Question:(II) A spring with\(k = {\bf{83 N/m}}\)hangs vertically next to a ruler. The end of the spring is next to the 15-cm mark on the ruler. If a 2.5-kg mass is now attached to the end of the spring, and the mass is allowed to fall, where will the end of the spring line up with the ruler marks when the mass is at its lowest position?

The stretched distance of the spring is measured against the ruler. From the energy conservation principle,the gravitational potential energyequals the spring potential energy after the mass hangs on the spring.

The given data can be listed below as:

• The spring constant value is\(k = 83{\bf{ }}{\rm{N/m}}\).
• The initial distance moved by the spring is\({y_1} = 15{\rm{ cm}}\).
• The mass attached to the spring is\(m = {\rm{2}}{\rm{.5 kg}}\).
• The acceleration due to gravity is \(g = 9.81{\rm{ m/}}{{\rm{s}}^2}\).

The diagram can be shown as:

Here,\({F_g}\)is the gravitational force acting on the spring, and k is the spring constant. The end of the spring lines up with the ruler marks when the mass is at its lowest position is \({y_2}\).

The gravitational potential energy of the spring is equal to the spring potential energy. The energy can be expressed as:

\(\begin{array}{c}{E_g} = {E_s}\\mg\left( {{y_2} - {y_1}} \right) = \frac{1}{2}k{\left( {{y_2} - {y_1}} \right)^2}\\mg = \frac{1}{2}k\left( {{y_2} - {y_1}} \right)\\\left( {{y_2} - {y_1}} \right) = \frac{{2mg}}{k}\end{array}\)

Here,\({E_g}\)is the gravitational potential energy and\({E_s}\)is the spring’s potential energy.

Substitute the values in the above expression.

\(\begin{array}{c}\left( {{y_2} - 15{\rm{ cm}}\left( {\frac{{1{\rm{ m}}}}{{100{\rm{ cm}}}}} \right)} \right) = \frac{{2 \times {\rm{2}}{\rm{.5 kg}} \times 9.81{\rm{ m/}}{{\rm{s}}^2}}}{{83{\bf{ }}{\rm{N/m}}\left( {\frac{{1{\rm{ kg/}}{{\rm{s}}^2}}}{{1{\rm{ N/m}}}}} \right)}}\\{y_2} - 0.15{\rm{ m}} = 0.59{\rm{ m}}\\{y_2} = 0.74{\rm{ m}}\left( {\frac{{100{\rm{ cm}}}}{{1{\rm{ m}}}}} \right)\\{y_2} = 74\;{\rm{cm}}\end{array}\)

Thus, the end of the spring will line up with the ruler marks at 74 cm when the mass is at its lowest position.