Driving Test
I’m currently stuck in the office while my third year students are tackling an exam I set. I have to wait by the telephone in case there’s a problem with the paper that I have to sort out.
As a quick diversion I thought I’d give my blog readers a little test of their own. Try this little poser:
Follow @telescoper
In the Dark 
May 16, 2012 at 1:58 pm
If it were to change the length of the day then you would have to be travelling east/west, limiting you to the m4, meaning traffic and contraflows would render driving on the other side of the road pointless….
May 16, 2012 at 2:11 pm
The change in planetary angular momentum due to roundabouts?
May 16, 2012 at 3:54 pm
That’s the right idea, but it’s not just roundabouts — it’s all two-way streets too. After all, take a roundabout and stretch it out until it’s very long and skinny, and it looks just like a two-way street.
To put it another way, suppose that you drive a regular route from home to work and then take the same roads back home again. You’re going around a long skinny closed loop in a clockwise direction (if you’re in the current UK or somewhere else that drives on the left). So you have some (average) angular momentum.
May 16, 2012 at 2:39 pm
Assuming the shape and angular velocity of the Earth is constant, yeah?
May 16, 2012 at 2:41 pm
It won’t have constant angular velocity if the length of the day is changing.
May 16, 2012 at 2:50 pm
Sure?
May 16, 2012 at 2:56 pm
Actually that doesn’t make sense. I’ll try again. Assuming the shape of the Earth is constant and system comprising of the Earth and everything on its surface is isolated from the rest of the universe.
May 16, 2012 at 3:12 pm
I think the day length increases. I await being shot down.
May 16, 2012 at 3:27 pm
If my back-of-the-envelope numbers are right the Earth’s angular momentum about its own axis is ~ 10^34 Js and the change in angular momentum of the cars is ~ 10^12 Js so the day length would change by 10^-17 s. I won’t spoil the game by giving away the sign though!
May 16, 2012 at 3:36 pm
Interesting. I wonder if there’s an anthropic reason why the angular moment of the Earth is of the same order of magnitude as 1/h?
May 17, 2012 at 12:05 am
Pulling out my copy of Barrow and Tipler (equations 5.48 and 5.49), the angular momentum of a typical planet is (m_e/m_p)^(3/4) alpha^2 alpha_G^-2 h_bar = 10^70 h_bar. An overestimate on earth of about 2 orders of magnitude, but not bad given the approximations used. (1/h has the wrong units to be the angular momentum of earth).
May 17, 2012 at 7:39 am
I know. I was just being facetious.
May 16, 2012 at 5:22 pm
It will stay the same because there is symmetry is swapping the direction of two way traffic so any effects of switching lanes will be cancelled out by the effects of oncoming traffic also switching lanes. No numbers necessary.
May 16, 2012 at 5:34 pm
There isn’t perfect symmetry. Latitude lines get shorter the further north you go…
May 16, 2012 at 11:23 pm
I’m trying my best not to make a really sarcastic comment, but I’m struggling. It might be best not to make any further comment.
May 16, 2012 at 5:32 pm
The day might *seem* longer, say if you drove to work.
May 16, 2012 at 6:42 pm
Even without roundabouts, perhaps the change between here and France would reduce the net Coriolis Force. Or some bollocks like that.
May 16, 2012 at 7:22 pm
Well of course there would be chaos on the roads and you’d spend all your time pretty much stationary on the road to work with all hell breaking loose around you. That would be a very long day.
May 16, 2012 at 9:13 pm
You need to calculate the total angular momentum of all the cars. Consider just the M25 (in the admittedly idealised case that anything at all is moving). The cars in the lane further from London are going clockwise; if they have the same speed as those in the inner lane, then the net angular momentum is positive – as if London was surrounded by a ring of mass rotating clockwise. So now you have to ask what happens to the Earth’s rotation if the cars stop and all this angular momentum gets transferred to the Earth. Neat question: if this wasn’t from the old Cambridge entrance exam, it should have been.
May 16, 2012 at 10:55 pm
That image works best if London were at the pole (feels like it at the moment). On the equator, it would induce a north-south rotation. The north-south component will change the period of precession, and therefore the length of the tropical year.
May 16, 2012 at 9:21 pm
It would feel a lot longer because of all of the traffic jams
May 16, 2012 at 10:44 pm
This is where I potentially embarrass myself. Consider an equal number of cars travelling east-to-west as travelling west-to-east. If all the cars have the same mass and travel at the same speed then – as we are in the northern hemisphere – the cars closer to the equator have more angular momentum than those further from the equator. Since total angular momentum is conserved (there is no external torque) whether the angular momentum of the earth increases or decreases then depends on whether the cars are driving on the right or on the left.
May 17, 2012 at 12:54 am
Driving in either lane has nothing to do with the time of light and day on the earth. If you are going to measure in joules what would happen if we got every motor, and all vehicles and turn them east and started them all up how many vehicles / motors would it take to affect the rotation of the earth?
May 17, 2012 at 12:56 am
Well, currently the westbound traffic is on the north side, and the eastbound is on the south side. This provides a counterclockwise torque during acceleration and a counterclockwise torque during deceleration. The sum of the two is zero on average, but there has been a certain amount of rotational inertia imparted by the cars currently in motion. This inertia is in the clockwise direction. Since the surface moves eastward more quickly as you go south, the traffic is currently making the earth rotate slightly faster. If the sides of the road were reversed, the earth would rotate slightly slower (Since the earth has 10^38 kg m^2/s of rotational inertia, and the cars would impart around 5×10^10 kg m^2/s of inertia either way, the change in day length would manifest in the 27th decimal place. This means the switch would make a year about 0.1 attoseconds longer).
Bonus points for anyone who figures out the effect on the orbital speed of Earth.
May 17, 2012 at 3:48 am
Torqueeeeee
May 17, 2012 at 4:22 am
Earth rotates counter-clockwise when seen from the North Pole.
Since London is in the northern hemisphere, a car that drives from point A to point B and back to A using the same route, staying on the right in both cases, also travels in a counter-clockwise direction, so, for the duration of the trip, it pushes the Earth clockwise, slowing down its rotation. While these cars are moving, the day is longer.
This presupposes that all or nearly all cars travel their routes in both directions. Which is actually a pretty big and possibly unjustified assumption (a single car that does a clockwise loop on M25 completely cancels effects of 1000 to 2000 cars driving the same roads in both directions.)
May 17, 2012 at 9:19 am
Is this possible: the Coriolis effect is associated with clockwise motion of projectiles in NH: since cars accelerate into a roundabout, clockwise movement, by Newton’s 2nd law, exerts a counterclockwise force on the earth. So a change to UK driving-side would exert a clockwise force. I’d imagine that this would speed up the earth, but my physics is a bit shaky / non-existent. (If I’m right, then, if we all stopped driving, the earth would also speed up, though by less). Fascinating question anyway: very Martin Gardner.
May 22, 2012 at 11:36 am
Hi I am a driving instructor so I spend a lot of time driving round. They say you turn left more often while driving so I think it would increase the distance traveled as you would be turning from the right hand side. Im sure someone will be able to correct me if im wrong.