Question

A ballistic pendulum consists of an arm of mass \(M\) and length \(L=0.480 \mathrm{~m} .\) 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 \(V=3.60 \mathrm{~m} / \mathrm{s}\). 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

In a ballistic pendulum system with a projectile of equal mass as the arm, the maximum angle to which the arm and attached mass will swing when the arm is treated as an ideal pendulum is approximately 77.1°. When the arm is treated as a thin rigid rod, the maximum angle is approximately 53.1°. These values were calculated using conservation of momentum and conservation of energy principles.

Step-by-step solution

Find the initial and final velocities of the system during collision

Use conservation of momentum principle: Initial total momentum = Final total momentum \(m_pV_i = (M+m_p)V_f \Rightarrow V_f = \frac{M}{2M} V_i = \frac{1}{2}V_i\) Here, \(m_p = M\) is the mass of the projectile.

Calculate the initial kinetic energy and final potential energy

We can use conservation of energy to find the maximum angle: Initial kinetic energy = Final potential energy \(E_k = E_p \Rightarrow \frac{1}{2} (2M)V_f^2 = Mgh\) Since \(h = L(1-\cos{\theta})\), we have: \(\frac{1}{2}(2M) (\frac{1}{2}V_i)^2 = MgL(1-\cos{\theta})\)

Find the maximum angle

Rearrange the energy conservation equation to solve for the maximum angle \(\theta\): \(1-\cos{\theta} = \frac{V_i^2}{4gL} \Rightarrow \cos{\theta} = 1-\frac{V_i^2}{4gL}\) Now, take the inverse cosine to find the maximum angle: \(\theta = \arccos(1-\frac{V_i^2}{4gL})\) Substitute the given values and calculate the maximum angle for this case: \(\theta = \arccos(1-\frac{(3.60)^2}{4\cdot9.81\cdot0.480})\) \(\theta \approx \boxed{77.7^{\circ}}\) Case b) Arm treated as a thin rigid rod

Find the initial and final velocities during collision

Use conservation of momentum principle: Initial total momentum = Final total momentum \(m_pV_i = (M+m_p) v_{cm} \Rightarrow v_{cm} = \frac{M}{2M} V_i = \frac{1}{2}V_i\) Here, \(v_{cm}\) is the center of mass velocity.

Calculate the initial kinetic energy and final potential energy of the arm

We can use conservation of energy to find the maximum angle: Initial kinetic energy = Final potential energy \(E_k = E_p \Rightarrow \frac{1}{2}(2M) (\frac{1}{2}V_i)^2 + \frac{1}{2}I\omega^2 = Mgh + \frac{1}{2}I\omega^2\) Since the center of mass is at \(L/2\) and \(I = \frac{1}{3}ML^2\) for a thin rod: \(\frac{1}{6}M(V_{cm} + V_i)^2 = MgL(1-\cos{\theta})\)

Find the maximum angle

Rearrange the energy conservation equation to solve for the maximum angle \(\theta\): \(1-\cos{\theta} = \frac{V_i^2}{6gL} \Rightarrow \cos{\theta} = 1-\frac{V_i^2}{6gL}\) Now, take the inverse cosine to find the maximum angle: \(\theta = \arccos(1-\frac{V_i^2}{6gL})\) Substitute the given values and calculate the maximum angle for this case: \(\theta = \arccos(1-\frac{(3.60)^2}{6\cdot9.81\cdot0.480})\) \(\theta \approx \boxed{53.5^{\circ}}\) So, the maximum angles in each case are approximately \(77.7^{\circ}\) and \(53.5^{\circ}\) respectively.

Key concepts

Conservation of Momentum

In the realm of physics, the conservation of momentum is a principle stating that the total momentum of a closed system remains constant if no external forces act on it. When a projectile collides with a pendulum, as in the exercise, they form a closed system during the impact.

The concept is pivotal in analyzing the ballistic pendulum scenario. Since the projectile sticks to the pendulum, the collision is inelastic, which means some kinetic energy is converted to other forms of energy such as heat or sound. However, momentum is conserved. We use this principle to find the velocity just after the collision. With the masses being equal, it simplifies the situation, resulting in the formula used in step 1 of the solution, yielding the velocity of the combined mass and projectile system just after impact.

Conservation of Energy

Conservation of Energy is another fundamental concept in physics, which tells us that the total energy in an isolated system cannot change; it is conserved. It implies energy can neither be created nor destroyed but transformed from one form to another.

In the context of the ballistic pendulum, once the projectile collides with the pendulum's arm and sticks to it, no external work is done on the system. Hence, we apply the conservation of energy to link the kinetic energy just after the collision to the potential energy at the peak of the swing. The kinetic energy which the system had just after impact (due to its velocity) converts into potential energy as the system reaches its maximum height (Step 2 in the solution). The equations clearly illustrate how this principle leads to finding the maximum angle the pendulum swings to.

Center of Mass

The center of mass is the point at which the mass of a body or system may be considered to be concentrated for the purpose of motion analysis. For symmetrical bodies with uniformly distributed mass, such as a uniform rod, the center of mass lies at the geometric center.

In the ballistic pendulum problem's second scenario (b), we regard the pendulum's arm as a thin rod with evenly distributed mass. This affects the motion since the arm will rotate about its pivot point after collision, and we need to consider the rotation's kinetic energy. We calculate the velocity of the center of mass (Step 4) and include the rotational kinetic energy due to the moment of inertia of the rod when calculating the total energy (Step 5). The conservation principles applied here consider both translational and rotational motion to determine the maximum angle.

Pendulum Motion Analysis

Pendulum motion analysis encompasses examining the movement of a pendulum system, often involving the impact of forces, energy transformation, and the properties of rotational motion.

In our ballistic pendulum problem, after the inelastic collision, the pendulum and projectile move as a single entity. The motion's analysis involves understanding how the kinetic energy transforms into potential energy, which allows us to find the height and, subsequently, the angle of the swing. In case (a), treating the arm as a point mass simplifies the motion to that of a simple pendulum, while in case (b), the distributed mass alters the moment of inertia, affecting the energy components and the subsequent angle calculation (Step 6). Pendulum motion analysis ultimately allows us to understand the system's behavior throughout its swing, up to its highest point.