The Evaporating Sun The Sun radiates energy at the prodigious rate of 3.90 ×10^26 W . (a) At wha

The Evaporating Sun The Sun radiates energy at the...

Key Concepts

Percentage and Relative Change Calculations

Determining what fraction of an original quantity has been altered involves calculating a percentage, a key skill in physics for comparing losses or changes over time. It provides a relative measure that is independent of the absolute magnitudes involved.

Temporal Integration and Unit Conversion

Calculating the total energy output of a system over its lifetime requires integrating power over time. This often necessitates converting time units (such as from years to seconds) to ensure consistency in the units used in equations and calculations.

Power as a Rate of Energy Transfer

Power is defined as the rate at which energy is transferred or converted, measured in watts (joules per second). Understanding this concept allows one to connect the instantaneous energy output of a system with the total energy produced over a period of time.

Mass-Energy Equivalence

This principle, most famously represented by Einstein’s equation E = mc², establishes that mass and energy are interchangeable. It implies that a given amount of energy corresponds to a specific amount of mass and is foundational for analyses where energy emission is directly related to mass loss, as in stellar processes.

Transcript

0:00 Question 60.

00:02 The sun reads energy at the prodigious rate of 3 .90 times 10 to 26 watts.

00:08 Question a, at what rate in kilograms per second does the sun convert mass into energy? we'll get the part b in a second, but we'll start with this is part a.

00:16 So we're given a power and we want to know how much, essentially mass per minute or per second, i guess, per second, the sun loses its mass, right? so we're given power in this scenario.

00:29 Know and you remember that power is simply energy over time and because you're dealing with chemicals or sorry atoms that are changing we can think of it as our our energy just being our mc squared term over time and we're asked to look for kilograms per second so we have our m over t or mass per time i'll do over here in the second line so we can rearrange this to have mass change in mass per time equal to power over energy oh sorry not power or energy power over c squared.

01:19 Let me get a little more obvious, yeah, power over c squared.

01:25 So we can substitute in our values now, 3 .90 times 10 to the 26 watts divided by 3 times 10 to the 8 meters per second, all squared.

01:47 And we find, again, with our units, if you were to check this, would cancel out this way.

01:53 Our mass per second loss is 4 .3 .3.

01:59 Times 10 to the 9 kilograms per second, which is a ton, well, i mean not a literal ton, but it's a huge, huge number.

02:12 But part b states it gives us the actual mass of the sun, so it would take a long time to lose all that mass.

02:17 But part b, assuming that the sun has irradiated at this same rate for its entire lifetime, 4 .5 times 10 to the 9 years, and the current mass of the sun is 2 times 10 to 30 kilogram, what percentage of its original mass has been converted to energy? so this part b kind of extends what we learn in part a.

02:36 How fast does it learn? does it leave its mass or reduce its mass? so we can think of this.

02:45 Our change in mass is our initial mass minus the mass at some other point in time.

02:55 So change in mass, especially here.

02:58 What we can do if we think of this as a rate, which is just kind of hinting at this question, divided by change in time, or it's not really changed, is only one time.

03:08 So over time, multiplied by time.

03:12 We're not changing with the left side of our equation here.

03:14 We're just showing again the rate that we have mass per time, which is what we calculated in part a.

03:23 And what i'll do, and just to make it a little more obvious in a second, i'm going to divide both sides by the original mass.

03:33 So that our right side of our equation here is now a percent difference of mass.

03:41 So we have our change in mass on top divided by original mass.

03:44 That's percent change in mass.

03:49 So we can just use the left side of our equation, our delta m, all right over here just as obvious.

03:54 What we're solving for, delta m over t times t.

04:03 Actually, i'm going to write this the other way around just so i have that.

04:06 So right over here, percent difference that we care about.

04:17 Is our change in mass per second.

04:21 That's the rate.

04:23 We'll multiply it by the time that it's been irradiating, which we have over its initial mass.

04:30 So these terms, this is what we have in part a.

04:35 This is its age, the sun's age, how long it's been eroding at that rate, and we have its original mass, original origina .mass.

04:47 Original orig.

04:47 So this is what we're looking for...

Please give Ace some feedback