Monday, May 19, 2008
Irodov Problem 1.163
There are no external forces acting on the (mass+disc) system in the horizontal direction and so the total momentum of the (mass disc) system will be conserved at all times. Initially the momentum of the system in the horizontal direction is mv. As soon as the disc starts to move up the curve, its starts to transfer it horizontal momentum to the mass. When the disc has completed the curve and is moving vertically upwards relative to the mass suppose that the mass has achieved a velocity v'. Since the disc is moving with the mass the net momentum of the (disc+mass) system in the horizontal direction will be (M+m)v'.
Hence applying conservation of momentum in the horizontal direction will be,
In addition to conservation of momentum, energy also will be conserved at all times. Initially since only the disc is moving, the total energy is the kinetic energy of the disc i.e. . When the disc leaves the mass and moves into the air it retains its horizontal component of velocity v'.
When it reaches its highest point in the air, it has no vertical component of velocity but has a horizontal component v'. The total energy of the system when the disc reaches it highest point is the sum of i) kinetic energy of the mass , ii) the kinetic energy of disc and iii) the gravitational potential energy gained by the disc mgh. Since energy must always be conserved,
Posted by Krishna Kant Chintalapudi at 11:55 AM