Calculating the Temperature Increase from the Work Done on a Substance: Truck Brakes Overheat on Downhill Runs

If the brakes are not applied, gravitational potential energy is converted into kinetic energy. When brakes are applied, gravitational potential energy is converted into internal energy of the brake material. We first calculate the gravitational potential energy $(\text{Mgh})$ that the entire truck loses in its descent and then find the temperature increase produced in the brake material alone.

1. Calculate the change in gravitational potential energy as the truck goes downhill
   $$\text{Mgh}=\left(\text{10,000 kg}\right)\left(9\text{.}\text{80 m}\mathrm{<msup><mtext>/s</mtext><mrow><mn>2</mn></mrow></msup>}\right)\left(\text{75.0 m}\right)=\text{7.}\text{35}\times {\text{10}}^6\text{J.}$$

   14.9
2. Calculate the temperature from the heat transferred using $Q\text{=}\text{Mgh}$ and
   $$\text{Δ}T=\frac{Q}{\text{mc}}\text{,}$$

   14.10

   where $m$ is the mass of the brake material. Insert the values $m=\text{100 kg}$ and $c=\text{800 J/kg}\cdot \text{ºC}$ to find

   $$\text{Δ}T=\frac{\left(7.35\times {\text{10}}^6\text{J}\right)}{\left(\text{100 kg}\right)\left(\text{800 J/kgºC}\right)}=\text{92ºC.}$$

   14.11

This same idea underlies the recent hybrid technology of cars, where mechanical energy (gravitational potential energy) is converted by the brakes into electrical energy (battery).