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 1280 kg automobile traveling at 30.8 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 change in temperature of each brake disk, we need to calculate the change in internal energy of the disks.
The change in internal energy (ΔU) can be calculated using the formula:
ΔU = m * cp * ΔT
Where:
m is the mass of the brake disk (3.5 kg)
cp is the specific heat capacity of iron (448 J/kg·°C)
ΔT is the change in temperature in Celsius
Since the kinetic energy is distributed equally to the internal energy of all four brake disks, we can divide the total change in internal energy by 4 to get the change in internal energy of each brake disk.
Now, let's calculate the total change in internal energy by converting the car's kinetic energy into Joules:
Kinetic energy (KE) = 0.5 * mass * velocity²
Substituting the given values:
KE = 0.5 * 1280 kg * (30.8 m/s)²
Now, we can find the total change in internal energy:
ΔU_total = KE
Remember that ΔU_total is the sum of the change in internal energy for all four disks.
Next, divide the ΔU_total by 4 to get the change in internal energy of each disk:
ΔU_disk = ΔU_total / 4
Finally, rearrange the formula for ΔU = m * cp * ΔT to solve for ΔT:
ΔT = ΔU_disk / (m * cp)
Substitute the values of ΔU_disk, m, and cp to calculate ΔT.