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This is my first time here, and I have a question regarding the calculation of the angle that an object would tip at, when a certain force is applied to its center of gravity.

(This question seems similar to this one, but that doesn't really answer my question; because its looking for a threshold height, but I'm looking for the angle of tipping.)

So, let's suppose we have a cylinder with a circular base of radius $r$, whose center of gravity is a perpendicular distance $h$ above its circular base. I am interested in looking at the angle that it would tip at, if we apply a horizontal force through its center of gravity.

The cylinder is placed on a horizontal surface, offering as much friction as is necessary to prevent the object from sliding (the friction acting is equal to the force applied).

This is the free body diagram I came up with.

FBD1

Here, if the torque caused by the weight force is greater than the torque by the applied force, it would not tip. However, if we apply a bigger force, we can assume that the object tips, and we have the following situation.

FBD2

Notice that $ER=h$ and $RB=r$.

The angle of tipping must be one that cancels out the torques of the weight and the applied force.

With some geometry, we have the following results.

$\angle EQR = \theta$. Hence $EQ=\dfrac{h}{\sin\theta}$, and $RQ=\dfrac{h\cos\theta}{\sin\theta}$.

$QB=RB-RQ=r-\dfrac{h\cos\theta}{\sin\theta}$.

Also, $\angle PQB=\theta$. Therefore, $PB=QB\sin\theta$, and $QP=QB\cos\theta$.

$PB=\left(r-\dfrac{h\cos\theta}{\sin\theta}\right)\sin\theta=r\sin\theta-h\cos\theta$.

The torque $\tau_W$ by the weight force is $mg\times PB$.


$$ \tau_W=mg(r\sin\theta-h\cos\theta)\qquad\qquad\qquad(1) $$


$QP=\left(r-\dfrac{h\cos\theta}{\sin\theta}\right)\cos\theta=r\cos\theta-\dfrac{h\cos^2\theta}{\sin\theta}$.

$EP=EQ+QP$

$\qquad=\dfrac{h}{\sin\theta}+r\cos\theta-\dfrac{h\cos^2\theta}{\sin\theta}$

$\qquad=r\cos\theta+\dfrac{h}{\sin\theta}(1-\cos^2\theta)=r\cos\theta+h\sin\theta$.

The torque by the applied force is $\tau_F$ is $EP\times F$.


$$ \tau_F=F(r\cos\theta+h\sin\theta)\qquad\qquad\qquad(2). $$


We need $(1)=(2)$ for equilibrium, so we have

\begin{align} mg(r\sin\theta-h\cos\theta)&=F(r\cos\theta+h\sin\theta)\\ \\ \sin\theta(mgr-Fh)&=\cos\theta(Fr+mgh)\\ \\ \tan\theta&=\dfrac{Fr+mgh}{mgr-Fh}\\ \\ \theta&=\arctan\left(\dfrac{Fr+mgh}{mgr-Fh}\right). \end{align}

However, this is clearly wrong, because as $F$ increases, $\theta$ increases, when it must in fact, decrease.

(The following is a variation of $\theta$ against $F$).

Angle against F 1

What is wrong here?


I also approached this problem in a different way, and got a more realistic answer.

We know that the friction force and normal reaction have a resultant, inclined at some angle to the horizontal. For the object to be in equilibrium, it must tip at some angle so that this resultant passes through its center of gravity.

FBD3

Suppose $\vec{R}$ is the resultant of the friction force and normal reaction.

We have that $f=F$ and $N=mg$, because the object is in equilibrium.

The angle $\angle EBS$ that $R$ makes with the horizontal is $\arctan \dfrac{N}{f}=\arctan \dfrac{mg}{F}$.

$\angle EBR=\arctan \dfrac{ER}{RB}=\arctan\dfrac{h}{r}$.

