Special Relativity Energy Problem a little confused

[Engineergirl2]

Homework Statement

At Earth's location, the intensity of sunlight is 1.5kW/m^2. if no energy escaped earth, by how much would Earth's mass increase in 1 day?

Homework Equations

ΔE=Δmc^2+ΔKE
R_earth=6.378x10^3

The Attempt at a Solution

I know that the change in kinetic energy does not change, so that value can go to 0. But I'm not sure what exactly I'm supposed to start with. The answer is supposed to be 1.83x10^5kg/day. Any help would REALLY be appreciated.

 

[nasu]

You need to calculate how much energy is received in 1 day by the entire Earth.
You know how much is received in one second (kW is energy per second) by 1 m^2 of the surface.

 

[Engineergirl2]

Do I use cross sectional area?

 

[nasu]

You can use the cross sectional area and assume the radiation is normal on it.
Integrating over the area of the (hemi-)sphere and taking into account the angle of incidence for each location will produce the same result.

 

[Nickie]

As a scientist, it is important to approach problems with a clear understanding of the relevant equations and concepts. In this case, the problem is related to special relativity and the relationship between energy and mass.

The first step is to understand the concept of mass-energy equivalence, as described by Einstein's famous equation, E=mc^2. This equation shows that energy and mass are equivalent and can be converted into one another.

In this problem, we are given the intensity of sunlight at Earth's location, which is a measure of energy per unit area. We also know that the Earth is not losing or gaining any energy, so we can assume that the change in kinetic energy is negligible. This means that the change in mass is due to the change in energy.

Using the formula ΔE=Δmc^2, we can calculate the change in mass by rearranging the equation to Δm=ΔE/c^2. Plugging in the given values, we get Δm=(1.5kW/m^2)(1 day)/(c^2). The speed of light, c, is equal to 299,792,458 m/s, so c^2=8.988x10^16 m^2/s^2. Plugging this in, we get Δm=(1.5x10^3 J/m^2)(1 day)/(8.988x10^16 m^2/s^2)=1.67x10^-14 kg.

However, this is the change in mass per second. To get the change in mass per day, we need to multiply by the number of seconds in a day. There are 86,400 seconds in a day, so the final answer is 1.83x10^5 kg/day.

In summary, by understanding the concept of mass-energy equivalence and using the relevant equations, we can calculate the change in mass of Earth due to the intensity of sunlight at its location. This type of problem highlights the intricate relationship between energy and mass, as described by special relativity.