Calculating Temperature Increase: Truck Brakes Overheat on Downhill Runs · Physics Playground

Example from Physics, 11.2 Heat, Specific Heat, and Heat Transfer

Calculating Temperature Increase: Truck Brakes Overheat on Downhill Runs

When a truck headed downhill brakes, the brakes must do work to convert the gravitational potential energy of the truck to internal energy of the brakes. This conversion prevents the gravitational potential energy from being converted into kinetic energy of the truck, and keeps the truck from speeding up and losing control. The increased internal energy of the brakes raises their temperature. When the hill is especially steep, the temperature increase may happen too quickly and cause the brakes to overheat.

Calculate the temperature increase of 100 kg of brake material with an average specific heat of 800 J/kg $\cdot °C$ from a 10,000 kg truck descending 75.0 m (in vertical displacement) at a constant speed.

A truck with smoking brakes barrels down a steep incline.

Work it out on paper first. Then open the solution one step at a time, and stop as soon as you can finish on your own.

Strategy
We first calculate the gravitational potential energy (Mgh) of the truck, and then find the temperature increase produced in the brakes.
2
Solution
1. Calculate the change in gravitational potential energy as the truck goes downhill.
  $M g h = (10, 000 kg) (9 {.80 m/s}^{2}) (75 .0 m) = 7.35 \times 10^{6} J$
  11.15
2. Calculate the temperature change from the heat transferred by rearranging the equation $Q = m c Δ T$ to solve for $Δ T .$
  $Δ T = \frac{Q}{m c},$
  11.16
  
  where m is the mass of the brake material (not the entire truck). Insert the values Q = 7.35×10⁶ J (since the heat transfer is equal to the change in gravitational potential energy), m $=$ 100 kg, and c $=$ 800 J/kg $\cdot$ $°C$ to find
  $Δ T = \frac{7.35 \times 10^{6} J}{(100 kg)(800 J / kg \cdot °C)} = 91.9 °C .$
  11.17

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