Conservation of angular momentum
DOWNLOAD
FREE
|
Rotational kinetic energy
Just as the kinetic energy of an object with linear momentum is [Equation goes here - download the original to see it.] so the rotational kinetic energy of an object with angular momentum will be [Equation goes here - download the original to see it.] Also, in any collision, kinetic energy must be conserved. Likewise, if a rotating object slows down, so that it loses rotational kinetic energy, then that energy is converted to other forms of energy, without loss. The use of energy in solving problems is illustrated by the next example. Example A uniform square lamina of mass 2m and side l is hinged along one edge, and thus is free to rotate about a fixed, smooth, horizontal axis which coincides with a side of the lamina. The lamina is hanging in equilibrium when a particle of mass 5m moving with speed v in a direction perpendicular to the plane of the lamina strikes it at its centre of mass. The particle sticks to the lamina. Find, in terms of v and a, the angular speed of the lamina immediately after the impact. Hence show that the lamina will perform complete revolutions if [Equation goes here - download the original to see it.] Solution [Diagram goes here - download the original to see it.] The side of the lamina is l so the distance of the centre of mass from the hinge is [Equation goes here - download the original to see it.] The moment of inertia of the lamina is given by the parallel axis theorem. [Equation goes here - download the original to see it.] [Equation goes here - download the original to see it.] The particle of mass 5m is travelling with a velocity of v. At impact its distance from the hinge is [Equation goes here - download the original to see it.] , so its angular momentum at that instant, using the equation moment of momentum = linear momentum x perpendicular distance is [Equation goes here - download the original to see it.] This is imparted to the angular momentum of the combined lamina and particle after the collision. By the principle of conservation of angular momentum we obtain [Equation goes here - download the original to see it.] [Equation goes here - download the original to see it.] To perform complete revolutions the lamina has to reach the uppermost position. At impact the moment of inertia of the particle is [Equation goes here - download the original to see it.] By the conservation of the energy we find that the rotational kinetic energy of the combined lamina and particle at impact must be greater than the gravitational potential energy of the centre of mass at the uppermost position. Thus [Equation goes here - download the original to see it.]
|
Contents of
Conservation of angular momentum
1 Conservation of Angular Momentum
2 Proof of the result (conservation of angular momentum)
3 Rotational kinetic energy
|
