## More Order-of-Magnitude Physics

A very busy day today so I thought I’d just do a quick post to give you a chance to test your brains with some more order-of-magnitude physics problems. I like using these in classes because they get people thinking about the physics behind problems without getting too bogged down in or turned off by complicated mathematics. If there’s any information missing that you need to solve the problem, make an *order-of-magnitude* estimate!

Give order of magnitude answers to the following questions:

- What is the maximum distance at which it could be possible for a car’s headlights to be resolved by the human eye?
- How much would a pendulum clock gain or lose (say which) in a week if moved from a warm room into a cold basement?
- What area would be needed for a terrestrial solar power station capable of producing 1GW of power?
- What mass of cold water could be brought to the boil using the energy dissipated when a motor car is brought to rest from 100 km/h?
- How many visible photons are emitted by a 100W light bulb during its lifetime?

There’s no prize involved, but feel free to post answers through the comments box. It would be helpful if you explained a bit about how you arrived at your answer!

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April 25, 2019 at 11:11 pm

What is this “100W lightbulb” of which you speak? I have in front of me a bulb labelled “810 lumen/9.5W” which I think is roughly equivalent to an outlawed bulb which used to consume 60W. Simple scaling means a modern 100W LED bulb is around 8500lumen. It says “2800K” but of course will have a narrow spectrum rather than a thermal one. I could carry on to a number of photons, but I felt a pedantry contribution will suffice for now! It claims 15000 hours life.

There are probably students out there who have never met an incandescent bulb. I’m used to having to explain vinyl records, but I felt old when a teenager asked “what’s a DVD?”

April 26, 2019 at 12:09 pm

If it consumes 100W and produces visible light then that’s enough to give an order-of-magnitude estimate, which is all you’re asked for! Whether it’s an incandescent bulb or not is neither here nor there.

April 26, 2019 at 7:31 am

Prasenjit Saha and Paul Taylor have written an entire book on order-of-magnitude astrophysics. Check it out!

April 26, 2019 at 3:05 pm

These are Fermi problems, as each physics knows.

I was waiting for the answers from the young people …

1. The minimum angular resolution for human is 30” that is (2*pi/360)*(30/3600) radians that is almost 1.5*10^-4 radians, the width of a car is approximately 1,5 m, so that the distance to see a car is 1,5/(1.5*10^-4)=10^4 meters=10 km

2. The difference of temperature between the two rooms is near to 20°, the linear expansion for a metal is near to 10^-5, so that the length variation is L(1+2*10^-4), the period variation is (the period is T=1 second for a pendulum) 2*pi*(sqrt(L(1+2*10^-4)/g)-sqrt(L/g))=T(sqrt((1+2*10^-4))-1)=10^-4 sec for second; so that a week is 7*86400 sec=6*10^5 sec and the variation in a week is 60 second

3.The Sun mean energy on the Earth is 500w/m^2, the efficiency of a solar panel is less of 30%, so that 150w/m^2 is the probable energy conversion; a 1Gw=10^9w, and S=10^9/150=6*10^6 m^2=6km^2

4. A weight of a car is 2t=2*10^3kg, the velocity of the car is (10^2/3.6) m/s=25m/s, so that the energy is 10^3*25^2=6.25*10^5 J, the specific heat capacity of the water is 4.18*10^3J/(°K*kg), and near 4.18*10^5 J to obtain hot water at 100 °K from 0 °K, so the energies are of the same order of magnitude.

5.If a light bulb last one year (lighting it only the night), then T=365*24*3600/2=3*10^7 sec is the total time of emission, the visible wavelength is 500nm, so that the energy of a photon is E=h v= 6*10^-34 *c/lambda= 6*10^-34*3*10^8/(5*10^2*10^-9)J=4*10^-19 J, and the total number of photons is 3*10^7*100/(4*10^-19)=75*10^26 photons.

Some small errors that compensate each other