If the sphere has a mass M of 29 kg a specific heat capacity cp of 6195 Jkg-K and a surface area of As = 1383 m2 what is the rate at which its temperature increases just after it is inserted into the

To find the rate at which the temperature of the sphere increases, we can use the formula:

Q = mcΔT

Where: Q is the heat transferred to the sphere (in Joules) m is the mass of the sphere (in kg) c is the specific heat capacity of the sphere (in J/kg-K) ΔT is the change in temperature (in K)

We can rearrange the formula to solve for ΔT:

ΔT = Q / (mc)

The heat transferred to the sphere can be calculated using the formula:

Q = PΔt

Where: P is the power of the furnace (in Watts) Δt is the time interval (in seconds)

The power of the furnace can be calculated using the formula:

P = σεA(Tf^4 - Ts^4)

Where: σ is the Stefan-Boltzmann constant (approximately 5.67 x 10^-8 W/m^2-K^4) ε is the emissivity of the sphere (assumed to be 1 for simplicity) A is the surface area of the sphere (in m^2) Tf is the temperature of the furnace (in K) Ts is the initial temperature of the sphere (in K)

Now, we can substitute the values given in the question:

m = 2.9 kg c = 6,195 J/kg-K A = 1.383 m^2

Assuming the sphere was at room temperature (Ts = 293 K), we need to find the temperature of the furnace (Tf) to calculate the power of the furnace.

Now: P = σεA(Tf^4 - Ts^4) = (5.67 x 10^-8)(1)(1.383)(Tf^4 - 293^4)

Let's assume that the sphere is inserted into the furnace for a time interval of Δt = 1 second.

Q = PΔt = (5.67 x 10^-8)(1)(1.383)(Tf^4 - 293^4)(1)

Now we can substitute the values into the formula for ΔT:

ΔT = Q / (mc) = (5.67 x 10^-8)(1)(1.383)(Tf^4 - 293^4)(1) / (2.9)(6,195)

Now we can calculate ΔT