Laser reflectors left on the surface of the Moon by the...

Laser reflectors left on the surface of the Moon by the Apollo astronauts show that the average distance from Earth to the Moon is increasing at the rate of $3.8 \mathrm{~cm}$ per year. (a) As a result, will the length of the month increase, decrease, or remain the same? (b) Choose the best explanation from among the following: A. The greater the radius of an orbit, the greater the period, which implies a longer month. B. The length of the month will remain the same because of conservation of angular momentum. C. The speed of the Moon increases with increasing orbital radius; therefore, the length of the month will be less.

Laser reflectors left on the surface of the Moon by the Apollo astronauts show that the average distance from Earth to the Moon is increasing at the rate of $3.8 \mathrm{~cm}$ per year. (a) As a result, will the length of the month increase, decrease, or remain the same? (b) Choose the best explanation from among the following:
A. The greater the radius of an orbit, the greater the period, which implies a longer month.
B. The length of the month will remain the same because of conservation of angular momentum.
C. The speed of the Moon increases with increasing orbital radius; therefore, the length of the month will be less.

Key Concepts

Orbital Period

The orbital period is the time it takes for an object to complete one full orbit. In orbital mechanics, as the radius of an orbit increases, the orbital period also increases because the object has a longer path to travel and experiences a weaker gravitational pull, leading to a slower orbital speed.

Kepler's Third Law

Kepler's Third Law states that the square of the period of any planet (or satellite) is proportional to the cube of the semi-major axis of its orbit. This law directly implies that if the distance between two orbiting bodies increases, the period of orbit will naturally increase as well, resulting in longer cycles or months.

Angular Momentum Conservation

The conservation of angular momentum is a fundamental principle in physics which states that the total angular momentum of a closed system does not change unless acted upon by an external torque. In orbital dynamics, even though individual angular momenta may redistribute (for example, through tidal interactions), the overall conservation law helps explain how changes in orbital radius affect the orbital period.

Transcript

00:01 So this question is asking about orbital motion, specifically the moon orbiting around the earth.

00:08 Now we can approach this in a number of different ways.

00:13 One of the ways we could approach it though is just to use kepler's law as modified to talking about the earth as the center of the system.

00:25 But the same idea still applies.

00:28 Kepler's law tells us that the period of something orbiting around a common object squared is proportional to the radius of that object's orbit cubed.

00:43 So this tells us that if the radius is increasing, and as in this problem that's what they tell us that's what's happening to our moon, that the moon's radius is increasing.

00:56 Therefore we would expect the period for the moon's motion to increase, and the period is what we mean of the moon is what we mean by a month, and so therefore we would expect the month to get longer.

01:09 Now this matches most closely answer explanation a.

01:13 The greater the radius of an orbit, the greater the period, which implies a longer month.

01:18 Let's look at the other two answers though and see what's wrong with reasoning in those.

01:22 The second answer says the length of month remain the same because the conservation of angular momentum.

01:27 Well we can do that angular momentum situation here.

01:31 For an object, a point object going around in a circular orbit, the angular momentum is just equal to m r squared, that's its rotational inertia if you will, times its angular velocity.

01:54 This is i omega.

01:55 You can get the same thing if you just take r cross m v, and v is just equal to r times omega, so that's why you get m r squared omega.

02:11 So you know if we have the m times r initial squared times omega initial being equal to, because angular momentum is going to change, there's no torque being exerted by gravity here, the mass of the planet at some time times the final radius of the orbit squared times omega final.

02:37 You can see that as r increases, omega, that is the angular velocity, is going to have to decrease.

02:48 And if the angular velocity decreases, that means that the time it takes to go around one full revolution is going to get bigger...

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