Hot or not?
The Sun puts out a steady stream of roughly
4
×
1
0
2
6
W of radiation and weighs about
2
×
1
0
3
0
kg, while the average human eats 2,000,000 calories (1 Calorie = 1,000 calories) or
≈
8,000,000 Joules a day, most of which is radiated as heat.
Calculate P / m , the rate of heat radiation per unit mass, for the Sun and for the human body. Approximately how many times greater is P / m for the Sun than for the human body?
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2 solutions
Moderator note:
A well presented, clean solution, with common sense approximations.
average weight of a human should have been given inthe question as an input
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The value you choose for the mass of the human in question is immaterial, since the difference in ratio is in the tens of thousands.
If you assume a (very light) 50 kg human, the P/m is 0.59 W/kg. If you assume a (very heavy) 200 kg human, the P/m is 2.35 W/kg.
Given that the P/m for the sun is approximately 1 0 0 0 0 2 , there's no answer closer than "10000 time less."
Try finding a living person with a mass, that sets the result of by 10000.
i dont get it!
The question did not mention how heavy a person is. so could not be solved
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They asked for an approximation. Read that as order of magnitude. An adult human will not weigh 7.5 kg or 750 kg. Therefore, using any reasonable weight will give an order of magnitude of 10^-4.
Agree that assumed weight of human should be included. Also the rate of the sun's radiation was not stated in terms of time. It appears from the solution that the rate is per second. I'm not sure how we were supposed to know that given that the rate of radiation for a human was given in terms of per day.
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Please look at Brian's or Dale's comments regarding the human weight. The options are designed in such a way that any reasonable estimate of the human weight will give the correct answer.
The question does state that the Sun outputs 4 × 1 0 2 6 Joules per second and the human outputs 8 × 1 0 6 Joules per day. We can convert from Joules per day to Joules per second by dividing by the number of seconds in a day.
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Of Course! Thanks for the clarification.
Not "joules per second". 1 joule is equal to 1 watt for 1 second. Over a 24-hour period, that amounts to 4 x 10^26 watts for 86,400 seconds, or 345.6 x 10^29 joules per day. Divide that by 2 x 10^30 kg (or, to simplify matters, 20 x 10^29 kg), and you get 17.28 j/kg. As for the "human" values, I took your 8 x 10^6 j for granted but used 65.5 kg as the average. That's because the "average man" is said to weigh 70 kg, whereas the "average woman" is said to weigh 61 kg. The arithmetic mean of these two values is 65.5 kg. That puts the human radiation per unit of mass at 122,137.4 j/kg--many times more than the sun's 17.28 j/kg, in fact, 7,068.137 times as much. For this reason, I chose "10,000 times more", which--while inaccurate--is the closest value provided.
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@Michael King – Well, 1 watt is defined as one joule per second which is the same as saying "1 joule is equal to 1 watt for 1 second". You have found the energy radiated in one day, and I have found the energy radiated in one second. The answer is the same either way.
Your figures look right. This question only asks for the approximate relation, not for the exact value. This question is designed such that it can be solved mentally by taking appropriate estimations.
@Michael King – Michael--- the answer should have been "10,000 times less" because the question asks for the relation of the sun to the human body. Your figures are correct, but you gave the relation of the human body to the sun--- the opposite of what the question asked.
I used a swift rule of thumb calculation. Mathematically the sun produces (as stated) 2 x 10^-4 W / kg whereas a human produces between 70W and 100W as proposed for heat gain in populated buildings. Therefore the sun produces LESS per kg than the human body - there is only one option to select LESS. QED
"Also the rate of the sun's radiation was not stated in terms of time" A watt is defined as 1 Joule/s so time is stated indirectly
Exact mass of a human isn't necessary since we are trying to calculate the order of magnitude of ratio between P/m of the Sun and a human, rather than an exact value. That could make a difference if we were to choose between a child and a really obese person.
In these kinds of questions, you don't even need exact calculations. Just work with the exponents. Then multiplication and division becomes addition and subtraction.
For the sun 26 - 30 = -4 so roughly 10^-4 W/kg
For the human 7 - 5 = 2 and 2 - 2 = 0 so roughly 10^0 = 1 W/kg
Assumptions: 8,000,000 J ~ 10^7 J; 1 Day ~ 10^5 sec; 1 person weighs ~ 10^2 kg
Sun
( m P ) Sun = 2 × 1 0 3 0 kg 4 × 1 0 2 6 W = 2 × 1 0 − 4 W/kg
Human Body
Heat radiated per second = 2 4 × 6 0 × 6 0 seconds 8 , 0 0 0 , 0 0 0 Joules ≈ 9 2 . 5 9 W
Let the mass of a person be 7 5 kg .
( m P ) Human ≈ 7 5 kg 9 2 . 5 9 W ≈ 1 . 2 3 W/kg
The ratio of heat radiated per second for Sun to Human is:
( m P ) Sun / ( m P ) Human ≈ 1 . 2 3 2 × 1 0 − 4 ≈ 6 0 0 0 1
The closest option is "10,000 times less". □