Building a trebuchet

In this Autumn, I had been working on a project of building a trebuchet. This project is required by a class I was taking at the University of Washington. This class is about studying science in ancient Greece and ancient Rome. The instructor of this class wanted the students to form groups and try to recreate an ancient technology. The ancient technology chosen by our group is the trebuchet. During the whole autumn quarter, our group worked on this project and tried to make a trebuchet that works properly. The whole process was challenging but fun. Although we encountered several challenges, the project was successful in the end. Trebuchet is an amazing invention. From the process of building the trebuchet, I realized that making a trebuchet work properly is not easy. There are many challenges to overcome during the construction and there are many factors that affect the launching. The mechanism of the trebuchet involves a lot of physics. Therefore, I am going to talk about the physics of the trebuchet in this article.

Trebuchet is war machine that launches projectiles. The mechanism of the trebuchet is quite simple. It basically uses the falling energy of the counterweight to throw the projectile. A trebuchet is consist of a beam attached to an axle that is suspended above the ground by a stable frame and base. The beam is attached in a way that makes one end longer and one end shorter. The counterweight is attached to the shorter end while a sling and the projectile is attached to the longer end. Before launching the trebuchet the projectile has to be loaded to a holder attached to the sling and the counterweight has to be lifted to a certain height. When the trigger is pulled, the counterweight will drop and rotate the beam. When the beam rotates to a certain angle, the projectile will be released. The reason for attaching the projectile to the longer arm is to give it a greater linear velocity. In order to launch the projectile at a high speed, the counterweight must be much heavier than the projectile. Various sources on the internet suggest that the optimal weight ratio is 1:133. (If the projectile is 1kg, the ideal weight of counterweight is 133kg ). However, I couldn't find a single source that explains why the optimal weight ratio is 1:133. Therefore, I tried to work it out myself. I end up coming up with my own explanation as follows.

 First, when the counterweight is falling without attaching to anything, it has an acceleration of 9.8m/s^2. When it is attached to the beam, however, it falls with an acceleration smaller than 9.8m/s^2 because the beam and the projectile as a whole have rotational inertia. From Newton's second law we know that: F=ma. Since the motion of the beam is rotational instead of translational, we need to adjust the formula to τ=Iα. (τ is the torque, I is the rotational inertia, and α is the angular acceleration). When the weight of the counterweight increases, the angular acceleration increases as well. However, the angular acceleration has a limit. (Because the greatest acceleration of the counterweight is 9.8m/s^2). When the counterweight is much heavier than the projectile, the acceleration of the counterweight will be very close to 9.8m/s^2, but the acceleration never exceeds that value. Therefore, increasing the weight of the counterweight will have little effect at certain point. That's why the optimal weight ratio is 1:133.

The design of the trebuchet includes a sling attached to the longer end of the beam. When I was constructing the trebuchet, I wondered what is the point of attaching the sling. I tried to find the answer on the internet but failed. Then I worked it out myself again and figured out that the point of attaching the sling is to increase the release height of the projectile. When the release height is increased, the projectile will stay in the air longer. The longer the projectile stays in the air, the further it will fly. Although there are other ways to increase the release height such as lengthening the beam and making the frame taller, simply attaching a sling is a much easier way to get the result more efficiently. 

The construction of the trebuchet is difficult and its launching mechanism is actually very complicated. Our team bade a trebuchet with a 3 feet* 2 feet base, 2 feet tall frame, 4 feet long beam, and 3 feet long sling. Because the launching of the trebuchet was affected by many factors, we adjusted the sling length and the position of the projectile a few time before we can launch it with nearly 45-degree angle (the optimal launching angle). In the end, we can shoot a softball about 50 feet far. We were satisfied with the result and enjoyed the process of making the trebuchet.