I understand that for momentum to be conserved, if I throw a tennis ball (or kick a football) the Earth must move in the opposite direction to the ball.
Obviously this is an infinitesimally small amount, but how could how calculate much would the Earth's rotation speeds up or slows down (depending on the direction of the kick/throw) by this action? Let's say that a football weighs 0.45kg and I kick it to a speed of 25 metres per second, can I work out the effect this has on the Earth?
This is similar to Can humans control rotation of the Earth? and How can you find the impact necessary to change the direction of Earth's spin?
Suppose the ball is kicked against the direction in which the Earth is rotating. This increases the speed of rotation of the Earth, reducing the rotation period (ie the length of 1 day).
The angular momentum imparted to the Earth by the ball's kick is :
$\Delta L$ = mass of ball x its velocity x radius of Earth = $mvR$.
This is related to the moment of inertia J of the Earth and the increase in angular speed $\Delta \omega$ by
$\Delta L = J\Delta \omega$.
Assuming the Earth is a sphere of uniform density with mass M and radius R then
$J = \frac25MR^2$.
The angular speed $\omega$ is related to the period of rotation of the Earth $T$ by $\omega=\frac{2\pi}{T}$ so $\Delta \omega = - \frac{2\pi}{T^2} \Delta T$.
Bringing this all together and rearranging we get
$mvR = (\frac25MR^2)(-\frac{2\pi}{T^2} \Delta T)$
$\Delta T = - \frac{5mvT^2}{\pi MR}$.
The LHS is the decrease (- sign) in the time it takes for the Earth to make one rotation.
You will need to look up values for M and R, and to convert $T=24$ to hours to seconds. While $vT \approx 2000km$, the distance travelled by the ball during one revolution of the Earth, is comparable with $R \approx 6000km$, the radius of the Earth, there is an enormous difference in the two masses $m$ and $M$. So $\Delta T$ will be very small compared with $T$.
First off I would recommend a very large boot. "Size billion" comes to mind.
Second I would query "in what direction are you planning to kick said ball with said billion sized boot, fine sir?" Since direction is not specified in this...ahem...matter...ahem...I find the issue at hand confounding indeed.
Still...we all could use a few more hours in our day here at Lake Wobegon so I think I speak for all parties when I say you'd goal is indeed a laudable one.
While I am not at liberty at this time to offer membership in our secret Lunar Lasoe Society I wish to say your incite into this matter has drawn you to the attention of our avuncular Committee of Desiree's and look forward to more pronouncements.
