 2014

### haruspex

Why do you say it is not correct?

8. Oct 26, 2014

### Satvik Pandey

Is the value of $\theta$ 33.69?

9. Oct 26, 2014

### haruspex

Yes, except that you had to make an assumption to arrive at that. Think about the given information you have not used.

10. Oct 27, 2014

### Satvik Pandey

I have not used value of friction because the frictional force passes through the point about which I have calculated torque. How should I include that?
Should I calculate torque about CoM of the cube?

11. Oct 27, 2014

### ehild

The coefficient of friction is quite big. What do you think, can the block slide on the slope ?

12. Oct 27, 2014

### Satvik Pandey

Well that depends on the value of $\theta$. It will not slide unless the value of $\theta$ is equal to the 60.

13. Oct 27, 2014

### Satvik Pandey

If the question asks us to to find the minimum value of $\mu$ (coefficient of static friction) required for block to topple for given $\theta$ then how should I approach to that question.
If I consider moment of forces about point A(please see figure in #post1) then I would not be able to get an expression involving the $\mu$. Should I consider torques about the CoM of the box?

14. Oct 27, 2014

### BvU

OK, so you found 33.69 degrees, or rather $\arctan{10\over 15}$ as the angle at which the thing topples -- provided the friction force is sufficient.
Since this is smaller than 60 degrees (or rather, $\arctan\sqrt 3$), the block will not slide but topple, so that's OK.

At an angle $\theta$ it will slide when the friction force $\mu_s mg \cos\theta$ is less than $mg \sin \theta$, in other words, $\mu_s \lt \ ?\$

Not necessary: all you want there is that it doesn't slide and that is about forces, not torques.

15. Oct 27, 2014

### haruspex

Quite so, but you're trying to answer whether it would slide, so you need to look at linear forces, not moments.

16. Oct 28, 2014

### voko

Why is the latter required? As far as I can see, if the block topples, it shall do so even if friction is absent.

17. Oct 28, 2014

### haruspex

Without friction, it will slide down the plane. What happens when it gets to the bottom we don't know. Maybe it slides right off, so won't topple.

18. Oct 28, 2014

### BvU

@voko: strong point. I imagine a gradual increase of theta until ... etc. There is no solid base for that perception, I agree. However, with $\theta > \arctan{10\over 15}$ and no friction, where can the torque to rotate around the c.o.m. come from ?

Now what about post 14 ? Makes things murky for me: no toppling unless $\theta > \arctan{10\over 15}$ and $\mu$ is irrelevant then ? Doesn't feel good.

Haru is on the same line I think.

19. Oct 28, 2014

### voko

Even without friction, the normal force and the weight can produce a non-zero couple when the angle is sufficiently great.

20. Oct 28, 2014

### BvU

Don't see a rotation around the c.o.m. coming up ...