When a driver brakes an automobile, friction between the brake disks and the brake pads converts part of the car's translational kinetic energy to internal energy. If a 1500 kg automobile traveling at 42 m/s comes to a halt after its brakes are applied, how much can the temperature rise in each of the four 3.5 kg steel brake disks? Assume the disks are made of iron (cp = 448 J/kg·°C) and that all of the kinetic energy is distributed in equal parts to the internal energy of the brakes.
To find the temperature rise in each of the four steel brake disks, we need to calculate the amount of kinetic energy converted to internal energy and then use the specific heat capacity of iron to find the temperature rise.
1. First, let's find the initial kinetic energy of the car:
Kinetic Energy = (1/2) * mass * velocity^2
Kinetic Energy = (1/2) * 1500 kg * (42 m/s)^2
2. Next, we need to find the total kinetic energy converted to internal energy when the car comes to a halt. Since all of the kinetic energy is distributed equally among the four brake disks, we'll divide the total by 4:
Total Internal Energy = Kinetic Energy / 4
3. Now, let's find the specific heat capacity of iron, given as cp = 448 J/kg·°C.
4. To find the temperature rise, we can use the formula:
Temperature Rise = Internal Energy / (mass * specific heat capacity)
Let's calculate the temperature rise in each brake disk:
Step 1:
Kinetic Energy = (1/2) * 1500 kg * (42 m/s)^2
Kinetic Energy = 1,470,000 J
Step 2:
Total Internal Energy = Kinetic Energy / 4
Total Internal Energy = 1,470,000 J / 4
Total Internal Energy = 367,500 J
Step 3:
Specific Heat Capacity of Iron (cp) = 448 J/kg·°C
Step 4:
Temperature Rise = Total Internal Energy / (mass * specific heat capacity)
Temperature Rise = 367,500 J / (4 * 3.5 kg * 448 J/kg·°C)
Calculating this expression will give us the temperature rise in each brake disk.