# Torque: The T-shaped stick problem

This is a problem that I have been unable to solve for some time.

Imaging a T-shaped stick, as shown in below image, which do not deform in any appreciable way and has pivot point at the tail of the "T".

There are two questions in my problem, the first is: given that the "T" is symmetrical, how would applying two units of force on one of the "T"'s horizontal appendage differ from two one unit of force applied on both appendages in opposite direction, in terms of the torque received at the pivot? (See image)

The second question is: how would the situation differ if the "T" is "italicized", but the forces are exerted perpendicular to the appendages? see image:

What is a systematic way to explain and describe the torque received at the pivot in terms of the lengths L and H and possibly the angles between them?

## 2 Answers

If a force $\vec{F}=(F_x,F_y)$ is applied at a location $\vec{r}=(x,y)$ then the torque at the origin is

$$\vec{\tau} = \vec{r} \times \vec{F} \\ \tau = x F_y - y F_x$$

All you need to do is sum up the torques at the pivot for the different situations in order to understand how this mechanism will move.

If you have two equal and opposite forces $F$ the net torque applied is going to be $F\,d$ where $d$ is the perpendicular distance between them. Care must be taken to consider if this torque is positive (counter-clock-wise) or negative (clock-wise).

The torque is done with a cross product and thus is going to be:$$\tau = |\vec F| |\vec r| \sin\theta$$where $\theta$ is the angle between the position vector $\vec r$ (which points from the pivot to the place where the force is applied) and $\vec F$ (which points however the force points).

You can also express this as $x F_y - y F_x$ and since your forces point in the $x$-direction, the only term is $-y F_x$. The following effects hold:

1. As long as the pivot is fixed, there is no torque difference between your two scenarios. The only difference is that when you only push with the $2~\rm N$ force on the top, the pivot needs to push back with a $2~\rm N$ force to be fixed in place -- so if you press too hard you could hypothetically break the pivot or the beam between, whereas the balanced $-1~\rm N$ at $+y$ and $+1~\rm N$ at $-y$ do not require any constraint force from the pivot.

2. The only difference when you "italicize" is that the sides come closer to the center-line of the T, as long as your forces remain in the $x$-direction. In fact if you lengthen the sides slightly so that these $y$-coordinates remain the same, there is no difference! As we can see that this lowers the "effort arm" of the lever that you're using, the torque will lessen if you "italicize" the T without thus lengthening it.