If we measure the angle $\angle RBS$ as the angle of tipping in this case, we have the following.

\begin{align} \angle RBS&=\angle EBS-\angle EBR\\ \\ &=\arctan\dfrac{mg}{F} -\arctan\dfrac{h}{r}\\ \\ &=\arctan \left(\dfrac{\dfrac{mg}{F}-\dfrac{h}{r}}{1-\dfrac{mgh}{Fr}}\right)\\ \\ &=\arctan\left(\dfrac{mgr-Fh}{Fr-mgh}\right). \end{align}

Now this makes sense, because as $F$ increases, $\angle RBS$ increases, as seen in the plot below.

Angle against F 2

So which way is actually correct? Why do the two approaches give different results?

(Please do answer in as simple terms as possible, because I'm still a beginner :) )

Thank you for taking your time.

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  • $\begingroup$ For F=0 $\theta$ must be 90 degrees? $\endgroup$ – Eli Dec 22 '19 at 11:36
  • $\begingroup$ No, for $F=0$, $\theta=\arctan\dfrac{h}{r}$. $\endgroup$ – Pulasthi Udugamasooriya Dec 28 '19 at 10:44
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Approach 1: ... because as F increases, θ increases, when it must in fact, decrease.

The $F$ in your problem is the external force required to keep the object in equilibrium at inclination $\theta$. And it makes sense that $\theta$ decreases as $F$ decreases: If I had two positions/orientations of cylinders (one at $\theta_1$ and another at $\theta_2$, where $\theta_2>\theta_1$) and I asked you to exert a force $F$ to maintain equilibrium, I think you would agree that less force is required to maintain the equilibrium for $\theta_1$ as compared to $\theta_2$.

In fact, $F$ is zero when the centre of gravity is directly above the point of contact with the floor (point B in your figure): This means that no external force is required to maintain equilibrium. This is the critical angle $\theta_c = \tan^{-1}(\frac{h}{r})$ after which $F$ changes sign.

Approach 2: We know that the friction force and normal reaction have a resultant, inclined at some angle to the horizontal. For the object to be in equilibrium, it must tip at some angle so that this resultant passes through its center of gravity.

They do yield consistent results. Check with the identity below. $$\tan^{-1}x-\tan^{-1}y=\tan^{-1}(\frac{x-y}{1+xy})$$ Also, you've mistakenly identified ($90^o-\theta$) as $\theta$ in approach 2: $\angle RBS = 90^o - \angle EQR$.

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  • $\begingroup$ Thanks a lot for this! How careless I had been to make such mistakes! $\endgroup$ – Pulasthi Udugamasooriya Dec 22 '19 at 6:45
  • $\begingroup$ What would happen if we apply a force greater than the maximum static friction; that is also big enough to cause it to tip? Would it move forward while tipping? How do I visualize that? $\endgroup$ – Pulasthi Udugamasooriya Dec 22 '19 at 15:03
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A constant horizontal force through the center of mass won't lead to an angle. It will either tip or not tip. If the force is sufficient to begin tipping, it will go all the way over.

Assuming a tipping corner, gravity (through the center of mass) provides a torque about that corner. When there is no applied force, the normal from the surface it sits upon provides the restoring torque and the object does not move.

As you apply a force from the side, the normal force shifts to reduce its torque. The limit of this shift is when the normal force acts directly through the tipping corner. At that point we can ignore its contribution. So the limit is reached when the torque from the applied force equals the torque from gravity.

If tipping begins, the center of mass will rise (increasing the torque from the applied force) and will move to be horizontally closer to the pivot (decreasing torque from gravity). So the tip situation is unstable and will continue to tip unless the force is decreased.

Related: Position of centre of mass during toppling

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I believe your second method has a mistake. In the last two equations of that method you have an expression with the arctan of a fraction. Shouldn't the minus sign in the denominator of the fraction actually be a plus sign? That is, in the final expression, the denominator $Fr-mgh$ should be $Fr+mgh$. This will make your two methods consistent.

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