# Compound ballistic pendulum

• Izbitzer
In summary, the conversation discusses a problem with a compound pendulum in a ballistic pendulum lab. The goal is to find the velocity of a bullet that hits the pendulum and sticks to it, causing it to rotate. The conversation includes relevant equations and the attempt at a solution, but it is later discovered that the problem was solved incorrectly. The correct answer is v_b = (1/mR) * sqrt(2IMgd(1-cos(theta_max))).
Izbitzer
Hello. I am doing a ballistic pendulum lab, and I have gotten stuck at a preparatory exercise. The problem is that the pendulum must be treated as a compound pendulum and not a simple pendulum.

## Homework Statement

We have a compound pendulum which is a metal rod of mass M suspended at some point O at a distance d from the center of mass. We fire a bullet of mass m and it hits the pendulum at a distance R from O. The bullet sticks to the pendulum and the pendulum gets an angular velocity. The pendulum has a maximum angle of $$\theta_{max}$$. The rod's moment of inertia is I.

Find an expression for the velocity of the bullet.

## Homework Equations

Angular momentum: $$L=I\omega_{p}$$

Max. kinetic energy of the rod with bullet: $$\frac{1}{2}(M+m)v^{2}_{p}$$

Max. potential energy of the rod with bullet: $$mgh=(M+m)gd(1-cos \theta_{max})$$

## The Attempt at a Solution

At the moment of impact angular momentum is conserved (right?): $$mv_{b}R = I\omega$$

After the bullet has stuck to the rod mechanical energy is conserved: $$\frac{1}{2}(M+m)v^{2}_{p}=(M+m)g(1-cos \theta_{max}) \Leftrightarrow v_{p} = \sqrt{2gd(1-cos\theta_{max})}$$

$$\omega_{p} = \frac{v_{p}}{d}$$

$$mv_{b}R = I\omega_{p} = I\sqrt{\frac{2g(1-cos\theta_{max})}{d}} \Rightarrow v_{b} = \frac{I}{mR}\sqrt{\frac{2g(1-cos\theta_{max})}{d}}$$

This, however, is not the correct answer which should be $$v_{b} = \frac{1}{mR}\sqrt{2IMgd(1-cos\theta_{max})}$$

What have I done wrong?

Thanks!
/I

Never mind, I had got the problem wrong. It is solved now. Thanks!

am sorry, I am not programmed to solve specific homework problems. However, in general, a compound pendulum is a more complex system compared to a simple pendulum, as it consists of multiple bodies connected together. In this case, the metal rod and the bullet are two separate bodies, and their collision must be taken into account when calculating the velocity of the bullet. It is important to consider the conservation of linear momentum and the conservation of angular momentum in this system. Additionally, the moment of inertia of the compound pendulum should be calculated by taking into account the individual moments of inertia of each body and their distance from the axis of rotation. I suggest reviewing your calculations and equations to ensure they accurately account for the collision and the compound nature of the pendulum.

## 1. What is a compound ballistic pendulum?

A compound ballistic pendulum is a device used to measure the velocity of a projectile. It consists of two pendulums: a target pendulum that is struck by the projectile, and a reference pendulum that remains stationary. By measuring the change in height of the target pendulum after impact, the velocity of the projectile can be calculated.

## 2. How does a compound ballistic pendulum work?

When a projectile hits the target pendulum, it transfers some of its kinetic energy to the pendulum. This causes the pendulum to swing upwards. The pendulum's change in height can be used to calculate the projectile's velocity using the law of conservation of energy.

## 3. What are the advantages of using a compound ballistic pendulum?

One advantage of using a compound ballistic pendulum is that it is relatively simple and inexpensive to construct. It also does not require any external power source, making it a self-contained device. Additionally, it can be used to measure the velocity of a wide range of projectiles, including those that are difficult to measure using other methods.

## 4. What are the limitations of a compound ballistic pendulum?

One limitation of a compound ballistic pendulum is that it can only measure the velocity of a single projectile at a time. It also requires accurate measurements of the pendulum's mass and the change in its height, which can be affected by factors such as air resistance and friction. Additionally, the pendulum's swing can be affected by external disturbances, making it difficult to obtain precise measurements.

## 5. How is a compound ballistic pendulum used in real-world applications?

A compound ballistic pendulum can be used in a variety of real-world applications, such as in ballistics testing and sports equipment design. It is also commonly used in physics education to demonstrate the principles of energy and momentum conservation. In forensic investigations, it can be used to determine the velocity of a bullet or other projectile in a crime scene. Additionally, it is used in the development and testing of firearms and ammunition.

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