Designing a Catapult: Calculating Mass Moment of Inertia & Launch Speed

[BuickBoy]

This is a catapult design project.

Homework Statement

The counterweight is 400lb, the boulder (projectile) is 50lb and the throwing arm is 100lb. The throwing arm is assumed to be rigid, uniform in density, and weight. The throwing arm is also 17ft in total length. The throwing arm starts at forty degrees and rotates about a pivot point counterclockwise, which launches the boulder from a chosen (optimal) angle. The pivot point is 12ft above the ground.

Determine the center mass of the launch subsystem, mass moment of inertia of the launch subsystem about its center of mass (assuming the throwing arm is a long slender rod and the counterweight and boulder are not point loads), mass moment of inertia of the launch subsystem about the pivot point, comparing results of the mass moments of inertia and comment on the results relative to the efficiency of a catapult system, usisng the concepts of work and energy , determine the boulders release speed, and calculate the maximum launching distance of your optimized catapult design.

The final steps include assuming W (omega) of the sling is equal to 1.5 x W(omega)of the launch arm. (This part I don't really need help with

Homework Equations

Work and Energy Equations

The Attempt at a Solution

Without having to type out my current work, I'm just trying to get started. The class is very vague. I started trying to calculate the velocity of the arm as it would rotate to the bottom of the support using the same concepts of a pendulum, I figured if I could assume each side of the pinned throwing arm was half of a pendulum the difference in the velocities would be accurate. I was told I'm wrong. Back to square 1.


Thanks in advance.

 

[KataKoniK]

Hello,

Thank you for sharing your project details. I would approach this catapult design project by first analyzing the forces involved and then applying the concepts of work and energy to determine the efficiency and maximum launching distance.

To start, let's consider the forces acting on the catapult system. We have the counterweight, the boulder, and the throwing arm. The counterweight and boulder will exert a downward force, while the throwing arm will exert a force in the direction of rotation. The pivot point will also experience a reaction force from the throwing arm.

Next, we can calculate the center of mass of the launch subsystem by taking the weighted average of the individual masses and their distances from the pivot point. This will give us the location of the center of mass, which will be important in determining the moment of inertia.

To calculate the moment of inertia of the launch subsystem, we can use the formula I = Σmr^2, where m is the mass of each component and r is the distance from the pivot point. We will need to consider the throwing arm as a long slender rod and use the parallel axis theorem to account for its rotation about the center of mass.

To compare the results of the moment of inertia about the center of mass and the pivot point, we can use the concept of work and energy. The efficiency of a catapult system can be measured by the ratio of the work done on the projectile to the work done by the throwing arm. A more efficient system will require less work to launch the projectile a certain distance.

To determine the release speed of the boulder, we can use the conservation of energy principle. The potential energy of the boulder at the top of the throwing arm will be converted into kinetic energy at the moment of release.

Finally, to calculate the maximum launching distance of the catapult, we can use the projectile motion equations, taking into account the release speed, angle of launch, and the gravitational acceleration.

I hope this helps you get started on your project. Let me know if you have any further questions. Good luck!

 

[Mene]

it is important to approach this project with a systematic and analytical mindset. First, it is important to gather all the necessary information and understand the goals of the project. In this case, the goal is to design a catapult that can launch a boulder at a chosen angle and distance.

Next, the mass and dimensions of all the components of the catapult must be determined. This includes the counterweight, throwing arm, and boulder. The mass of the throwing arm and boulder can be easily obtained, but the mass of the counterweight may need to be measured. It is also important to determine the dimensions of the throwing arm, such as its length and the distance from the pivot point to the center of mass of the launch subsystem.

Once all the necessary information is gathered, the center of mass of the launch subsystem can be calculated using the formula:
xcm = (m1x1 + m2x2 + ... + mnxn) / (m1 + m2 + ... + mn)
where xcm is the center of mass, m is the mass, and x is the distance from the pivot point.

The mass moment of inertia of the launch subsystem about its center of mass can be calculated using the formula:
Icm = m1r1^2 + m2r2^2 + ... + mnrn^2
where Icm is the mass moment of inertia, m is the mass, and r is the distance from the center of mass.

Similarly, the mass moment of inertia of the launch subsystem about the pivot point can be calculated using the parallel axis theorem:
Ip = Icm + md^2
where Ip is the mass moment of inertia about the pivot point, Icm is the mass moment of inertia about the center of mass, m is the mass, and d is the distance between the pivot point and the center of mass.

Comparing the results of the two mass moments of inertia can provide insight into the efficiency of the catapult system. A higher mass moment of inertia about the pivot point would indicate that more energy is required to rotate the throwing arm, which could result in a slower launch speed and shorter distance.

Using the concepts of work and energy, the launch speed of the boulder can be calculated by equating the kinetic energy of the boulder at launch to the work done by the throwing arm:
0.5mv^2 = Fd
where m is the mass of the boulder