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6

We want the Newtonian limit of the Einstein Field equations for nonzero vacuum energy(=cosmological constant). As $\rho_\mathrm{vac}=\Lambda/4\pi G$ is a mass(=energy) density, Poisson equation is $$\Delta\Phi=4\pi G\rho(\boldsymbol r)-\Lambda \tag{1}$$ If we assume spherical symmetry, and point-like source $\rho\sim\delta(\boldsymbol r)$, the ...

3

You get an extra term that increases with r: $$a = -\frac{G\cdot M}{r^2} + j\cdot r$$ with j as the repulsive component.

2

An infinite "universe full of water" is actually very close to how the real universe is typically modeled, except that instead of water it's the right mix of ordinary matter, dark matter, radiation, curvature and cosmological constant. On large scales it's reasonable to assume that the density of each component is everywhere equal at a given time. And ...

2

The only condition for free fall as you said is that the motion of the body should be only under the influence of gravity alone. There should not be any effect of other forces like air resistance, viscous drag etc. The condition depends on the property of the material under free fall. For example, if the body has a certain mass as well as charged, it causes ...

2

If a body of mass m hanged on a string is moving, let uniformly, on a circle fixed relatively to the ground, then an observer G on the ground uses the 2nd Newton Law : $$\mathbf{F}=m\cdot \mathbf{a} \tag{01}$$ and finds the relation between the force $\mathbf{F}$ and the acceleration $\mathbf{a}$. For observer G there exists a "real" force, the tension ...

2

As far as we know the classical (i.e. non-quantum) laws of gravity apply at all length scales. There are theoretical reasons to suppose that the classical description fails at scales approaching a Planck length, but this is far, far smaller than the size of a neutron. So inside a neutron we would expect the classical laws of gravity to apply, and in ...

1

I think the two big factors would be that the Earth would 'want to' become tidally-locked to the Moon and the Sun. The Moon would win here, which is easy to see because tides are caused more strongly by the Moon than by the Sun. So in due course the Earth would end up tidally-locked to the Moon with a rotation period which would be the same as a lunar ...

1

Note that you have to swing the pail with a certain minimum speed for the water to stay in. That minimum speed is such that when the pail is at the top of the arc, the rope accelerates the pail downward faster than gravity accelerates the water downward. Otherwise, the water falls out.

1

The chain looks pretty long, only a meter or so is actually in the air. The gravitational force on the runner by the chain is not that big. The runner is bulled back/down in the angle that the chain attaches to the back. The chain cannot convey any torque or transversal force. So there are only a few kilograms of weight directly pulling on the runner. A lot ...

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