Chapter 6, Problem 47P

To determine

The amount of thermal energy generated due to friction in the process.

Answer to Problem 47P

Solution:

  $\text{332}\text{J}$

Explanation of Solution

Given:

Mass of child $=16.0$ kg

Initial height of child $\left(h_1\right)\text{=2}\text{.20}\text{m}$

Initial velocity of child $\left(V_1\right)\text{=0}\text{.0}\text{m/s}$

Final height of child $\left(h_2\right)\text{=0}\text{.0}\text{m/s}$

(Reference level)

Final speed of child $\left(V_2\right)\text{=1}\text{.25}\text{m/s}$

Formulaused:

  $\begin{array}{c}{W}_{NC}=\Delta KE+\Delta PE\\ =\left(\frac{1}{2}m{V}_2{}^2-\frac{1}{2}m{V}_1{}^2\right)+\left(mg{h}_2-mg{h}_1\right)\end{array}$

Work done by friction force converted into thermal energy

  $W_{NC}$: Work done by non-conservative force (friction)

Calculation:

  $\begin{array}{c}P.E_1=mg{h}_1\\ \text{=}\left(\text{16}\text{.0}\text{kg}\right)\left(\text{9}\text{.8}{\text{m/s}}^2\right)\text{×}\left(\text{2}\text{.20}\text{m}\right)\\ K.E_1=\frac{1}{2}m{V}_1{}^2\\ \text{=}\frac{\text{1}}{\text{2}}\text{×}\left(\text{16}\text{.0}\text{kg}\right)\left(\text{0}\text{m/s}\right)=0\\ P.E_2=mg{h}_2\\ \text{=}\left(\text{16}\text{.0}\text{kg}\right)\left(\text{9}\text{.80}{\text{m/s}}^{\text{2}}\right)\left(\text{0}\text{m}\right)\\ \text{ =0 }\\ K.E_2=\frac{1}{2}m{V}_2{}^2\\ \text{=}\frac{\text{1}}{\text{2}}\text{×}\left(\text{16}\text{.0}\text{kg}\right){\left(\text{1}\text{.25}\text{m/s}\right)}^{\text{2}}\\ W_{NC}=\Delta KE+\Delta PE\\ =\left(\frac{1}{2}m{V}_2{}^2-\frac{1}{2}m{V}_1{}^2\right)+\left(mg{h}_2-mg{h}_1\right)\\ \text{ =}\left(\frac{\text{1}}{\text{2}}\text{×}\left(\text{16}\text{.0}\text{kg}\right)\text{×}{\left(\text{1}\text{.25}\text{m/s}\right)}^2-0\right)\text{+}\left(\text{0-}\left(\text{16}\text{.0}\text{kg}\right)\text{×}\left(\text{9}\text{.80}{\text{m/s}}^{\text{2}}\right)\left(\text{2}\text{.20}\text{m}\right)\right)\\ =\text{+12}\text{.5}\text{J-344}\text{.96}\text{J}\\ =-\text{332}\text{.46}\text{J}\end{array}$

Hence, work done by friction force (non-conservative) $\text{=332}\text{J}$

This means that $\text{=332}\text{J}$ of mechanical energy is converted into thermal energy.

Thermal energy due to friction was generated $\text{=332}\text{J}$

Conclusion: Friction force or air resistance convert mechanical energy into thermal energy

Thermal energy generated due to friction $\text{=332}\text{J}$ .