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# Gravitational Physics

Here we lay out Newton's law of gravity and crack open the universe of consequences that spring from it.

To talk about gravity, we need a solid conception of mass itself. In Newtonian gravity, mass appears in two central roles that we will review in this quiz. In fact the mysterious dual role of mass is the point of departure for Einstein’s celebrated theory of gravity, general relativity.

Without further ado, let’s begin our exploration of gravity with the mystery of the white room.

Your eyes blink open in an intensely white room. Beside you lay two cubes, one white and one black. You have no idea how you got here. You shakily lift yourself from a supine position.

A syrupy voice informs you that you have fallen into a sensory test chamber. Beside you, one cube is real matter and the other is a sensory distortion.

"*What?*" you yell, but there is no response.

You examine the cubes, picking up one in each hand. They both feel pretty real to you.

Suddenly, the voice says menacingly: "This test will end only if you can name the cube that is not real. You get one try."

Startled, you drop both cubes. You notice that they do not hit the ground at the same time.

Both cubes fall from the same height with identical initial speeds. If they do not hit the ground at the same time, what do we know about their accelerations?

Puzzled, you drop the cubes a second time, and again, they do not strike the ground at the same time. What can we say about their masses?

**Details & Assumptions**

- You are very careful to drop the cubes from the same height with identical initial speeds.
- Assume that during the fall, the effect of
**air drag**is unnoticeable.

In Newton's theory of gravity, Earth's mass generates a gravitational field \(\mathbf{g}\) that pulls the cubes toward its center. At the surface, Earth's gravitational field has a magnitude \(g=\SI[per-mode=symbol]{9.8}{\meter\per\second\squared}.\)

The gravitational field has the same units as acceleration because, as we saw in the previous question, any mass in **free-fall** has the same acceleration.

What force \(F_g\) would produce this acceleration on a mass \(m\) during free-fall near Earth's surface?

**Hint:** Use **Newton's second law**.

To begin testing for differences between the cubes, you attach a string (from your pocket) to both cubes, and hang them over one of the horizontal supports above you to compare their weights.

It turns out they **balance**; neither one accelerates when you let go.

Assuming there is negligible friction between the string and the support, which cube weighs more?

Before, you observed a difference between the cubes while they were moving, so you decide to tie the string tightly around the horizontal support. The cubes can now swing like pendulums.

When you pull both blocks back to the same height and release them, which block feels a larger net force \(F_\text{net}\) the instant you let go?

After you let go, the cubes oscillate side-by-side (without rotating or bumping each other).

You notice that the white cube has a distinctly longer **oscillation period**.

Although both cubes feel the same net force as you let go, the somehow have difference accelerations. Which cube has the smaller acceleration?

**Details & Assumptions**

- The angle is small enough that the motion of both cubes can be considered simple harmonic, with position \(s(t)\propto \cos\omega t.\)
- The angular frequency \(\omega\) and the period \(T\) are related by \(\omega^2=2\pi/T.\)
- The lengths of string connecting each cube to the support are equal.

Both cubes have the **same weight** in Earth's gravitational field, yet they have **different accelerations** under the same net force. One of the cubes is violating the ordinary Newtonian physical rules, but which rule?

In our testing, we have identified two roles of an object's mass \(m\):

- determining the magnitude of the gravitational force, as in \(F_g = mg.\)
- representing an object's
**inertia**in Newton's second law, as in \(F=ma\).

The **equivalence principle** asserts that an object's gravitational mass is identical to its inertial mass. This is an often-overlooked postulate of Newtonian physics derived from observation. Perhaps surprisingly, it is not derivable from Newton's laws!

You cannot identify which cube is real without comparing the cubes to another object with a known physical existence.

Fortunately, in your pocket, you carry a lucky (and very real) coin. You lay it on top of the black cube, then you drop the cube. The cube and the coin do not fall together.

Which cube is real?

You call out your deduction, and the door opens flooding the room with red light. You slowly walk across the threshold.

Gravity's grip extends far beyond Earth's surface. It pulses through the emptiness of space, sculpting unfathomably large galaxies around objects so massive that light cannot escape from them. Mass is both the source and the subject of gravity's action. How mass is distributed throughout the universe is inexplicable without understanding gravity's action.

This course is your portal to understanding gravity's far-reaching effects. The focus of this first chapter is to introduce Newton's revolutionary idea of a universal force between masses.

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