Carry out the same calculation as in Section 19.2 for the...

Name: Problem 3, Chapter 19: Pathways to Astronomy
Uploaded: 2020-07-21T03:30:44-08:00
Duration: 4 min 26 s
Channel: Matthew Miranda
Description: Carry out the same calculation as in Section 19.2 for the tidal force on the far side of the Earth. Show that this tidal force has approximately the same strength as the tidal force on the near side but points in the opposite direction.

Carry out the same calculation as in Section 19.2 for the tidal force on the far side of the Earth. Show that this tidal force has approximately the same strength as the tidal force on the near side but points in the opposite direction.

Key Concepts

Gravitational Force

This is the fundamental interaction described by Newton’s law of gravitation, which states that any two masses exert an attractive force on one another that is proportional to the product of their masses and inversely proportional to the square of the distance between their centers. In the context of tidal forces, gravitational force provides the baseline from which differential effects are measured.

Tidal Forces

Tidal forces arise when there is a spatial variation in gravitational force across an extended object. Because one side of the object (i.e., the near side) experiences a slightly stronger gravitational pull than the other side (i.e., the far side), a stretching effect occurs. This concept is crucial in explaining phenomena in astrophysics such as ocean tides, deformation of celestial bodies, and even orbital evolution.

Gravitational Field Gradient

The gravitational field gradient is the rate at which the gravitational force changes with distance. For an extended object like Earth, differences in distance from the gravitating body (such as the Moon) result in varying forces across the object. This gradient is the mathematical basis for understanding and calculating the tidal forces by determining how much the force differs from one side of the object to the other.

Differential Acceleration and Series Expansion

Differential acceleration is the concept that different parts of an extended object accelerate differently under a non-uniform gravitational field. Taylor series expansion is often used as a mathematical tool to approximate the difference in gravitational forces over a small distance compared to the total distance to the gravitating body. This approximation reveals that the tidal forces on the far side of the Earth have nearly the same magnitude as those on the near side but are directed oppositely, which is key to understanding the symmetry of tidal phenomena.

Transcript

00:01 Question number 37 is asking us to compare the tidal forces between jupiter and i .o.

00:07 With those between earth and the moon.

00:10 Now, the equation for tidal forces, f subtitle, tidal forces equal to two times the gravitational constant times the mass of the planet, times the mass of the orbiting moon, times the orbiting moon's radius, divided by the cube of the distance between the planet and the moon.

00:37 Now, in comparing the tidal forces on io from jupiter with those between earth and the moon, we can do a little bit of early cancellation by just not even putting numbers into the equation yet and just putting down the variables and doing a few cancellations.

01:01 So, first we've got the tidal forces between jupiter and i .o., which is equal to two times the gravitational constant, times the mass of jupiter times the mass of i .o.

01:15 Times i .o's radius divided by the cube of the distance between jupiter and io.

01:20 So we can multiply that by the reciprocal of the tidal forces between earth and the moon, which is equal to the distance between earth and the moon cubed, divided by two times the gravitational constant, times earth's mass, times the moon's mass, times the moon's radius.

01:47 Now, right away, this gets rid of our 2g.

01:53 So now, what we can do is rewrite the equation into a little bit more of a simple form.

02:02 So now, f title for i .o.

02:05 Divided by f title for the moon turns into the mass of jupiter times the mass of i .o.

02:14 Times ios radius, times the cube of the distance between earth and the moon, divided by the mass of earth times the mass of the moon, times the radius of the moon, times the distance between jupiter and i .o.

02:36 Cubed...

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