Counterweight Stall

# Definition

The Counterweight Stall is when the counterweight stops moving.

# Significance

Once the counterweight has stopped moving, it no longer has any kinetic energy. That means any kinetic energy that was in the counterweight has been transfered, usually to the arm.

# Background: Energy Transfers

Trebuchets, before they throw, they are cocked. A cocked trebuchet has no kinetic energy (nothing is moving), but it does have lots of potential energy stored in the counterweight's potential drop height. During the throw energy is preserved (conservation of energy). Once the trigger is pulled, the potential energy of the counterweight begins to be transformed into kinetic also stored in the counterweight. As the counterweight falls it is slowed by the arm. Thusly, it transfers the kinetic energy to the arm. If all the kinetic energy is transfered, the counterweight will no longer be moving (no kinetic energy) and is considered stalled. (Counterweight Stall)
Just as the counterweight is slowed by its transfer of energy to the arm, the arm is slowed by its transfer through the sling to the projectile. If all the kinetic energy from the arm is transfered to the projectile, the arm will no longer be moving (no kinetic energy) and is considered stalled. (Arm Stall)
Sometimes energy is stored in other ways during a throw such as the kinetic energy of the frame or floating axle. This adds the concepts of frame and axle stall to some types of trebuchets.

# Counterweight Energy and Velocity

The counterweight's kinetic energy is stored in its velocity. Below is the equation where $E_k$ is the kinetic energy, $v$ is the velocity and $m$ is the mass.

(1)
\begin{align} E_k = \begin{matrix} \frac{1}{2} \end{matrix} mv^2 \end{align}

The most important fact here is that kinetic energy is zero if the velocity is zero. This means that transferring the kinetic energy to the arm simply requires stopping the counterweight (reducing its velocity to zero) with the arm. When doing this where are two parts of the velocity to consider, the the tangential component and the normal (perpendicular) component.

## The Normal Component

The normal component of the counterweights velocity is the part that is going toward or away from the center of rotation (axle). If the counterweight is moving straight away from the axle, its normal component contains all the velocity.

## The Tangential Component

The tangential component of the counterweights velocity is the part that is tangent to the circle centered at the center of rotation (axle or hanger axle). If the counterweight is moving around the axle, like a fixed counterweight trebuchet, its tangential component contains all the velocity.

## Changing the Components

There are several ways the components changed produced.

### Gravity

Gravity accelerates the counterweight downward converting the potential energy to kinetic energy. As long as the counterweight is not directly to the side (at the same height of) the counterweight, some of gravity will be a normal force, which, depending on the directions the counterweight can move, can cause an increase in normal velocity. The same applies for the tangential component except it requires the counterweight to not be above or below the axle. If the counterweight is going up, gravity subtracts from the velocity and adds to the potential energy.

### Counterweight Movement

As the counterweight moves, the direction toward the axle changes and thus, one of the velocity's components can be converted into the other simply by the change in location of the counterweight. One some what complicated effect of this is the centripetal force (which is a normal force) which happens then the transition to normal velocity because of counterweight movement is prevented as in a fixed counterweight trebuchet.

### Axle Movement

The direction from the counterweight to the axle can also be changed by axle movement which works the same way as counterweight movement above. If the counterweight is moving down and the axle moves over it (such as in a Floating Arm Trebuchet when the arm goes vertical, and to a lesser extent in any wheeled trebuchet) all the counterweights velocity will be in its normal component and none in the tangential component. If the counterweight is moving down and the axle moves to the side of it (such as in a Floating Arm Trebuchet when the arm is horizontal) all the counterweights velocity will be in its tangential component and none in the normal component.

### Transferring Energy to or from the Arm

The components can also be changed through transferring energy to or from the arm. This is discussed below under "Stalling the Components".

# Stalling the Components

The two components of the counterweight's velocity can be stalled independently.

## Stalling the Tangential Component

The tangential component of the counterweight's velocity is usually stalled (reduced to zero) in two ways: transferring it to the normal component, or accelerating the arm.

### Accelerating the Arm with the Tangential Component

The tangential velocity can be used to move the arm at the same rotational speed as the counterweight (both rotating around the axle). This effectively joins the tangential energy stored in the counterweight with the rotational energy of the arm. This is because slowing the arm down slows the counterweight's tangential velocity. This is simplest with Fixed Counterweight Trebuchets. This does make the the counterweight's tangential velocity available to the sling, but it does not speed up the arm (it simply makes it heavier). The sling is not able to such as much energy out of the arm at the lower speed and thus when the arm is effectively heavier instead of faster the sling is unable to use as much energy and the efficiency decreases. This is why Fixed Counterweight Trebuchets lose lots of efficiency as the mass ratio increases.

### Converting the Tangential Velocity into Normal Velocity

With direct transfer or tangential counterweight energy to the arm and sling (described above) not working very well in many cases (high mass ratios or short durations especially), trebuchets often convert tangential velocity into normal velocity (using counterweight and frame movement as described above) so it can me transfered to the arm with the methods below.

## Stalling the Normal Component

### Arm hanger Alignment

Frequently hinged counterweight trebuchet's (including all trebs with hangers like whippers and King Arthurs) stalls work by converting the counterweight's velocity entirely into the normal component and stalling it through arm/hanger alignment. Arm hanger alignment is where the arm and hanger become in a line. When this happens the counterweight is as far as possible from the axle and therefore can not move ant further away. This means that its normal velocity must be zero. The energy has to go somewhere and it does: it does into the arm. Here is how: as the arm and hanger approach alignment, the angle that the arm rotates for each unit distance the counterweight moves away from the axle approaches infinity (Graphs and equations coming soon). The Same thing happens with Floating Arm Trebuchets, but it happens through axle movement.

# Counterweight Stall Location

Ideally, at release, the counterweight (and arm) will have no energy, potential or kinetic. This means that the ideal release is at counterweight stall, and where the counterweight is at its lowest point.
-to be continued-

page revision: 9, last edited: 17 Jun 2008 03:30