Q4.15 Balance (K)

A light horizontal platform is supported at position x=0 by a spring of spring constant k=250 Nm-1. A mass m=5 kg is dropped onto the platform. After a time the oscillations die away and the platform comes to rest at a point x=-a.
  1. Evaluate a. [Take g=10 ms-2]
  2. Decide (without calculation) on the signs of:
    1. the work done by gravity on the mass
    2. the change in gravitational potential energy of the mass
    3. the work done by the spring force
    4. the change in the potential energy stored in the spring
  3. Evaluate each of the above.
  4. Compare the two potential energies you have calculated and comment.
Graphic - No title - spring
Hide

Hint

Compare potential energies before and after the mass is dropped and come to rest.
 

Solution

Reveal
Hide

Solution

  1. After the oscillations have died away, the mass is in static equilibrium. Thus
     mg=ka; a=mg/k=0.2 m.
  2. (a) positive (movement of mass in direction of gravitational force) ; (b) negative (the mass moves down, hence hfinal<hinitial); (c) negative (mass moves in opposite direction to spring force); (d) positive.
  3. Gravity does work mga=10 J; the gravitational potential energy decreases by 10 J.

    The spring force does work $-\frac{1}{2}kx^{2}=-5\,J$; the potential energy stored by the spring increases by 5 J.

  4. The total potential energy decreases by 5 J. This energy is dissipated by internal friction in the spring and air drag on the mass as the oscillations die away.