Irodov Problem 1.140

When the thread holding the mass to the wall is burnt, the free hanging mass drags the mass on the horizontal plane to the right. This elongates the spring which starts to resist the movement of the mass with a restoring force F. During the process, the spring also inclines at an angle to the vertical and F is inclined with the vertical at an angle. The vertical component of F tends to lift the mass upwards and will do so when its strong enough to resist the weight of mass mg.


Suppose that the elongation of the spring is x at the break-off point so that the total length of the spring is now l+x. Then, we have,




There are three forces acting in the vertical direction on the mass on the horizontal plane i) the normal reaction N, ii) the component of the force exerted by the spring and iii) the gravitational force mg. The mass has no acceleration along this direction prior to the break-off from surface. Further, at break-off the normal reaction N=0. So we have,











Now we need to determine the mass's velocity just before break-off. We shall solve this using the conservation of energy principle. The hanging mass, as it descends, looses potential energy. This loss in potential energy is converted into the i) energy stored in the spring and ii) the kinetic energy of the two masses. Just before break-off, the horizontal mass moves a distance,




and this is the same distance that the hanging mass falls.

Let the velocity of the masses just before break-off be v. Then by conservation of energy we have,