Conservation of Momentum and Angular Momentum

Imagine that you go for shopping in a big store with a little child. You take a trolley and put the child in it.

You are busy looking for something on the shelves and therefore you have left the trolley alone.

Meanwhile, the child sees something that interests her. This thing is at some distance. To get it, the child tries pushing the trolley in that direction. Will the trolley move?

In most modern stores, the floor is quite smooth. So we can ignore friction between the trolley and the floor. The child and the trolley form one system. The force by the child and the reaction of the trolley are internal forces. Can they move the trolley?

Conservation of MomentumNo. Only external forces can move the system.

In the absence of such forces the momentum of the system will be conserved. The CM of the system stays put where it is.

The poor child is frustrated due to the law of conservation of momentum.

The important point to remember is that the law applies only to closed systems.

The students generally forget this. You have to remind them every time you talk about the law of conservation of momentum.

A common question:

When a body is falling freely, why is momentum not conserved?

Remind them that even when a body falls freely, there is a force acting on it. So, its momentum cannot be conserved. However, if you consider the body and the earth forming one system, the momentum of the system is conserved.

Moment of InertiaLet us talk briefly about Moment of Inertia.

You know its physical significance, so I would not talk about it.

The important thing is that the MI depends on the distribution of matter round the axis of rotation.

When we change the axis of rotation, or change the distribution around the same axis, the MI changes.

MI=(1/12)ML2

MI=(1/3)ML2

Another exampleIf we reshape a candle, keeping its mass the same, the Moment of Inertia of the candle about the same axis of rotation will change.

An Interesting ExampleAn interesting example is the shifting of large

air masses in the earth’s atmosphere and shifting of water masses by ocean currents.

Does the distribution of matter around the axis of rotation of the earth change?

Source of Fig: ess.geology.ufl.edu

Moment of Inertia and Angular VelocityThe MI around the axis of rotation changes. You know that

To conserve angular momentum, therefore, the angular velocity must change.

IL

Change in the Length of the DayThe change in angular velocity means that the length of the day must change. These changes are, of course, sporadic.

They are irregular and have no pattern.

Source: tycho.usno.navy.mil/gif/bulletinA.jpg

Black: Change in the length of the dayRed: Change in the atmospheric angular momentum

Source: celebrating200years.noaa.gov

Slow Increase in the length of the DayApart from these sporadic changes, there is a continuous slow increase in the length of the day.

Figure shows how the length (measured in hours of today) of the day depends on the time t. The solid line shown the length of the sidereal day for model A, the dashed line the sidereal day for model B, and the dotted lines the corresponding lengths of the synodical day (solar day).

Source: www.astro.uu.nl/~strous/AA/pic/tide4.gif

Today

TidesYou are familiar with the tides. The major factor in causing the tides is the gravitational pull of the moon.

Tides at Puri

The Tides

The Sun also exerts gravitational pull on the waters of the ocean. In fact, its force is greater than that exerted by the mooon. So, why do we say that the moon is resposible for the tides?

The magnitude of the gravitational force:

2r

GMmF

Let us compare the forces due to the Sun and the Moon on an element m on the earth.

Moon

EarthMoon

EarthSun

Sun

Moon

Sun

M

r

r

M

F

F 2

2

)(

)(

222

28

211

30

10103.7

)108.3(

)105.1(

102

Moon

Sun

F

F

Then why do we attribute the tides to the influence of moon?

What is important is to consider the diffrence between the forces exerted at the two neighbouring points and not the force at a point.

The difference between the force at two neighbouring points is:

3

1

rdF

You can show that dF for the Sun is much smaller than that for the moon. That is the reason we say that tides are caused by the moon.

Effects of TidesThe moon attracts the water of the oceans, while the earth tends to rotate on its axis. So there is friction at the interface between the water and the solid earth beneath.

The friction causes not only the loss of energy, but also acts against the rotational motion of the earth.

Increase in the length of the day due to tidesThe earth slows down in its rotation.

The length of the day increases slowly. Today the length of the day is much longer than a few billion years ago.

Source: www.astro.uu.nl/~strous/AA/pic/tide4.gif

Length of the Year does not change

As a result, throughout Earth’s history the length of a day has steadily increased by about 0.002 seconds per century. Since the slowing of Earth’s spin has no effect on the planet’s revolution around the sun, however, the length of a year remains unchanged.

This slowdown may not seem like much, but remember that the Earth has been around for 4.6 billion years, and given long periods of time, the slowdown adds up.

In the middle Devonian period, about 375 million years ago, days were about 21.9 hours long, and there were about 400 days in a year. This estimate has been confirmed by counting daily growth rings in fossil corals and clams.

Some coral species exhibit banding in their skeletons resulting from annual variations in their growth rate. In fossil and modern corals these bands allow geologists to construct year-by-year chronologies, a form of incremental dating, which can provide high-resolution records of past climatic and environmental changes when combined with geochemical analysis of each band.

Coral and Clams

The Effect on the MoonThe moon also feels the effect of the this gravitational locking by the earth. The rotation of the moon slows down. In fact, it has already slowed down so much that its rotational period is equal to its period of revolution.

So, we see always the same face of the moon.

What about the conservation of angular momentum?

Since the earth slows down, there is a loss of its angular momentum.

Since the earth and the moon form one closed system, the moon gains an equal angular momentum.

However, the increase in the angular momentum of the moon cannot cause any slow down of its rotation or of revolution.So, what happens to the gained angular momentum?

The angular momentum of the moon due to its revolution round the earth is

2RML moon

Here R is the distance between the earth and the moon.

Since the angular speed cannot change, R increases in response to the gain of angular momentum. That means that the moon drifts away from the earth.

As a spinning ice skater stretches out his arms, he slows down. Similarly, as the spinning Earth slows in its orbit, the Moon moves farther away, at about 4 cm per year (4 km every million years).

The distance of the Moon (1 Mm = 1000 km) for time t in units of millions of years since today. The solid line fits with the current recession speed of the Moon (model A, 3.7 cm per year today), and the dashed line goes with a reduced recession speed (model B, 1.1 cm per million years today).

Source: www.astro.uu.nl/~strous/AA/pic/tide4.gif

Today

If we take the present rate of recession, in middle Devonian time (about 400 million years ago), the Moon was about 1500 km closer, and at the beginning of the Archean (about 3.9 Ga), it may have been over 15000 km closer.