Two spheres S1 and S2 of the same metal and having the same type of surface, have radii 20cm and 10cm respectively. Compare their rates of fall of temperature as they cool (a) when S1 is at 50 degree centigrade, S2 is at 40 degree centigrade, the surroundings are at 20 degree centigrade and Newton's law applies (b) when S1 is at 400 degree centigrade, S2 at 300 degree centigrade the surroundings are at 200 degree centigrade and stefan's law applies.

For the Newton's Law case,

the heat loss rate rate is h*A(T - 20). and this equals M C dT/dt. M is the mass, C the specific heat and
A is the area of the sphere(s). The differential equation can be solved for T(t) in terms of A, h, M and C. It results in an exponential approach to T = 20 C.

You have not been told the value of the "film heat transfer coefficient", h, although one could calculate it for natural of forced convection using heat transfer engineering formulas that are known. You may have to answer this question using equations rather than graphs with real numbers. Try to use dimensionless groups wherever you can.

In the "Stefan", more propertly called the Stefan-Boltzmann case,
e*sigma*A*(T^4 - 473^4) = M C dT/dt,
where T must be expressed in Kelvin.

e is the emissivity, which they may want you to assume is 1, and sigma is the Stefan Boltzmann constant.

I am going to have to leave the solving of the differential equations, and the "comparing" up to you. They are not difficult.

In a real situation, both types of heat transfer will usually be present, with the radiative "Stefan" type dominating at higher tempertures.

To compare the rates of fall of temperature for spheres S1 and S2 under different conditions, let's examine the two scenarios separately:

(a) Using Newton's law of cooling:
Newton's law states that the rate of change of temperature of an object is directly proportional to the temperature difference between the object and its surroundings. Mathematically, this law can be expressed as:

dT/dt = -k(T - Ts)

Where dT/dt is the rate of change of temperature, k is the cooling constant, T is the temperature of the object, and Ts is the temperature of the surroundings.

For spheres S1 and S2, we have the following given data:
S1: T1 = 50°C, Ts = 20°C, radii = 20 cm
S2: T2 = 40°C, Ts = 20°C, radii = 10 cm

To compare their rates of fall of temperature, we need to calculate the values of k for both spheres using the given data. The value of k can be determined experimentally or obtained from the material's cooling properties.

Once we have the values of k for both spheres, we can calculate the rate of change of temperature (dT/dt) for each sphere using the formula above. The sphere with a higher dT/dt value will cool down faster.

(b) Using Stefan's law of cooling:
Stefan's law states that the rate of heat loss by radiation is directly proportional to the surface area of an object and the fourth power of its absolute temperature. Mathematically, this law can be expressed as:

dE/dt = -σA(T^4 - Ts^4)

Where dE/dt is the rate of change of energy, σ is the Stefan-Boltzmann constant, A is the surface area of the object, and T and Ts are the temperatures of the object and surroundings in Kelvin.

To compare the rates of fall of temperature for spheres S1 and S2 under these conditions, we need to convert the temperatures from Celsius to Kelvin (T(K) = T(°C) + 273.15) and calculate the difference in energy loss for each sphere using the formula above. The sphere with a higher rate of energy loss will cool down faster.

Note: It's important to obtain accurate values for the cooling constants and the Stefan-Boltzmann constant in order to get precise results. The values provided here are just examples to explain the process.