Final answer:
To calculate the time required for the steel ball to cool down to 70°C, use the Stefan-Boltzmann law to find the rate of heat loss by radiation. Then, use the heat energy equation to find the time. Convert the given values and plug them into the equations to calculate the time.
Step-by-step explanation:
To calculate the time required for the steel ball to cool down to 70°C, we can use the Stefan-Boltzmann law, which states that the rate of heat loss by radiation is proportional to the fourth power of the temperature difference between the object and its surroundings.
The formula for the rate of heat loss by radiation is:
Q = ε σ A (T^4 - T_s^4)
Where:
Q is the rate of heat loss by radiation
ε is the emissivity of the object (assumed to be 1 for a black body)
σ is the Stefan-Boltzmann constant (5.67 × 10^-8 W/m^2K^4)
A is the surface area of the object (4πr^2 for a sphere)
T is the temperature of the object (in Kelvin)
T_s is the temperature of the surroundings (in Kelvin)
Solving for time:
Q = m c ΔT
Where:
Q is the heat energy
m is the mass of the object
c is the specific heat capacity of the object
ΔT is the change in temperature
Given the diameter of the steel ball (20 cm), we can calculate its radius (10 cm) and surface area (4πr^2). We can also convert the temperatures to Kelvin and use the given density and specific heat capacity of steel to calculate the mass of the ball.
Finally, we can plug the values into the equations to calculate the rate of heat loss and then use it to find the time required for the ball to cool down to 70°C.