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A thin plank of mass $M$ and length $l$ is pivoted at one end. The plank is released at 60$^{\circ}$ from the vertical. What is the magnitude and direction of the force on the pivot when the plank is horizontal?

I have a doubt: does the pivot exert an upwards force on the plank when the plank is horizontal? I think the answer to this is yes, because:

The torque on the plank about the pivot, when the plank is horizontal, is

$$\tau=\frac{Mgl}{2}$$

And the plank's angular acceleration at that point will be

$$\alpha=\frac{\tau}{I}=\frac{3g}{2l}$$

So the vertical acceleration of the plank's CM will be

$$a=\alpha R=\frac{3g}{4} $$

Therefore, the upwards force exerted by the pivot on the plank is

$$F_{up}=\frac{mg}{4}$$

Is this reasoning ok for this case? (I'm doubtful of this part because there was a post on PF where this force was not mentioned.)

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  • $\begingroup$ There is post in here somewhere asking if there is plank held up by two wires on each end, and one gets cut to find the tension on the other and the answer was $\frac{m g}{4}$ also. $\endgroup$ – ja72 Sep 27 '15 at 23:32
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Yes, spot on! Another way of proving that this is right is through d'Alembert's principle. This involves the drastic transformation of Newton's 2nd Law of:

$$F_{net} = ma_G$$

(where $a_G$ is the acceleration of the centre of mass)

into:

$$F_{net} - ma_G = 0$$

Indeed this may seem very little has changed by this algebraic manipulation. However, it is worth noting that the equation is now in the following form:

$$F' = 0$$

In other words, it is possible to treat this dynamics problem as a statics problem by defining an equivalent but stationary system with a net force of $F' = F_{net} - ma_G$. Therefore, we can solve this new system by enforcing equilibrium equations in order to determine the dynamics of the original system.

This can be extended to the rotational version of 2nd law, for the equilibrium equation for torque:

$$\tau' = \tau_{net} - J\alpha = 0$$

For the problem in the question, the usual dynamic system is given by:

enter image description here

The equivalent stationary system is:

enter image description here

Note that forces and moments are shown in red and accelerations are shown in blue.

$I_G$ is the moment of inertia of the rod taken about its centre of mass.

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  • $\begingroup$ Newton's 2nd law is not $F_{net} = ma_G$, it's $F_{net} = ma_{net}$. Gravity really is one of the forces contributing to $F_{net}$ on the left side. $F_G = ma_G$ is a separate formula, and its superficial similarity to N's 2nd is something of a mystery. I know how pedantic I sound, but IMO it's important to realize that gravity is just another force. $\endgroup$ – LastStar007 Dec 15 '16 at 22:33
  • $\begingroup$ Sorry, my notation is a bit confusing: $a_G$ is the (net) acceleration of the centre of gravity as opposed to being the acceleration due to gravity. I realise that gravity is indeed a force (or inertial effect, thanks to the equivalence principle). My point is that you can treat a dynamics problem $F=ma$ as a static equilibrium problem $F'=0$ by defining an equivalent stationary system with a net force $F' = F-ma$ acting on it. $\endgroup$ – Involutius Dec 16 '16 at 10:18
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A free body diagram is your friend ($c= \frac{\ell}{2}$)

fbd

The general case above yields the following equations of motion

$$\begin{align} A_x & = m \ddot{x}_C \\ A_y - m g & = m \ddot{y}_C \\ A_x c \sin \theta - A_y c \cos \theta &= I_C \ddot{\theta} \end{align} $$

Since the pivot does not move, the acceleration of the center of mass $(\ddot{x}_C,\,\ddot{y}_C)$ is

$$ \begin{align} \ddot{x}_C & = - c\dot{\theta} \cos\theta - c \ddot{\theta} \sin\theta \\ \ddot{y}_C & = - c \dot{\theta}^2 \sin \theta + c \ddot{\theta} \cos\theta\end{align} $$

Using the motion of the center of mass in the equations of motion, and setting $\theta=0$ yields

$$ \left. \begin{align} A_x & = - m c \dot{\theta}^2 \\ A_y &= m c \ddot{\theta} + m g \\ -A_y c &= I_C \ddot{\theta} \end{align} \right\} \begin{aligned} \ddot{\theta} &= -\frac{m c g}{I_C + m c^2} \\ A_x &= -m c \dot{\theta}^2 \\ A_y &= \frac{I_C}{I_C+m c^2} m g \end{aligned} $$

using $c=\frac{\ell}{2}$ and $I_C =\frac{m}{12} \ell^2$ yields a familiar value $$ A_y = \frac{m g}{4}$$

To find the rotational speed $\dot{\theta}$ you use conservation of energy to relate initial potential energy to final kinetic energy.

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  • $\begingroup$ What did you use to make the FBD? $\endgroup$ – LastStar007 Dec 15 '16 at 22:27
  • $\begingroup$ @LastStar007 I used PowerPoint with the IguanaTex addin for the math equations. $\endgroup$ – ja72 Dec 15 '16 at 22:29
  • $\begingroup$ Lol it came out looking very clean. I was hoping there was something more physics-oriented :/ $\endgroup$ – LastStar007 Dec 15 '16 at 22:38
  • $\begingroup$ Me too. I was hoping there is a easy sketch maker, but alas. Haven't found it yet. $\endgroup$ – ja72 Dec 16 '16 at 3:19
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Yes! Your reasoning is absolutely correct! There will be two forces exerted by the pivot: one in the horizontal direction and the other in the vertical direction!

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