In the figure is shown a system used to arrest the motion of...

In the figure is shown a system used to arrest the motion of an airplane landing on a field of restricted length. The plane of mass $m$ rolling freely with a velocity $v_{0}$ engages a hook which pulls the ends of two heavy chains, each of length $L$ and mass $\rho$ per unit length, in the manner shown. A conservative calculation of the effectiveness of the device neglects the retardation of chain friction on the ground and any other resistance to the motion of the airplane. With these assumptions, compute the velocity $v$ of the airplane at the instant when the last link of each chain is put in motion. Also determine the relation between the displacement $x$ and the time $t$ after contact with the chain. Assume each link of the chain acquires its velocity $v$ suddenly upon contact with the moving links.

Key Concepts

Distributed Mass Dynamics

Systems with distributed mass, like chains or cables, have their mass spread continuously over a length rather than concentrated at a point. Analyzing such systems involves integrating the contributions of small mass elements to calculate overall energy or momentum, which is critical for understanding events where every segment of the mass starts moving at a specific moment.

Differential Equations and Kinematics

When the motion of a system is influenced by continuously changing parameters, such as gradually increasing mass in motion, differential equations become essential. By formulating and integrating these equations, one can derive relationships between displacement, velocity, and time, providing a detailed description of the system's temporal evolution.

Conservation of Energy

This principle states that in a system with only conservative forces acting, the total mechanical energy remains constant. In dynamic problems, such as those where moving bodies engage additional mass, the initial kinetic energy is redistributed among the system’s components without losses due to friction or other nonconservative forces.

Variable Mass Systems

In dynamics, variable mass systems are those in which the mass changes over time due to the addition or loss of material. Handling such systems requires careful application of energy conservation or momentum principles, as the changing mass influences both the kinetic energy and the motion, often leading to differential equations governing the system’s behavior.

Transcript

00:01 In this question, we are given this setup.

00:04 So initially, we have a horizontal air track, and then there's a glider, and then on the glider that is a pendulum.

00:17 And initially, this is at the center.

00:21 Okay, and then a force is pushed onto the glider.

00:27 Yeah, and then we are also given that the pendulum is mass m.

00:36 The glider is also a mass m and then after some time the force pushes the system so i'm going to denote this as the initial position of the glider and after some time the glider moves some distance and this is x1 and then the pendulum would move just before the force is removed, the pendulum would swing in this manner.

01:19 So this is our x2, and x1 and x2 are measured from the initial position to the center of that object.

01:40 So in part a, we want to find a horizontal components of the v -cm, of the glider sphere system.

01:52 Okay, so to do this first we want to find xcm.

02:00 So this is m1, x1 plus m2, x2, divide by m1 plus m2.

02:09 Since m1 and m2 are the same, so xm will just be half times x1 plus x2.

02:18 Then we also want the acm, so from newton's second law.

02:30 So f net equals to the total mass times acm.

02:37 So this gives acm equals to f over 2m.

02:46 Then we'll be using the kinematics equation.

02:56 So we find a square equals to v i square plus to a delta x.

03:02 So a is our acm, delta x is our xcm...