The carriage of mass 2 m is free to roll along the...

The carriage of mass $2 m$ is free to roll along the horizontal rails and carries the two spheres, each of mass $m,$ mounted on rods of length $l$ and negligible mass. The shaft to which the rods are secured is mounted in the carriage and is free to rotate. If the system is released from rest with the rods in the vertical position where $\theta=0,$ determine the velocity $v_{x}$ of the carriage and the angular velocity $\dot{\theta}$ of the rods for the instant when $\theta=180^{\circ} .$ Treat the carriage and the spheres as particles and neglect any friction.

Key Concepts

Conservation of Linear Momentum

In a frictionless system with no external horizontal forces, the total linear momentum remains constant. This principle is crucial when analyzing the motion of interconnected components, such as a moving carriage and attached rotating masses. Any momentum gained by one part of the system must be balanced by an opposite momentum in another, ensuring the overall center-of-mass motion is maintained.

Conservation of Mechanical Energy

When non-conservative forces like friction are absent, the total mechanical energy of the system (the sum of gravitational potential energy and kinetic energy) remains constant. As the system evolves from one configuration to another, potential energy lost by the rotating masses converts into translational kinetic energy of the carriage and rotational kinetic energy of the masses, allowing one to relate initial and final states to determine velocities.

Rotational Kinematics

Rotational kinematics deals with the motion of bodies that rotate about an axis, establishing a relationship between angular displacement, angular velocity, and the linear motion of points on the body. For a rod of fixed length rotating about an axis, the angular velocity directly influences the tangential speed of masses attached to the rod. Understanding these relationships is key to linking the rotational motion to the linear motion of the entire mechanical system.

Transcript

00:01 So here on this problem we are given that the carriage that has a mass of 2m is free to roll on this horizontal rail and we have to assume the surface to be frictionless the surface is frictionless and the two spheres are mounted on this carriage and the mass of two sphere is given as m now this length of the road is given as l this is also l now initially the system is at rest and the roads are vertical with theta equals to zero degree now we have to calculate the velocity of the carriage and the angular speed of the road that means vx we have to calculate for the carriage and theta dot or the angular velocity for the road we have to calculate when theta becomes 180 degrees this is what given in the question now to solve this problem we'll be using two things first the conservation of linear momentum second the conservation of energy so we if you consider for the for the entire system we can say that net force acting on the system external net force acting on the system is zero so we can conserve the moment in x direction that is delta g of x equals to zero now let us see the initial and the final position for the for the system of masses so this is the initial position when theta equals to zero and this is the final position for the system of masses when the theta becomes over 180 degrees so we can consider the momentum in x direction this is the x direction now since the initially the system was at rest so initial momentum would be zero so we can directly write the final momentum some final momentum would be equals to 2 times m vx that means the mass of the carriage times it to velocity so final velocity we're assuming to be vx plus there are two masses so 2 m times the velocity now in this case we lost the velocity of the sphere have two velocities first the the sphere has a linear velocity of vx towards this direction and second it have an angular velocity of l theta dot towards this direction so its momentum would be two times m times vx minus l theta dot and it should be equals to zero so from here we'll get two times v or of x is equals to l theta dot or we can write that vx is equals to l theta dot divide by two let this be the equation number one now next i will be applying the conservation of energy so here also we can say that the change in energy that is the change in energy for the kinetic change in…

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