Curving of gravitational field lines

[pgirl1729]

Homework Statement

My textbook mentions "the tangential drawn to any point on a field line can be used to determine the direction of the gravitational field intensity at that point."

Relevant Equations

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My textbook mentions "the tangential drawn to any point on a field line can be used to determine the direction of the gravitational field intensity at that point." Aren't gravitational field lines always perpendicular to the center of they body? why do you have to draw a tangent? Why would the lines get curved? If so how would they curve? like in what direction?

 

[Ibix]

I think you are thinking of the gravitational field of a spherically symmetric mass like (approximately) the Earth, and in that case you are correct. But what about other mass distributions? For example, think of the Earth and Moon. Does gravity always point to their common center of mass?

 

[PeroK]

The field lines may be curved when two or more bodies are involved. See here, for example:

https://en.wikipedia.org/wiki/Gravitational_field

 

[Steve4Physics]

Aren't gravitational field lines always perpendicular to the center of they [sic] body?

A line can't be perpendicular to a point. You probably meant to ask:
"Aren't gravitational field lines always straight and directed towards a body's centre of mass?".

To add to what’s already been said, even single objects can have curved gravitational field lines. E.g. the shape of the earth is better approximated as an ellipsoid rather than a sphere, so its field lines will be slightly curved - it's an interesting exercise to try and sketch them.

 

[kuruman]

To determine the direction of the gravitational field, also known as "down", you suspend a mass from a string and you deduce that "down" is the direction from the point of support to the attached mass at equilibrium. This device is also known as a plumb-bob.

The picture below is modified from the one suggested by @PeroK. If you attach the free end of the plumb-bob at point A, and let the mass go to its equilibrium position, the length of the string will be tangent to the field line passing through point A. It shows that the direction of the net gravitational force at point A is neither towards the center of the Moon nor the center of the Earth. BTW, this also shows that field lines cannot cross.

 

[jbriggs444]

As the graphic shown by both @PeroK and @kuruman suggests, the "field lines" in blue are perpendicular to the "equipotential surfaces" in red.

For a point mass (or a spherically symmetric mass distribution), the equipotential surfaces will be concentric spherical shells and the field lines will point straight at the center of mass.

In vector calculus, the vector corresponding to the [tangent to the] field line at a point is the "gradient" of the potential field. It points in the direction where potential is [locally] decreasing most rapidly. Its magnitude corresponds to the rate of change of the potential in that direction.

 

[Herman Trivilino]

Aren't gravitational field lines always perpendicular to the center of they body?

They're radial. If you have a spherically symmetric body the field lines are perpendicular to the surface of the body, which is the radial direction (towards the center of the body).

 

[wrobel]

Sometimes students think that a mass point can move along a potential line. In general It is not so.

 

[limaster]

It sounds like you’re picturing the simple case of a perfectly spherical mass, where the field lines do point straight toward the center so in that situation the “tangent” rule does feel redundant.

But the textbook is talking in general. Gravitational field lines only stay radial when the mass distribution is perfectly symmetric. The moment you have more than one body (Earth + Moon), or a mass that isn’t spherical, the field becomes distorted and the lines bend toward the region where the pull is stronger.

So the tangent is just a universal way to read the local direction of the field.

Bạn curious about how they look near two bodies? The classic “peanut-shaped” pattern shows the curvature really clearly.

 

[tech99]

Is not the shape of the field simply the superposition at every point of radial fields from more than one body? This gives the curvature.