As shown in Fig. 3.1, a vertical rod is attached to the horizontal top of a table. A spring of mass $7.5\,\text{g}$ can move freely along the entire length of the rod. Initially, the spring is pushed against the table surface so that it is compressed by $2.1\,\text{cm}$. It is then released suddenly, leaving the table surface with kinetic energy $0.048\,\text{J}$ before travelling upwards along the rod. Assume that Hooke’s law applies to the spring and that the elastic potential energy stored in the compressed spring initially is equal to the kinetic energy of the spring as it leaves the table surface. Ignore air resistance.
(a)[2]
Using the spring’s initial elastic potential energy, calculate the spring constant.
(b)[2]
Calculate the speed of the spring as it leaves the surface of the table.
(c(i))[2]
The spring reaches its greatest height up the rod above the table surface. As a result, the gravitational potential energy of the spring increases by $0.039\,\text{J}$. Calculate, for this motion of the spring, the rise in height of the spring after it has left the table surface.
(c(ii))[2]
Calculate the average frictional force exerted by the rod on the spring as it rises.
(d)[2]
A different rod is used, and this rod provides negligible frictional force on the spring as it moves. The initial compression $x$ of the spring is now changed so that the greatest increase in height $\Delta h$ of the spring after it leaves the table surface also changes. Assume Hooke’s law remains valid for every compression. On Fig. 3.2, sketch a graph to show how $\Delta h$ varies with $x$. You do not need to label any numerical values.
Worked solution & mark scheme
This 10-mark question has a full step-by-step worked solution and mark scheme. One marking point: “$E_p=\tfrac12kx^2$ or $E_p=\tfrac12Fx$ with $F=kx$” …