Tidal effects of the moon on the earth have caused the...

Tidal effects of the moon on the earth have caused the earth's rotation rate to slow, thus reducing the spin angular momentum of the earth, leading to an increase in earth's day by $0.1 \mathrm{~s}$ in the past 3800 years. This reduction has been made up for by an increase in the orbital angular momentum of the moon around the earth. Therefore how much farther is the moon now from the earth than it was 3800 years ago? (Note that the moment of inertia of the earth about its axis is $I=0.33 M R^2$ and that the mean earth-moon distance is $3.8 \times 10^5 \mathrm{~km}$.)

Key Concepts

Conservation of Angular Momentum

This principle asserts that within a closed system and in the absence of external torques, the total angular momentum remains constant over time. In astrophysical contexts, such as the Earth-Moon system, any decrease in one form of angular momentum (for example, Earth’s rotational angular momentum due to tidal friction) must be balanced by a corresponding increase in another form (the Moon’s orbital angular momentum).

Tidal Interactions

Tidal interactions occur due to the gravitational pull between celestial bodies, which leads to distortions or bulges in the bodies involved. These deformations result in a transfer of energy and angular momentum between the bodies, thereby impacting their rotational speeds and orbital distances over long periods.

Moment of Inertia

The moment of inertia is a measure of an object's resistance to changes in its rotational motion, depending on how its mass is distributed relative to the axis of rotation. It plays a critical role in determining the angular momentum of rotating bodies, linking the distribution of mass to the rotational dynamics of the system.

Orbital Dynamics

Orbital dynamics deals with the motions of objects under the influence of gravitational forces, including their trajectories, velocities, and changes in orbital parameters like radius and period. In the context of angular momentum exchanges, orbital dynamics helps explain how the Moon's orbit can adjust in response to changes in Earth's spin caused by tidal effects.

Transcript

00:03 In this case, we have to calculate the rate of change of distance in the earth -moon system due to the change of moon's orbital period due to the tidal effects.

00:14 So we need to calculate delta r divide by delta t.

00:20 We can write the change in the distance as delta r is equals to r2 minus r1.

00:27 So this will be our equation number one.

00:29 Now, using the kepler's period law, we can write this r1 as r1 is equal to the equation.

00:35 Equals to c t1 square whole raise power 1 divided by 3 so this will be our equation number 2 similarly we can write r 2 as r 2 is equals to c t 2 2 r s power 2 whole power 1 divided by 3 and this is our equation number 3 in this case this c is written as c is equals to capital g m e divided by 4 pi square and we call it equation number a let's calculate this c using the equation number a so we can write here c is equal to 6 .67 multiplied by turnous power minus 11 newton meter square per kg square into the mess of the earth which is equals to 5 .97 multiply by turner as power 24 kg divided by 4 pi square from here we will get the value for this c as c is equals to 1 .009 multiply by turnus power 13 newton meter square per kg.

01:51 Now let's calculate t1 and t2.

01:55 So t1 is equal to 27 .3 dash.

02:02 Sorry, here we have 27 .3 dash.

02:06 So we can convert these dash into seconds by multiplying it with with 86 ,000 and 400 seconds per day.

02:17 So from here we will get this t1 as 2 .35872 872, multiply by 10th power 6 seconds.

02:30 So this is the t1.

02:32 We can write t2 as t2 is equal to 2.

02:38 2 .35872.

02:44 Multiply by turnous power 6 plus 35 multiply by turn 10 minus 3 seconds.

02:54 So here we have seconds.

02:55 From here we will get the value for this t2 as 2 .35872 0 .035 multiplied 102 035 multiplied power 6 seconds...