Question

A ballistic pendulum consists of an arm of mass \(M\) and length \(L=0.48 \mathrm{~m} .\) One end of the arm is pivoted so that the arm rotates freely in a vertical plane. Initially, the arm is motionless and hangs vertically from the pivot point. A projectile of the same mass \(M\) hits the lower end of the arm with a horizontal velocity of \(V=3.6 \mathrm{~m} / \mathrm{s}\). The projectile remains stuck to the free end of the arm during their subsequent motion. Find the maximum angle to which the arm and attached mass will swing in each case: a) The arm is treated as an ideal pendulum, with all of its mass concentrated as a point mass at the free end. b) The arm is treated as a thin rigid rod, with its mass evenly distributed along its length.

Short answer

Answer: a) In the case of an ideal pendulum, the maximum angle is approximately \(18.38^\circ\). b) In the case of a thin, rigid rod, the maximum angle is approximately \(11.11^\circ\).

Step-by-step solution

Case a) Ideal Pendulum

In this case, we can assume that all the mass of the arm is concentrated at the free end. The initial energy of the system is entirely kinetic energy from the projectile: \(E_{i}=\frac{1}{2}Mv^2\) When the pendulum reaches its maximum height, all kinetic energy has been converted to potential energy: \(E_{f}=Mgh\) Where \(h\) is the increase in height of the center of mass. We can find the height by using geometry. When the pendulum is at its maximum angle, we have: \(h=L(1-\cos(\theta_{max}))\) Now, we use the conservation of energy principle: \(E_i = E_f\) \(\frac{1}{2}Mv^2 = MgL(1-\cos(\theta_{max}))\) Since every term in the equation has \(M\), we can cancel it out: \(\frac{1}{2}v^2 = gL(1-\cos(\theta_{max}))\) Now, by rearranging the terms and plugging in the given values, we can solve for \(\theta_{max}\): \(\theta_{max}=\cos^{-1}\left(1-\frac{v^2}{2gL}\right)\) \(\theta_{max}=\cos^{-1}\left(1-\frac{(3.6)^2}{2\times9.81\times0.48}\right) \approx 18.38^\circ\)

Case b) Thin, Rigid Rod

In this case, the arm is treated as a thin, rigid rod whose mass is distributed evenly along its length. We will still use the conservation of energy principle. The initial energy is still the kinetic energy of the projectile: \(E_{i}=\frac{1}{2}Mv^2\) Now, when the arm reaches its maximum height, the center of mass will be at a different position compared to when it was an ideal pendulum. Since the mass is evenly distributed, the center of mass will be located in the middle of the arm. The height of the center of mass can be expressed as: \(h=L\frac{1}{2}(1-\cos(\theta_{max}))\) The final energy, which will be the potential energy of the center of mass, is given by: \(E_{f}=\frac{3}{2}Mgh\) Using the conservation of energy principle, we have: \(E_i = E_f\) \(\frac{1}{2}Mv^2 = \frac{3}{2}MgL\frac{1}{2}(1-\cos(\theta_{max}))\) After canceling out \(M\), we can rearrange the terms and solve for \(\theta_{max}\): \(\theta_{max}=\cos^{-1}\left(1-\frac{v^2}{3gL}\right)\) \(\theta_{max}=\cos^{-1}\left(1-\frac{(3.6)^2}{3\times9.81\times0.48}\right) \approx 11.11^\circ\) So, the maximum angles for the two cases are as follows: a) \(\theta_{max} \approx 18.38\,^\circ\) (Ideal Pendulum) b) \(\theta_{max} \approx 11.11\,^\circ\) (Thin, Rigid Rod)

Key concepts

Conservation of Energy

The principle of conservation of energy is a critical concept encompassing not just physics exercises like the ballistic pendulum, but the entire realm of classical mechanics. In its essence, it states that energy cannot be created or destroyed in an isolated system; it can only transform from one form into another.

This concept is showcased in the ballistic pendulum problem, where the initial kinetic energy of the projectile, \( \frac{1}{2}Mv^2 \), is wholly transferred into potential energy at the system's maximum height. One of the key insights from this example is how, regardless of the pendulum's configuration (ideal or rigid rod), the energy at the beginning and height of the swing remains constant - aligning with the conservation of energy. To better understand and solve such exercises, always equate the initial and final energy keeping the conservation principle in mind, and watch how mass, gravity, and height play pivotal roles in the phenomena.

Kinetic Energy

Kinetic energy, often symbolized as KE, refers to the energy possessed by an object due to its motion. Mathematically, it's represented as \(\frac{1}{2}mv^2\), where \(m\) is the mass and \(v\) is the velocity of the object.

In the context of our ballistic pendulum exercise, the projectile's kinetic energy is the main ingredient in the initial state of the system, just before the collision. This energy is what propels the arm and the attached mass to their maximum height. Understanding kinetic energy's dependency on the square of velocity (\(v^2\)) is key—a small increase in speed results in a substantially greater kinetic energy, influencing the whole system's eventual height.

Potential Energy

On the flip side of kinetic energy is potential energy (PE), the stored energy of an object based on its position or state. For objects near Earth's surface, gravitational PE can be calculated with \(PE = mgh\), where \(m\) is mass, \(g\) is acceleration due to gravity, and \(h\) is the height above the reference point.

In the ballistic pendulum scenario, once the initial kinetic energy propels the pendulum up to its apex, it becomes fully converted into gravitational potential energy. Height \(h\) is determined differently for the ideal pendulum and the rigid rod due to their distinct center of mass locations, causing different PE calculations for the two cases. This conversion from kinetic to potential energy, and its impact on the final state of the pendulum, illustrates the practical application of energy transformation in physical systems.

Center of Mass

The center of mass of an object is the point where mass is equally distributed in all directions—it’s the object’s mean position of mass. For the ballistic pendulum, understanding the center of mass is critical as it influences the potential energy at the highest point of the swing.

In our solution, we treated the problem in two different ways. For the ideal pendulum case, the arm's mass was considered to be concentrated at a single point—simplifying calculations by treating the entire mass as if it's located at the end of the arm. However, for the rigid rod, the mass was distributed along its length, with the center of mass sitting halfway along the arm. This changes the dynamics considerably, resulting in differences in both the potential energy and the maximum swing angle. Seeing how different configurations of the pendulum's mass affect its motion is a brilliant exercise in grasping how the center of mass can dictate the behavior of physical systems.