Question

Short answer

The rise in temperature is \(0.38\circ {\rm{C}}\).

Step-by-step solution

Given data

The mass of the sphere is\(m = 0.095\;{\rm{kg}}\).

The height of the building is\(h = 55\;{\rm{m}}\).

The efficiency is \(\eta = 65\% \).

Understanding the potential energy of the sphere

In this problem, the potential energy will take place if the sphere is dropped from a certain height. Only 65% of original potential energy is consumed for heating the aluminum sphere.

Calculation of the change in temperature

The relation to find the temperature is given by:

\(\begin{array}{c}0.65PE = Q\\0.65mgh = mc\Delta T\\0.65gh = c\Delta T\end{array}\)

Here, cis the specific heat of aluminum,\(\Delta T\)is the change in temperature, Qis the heat transfer, and PE is the potential energy.

On plugging the values in the above relation, you get:

\(\begin{array}{c}0.65\left( {9.8\,{\rm{m/}}{{\rm{s}}2}} \right)\left( {55\;{\rm{m}}} \right) = \left( {900\;{\rm{J/kg}} \cdot \circ {\rm{C}}} \right)\Delta T\\\Delta T = 0.38\circ {\rm{C}}\end{array}\)

Thus, \(\Delta T = 0.38\circ {\rm{C}}\) is the required rise in temperature.