Showing posts with label Gravitational Energy. Show all posts
Showing posts with label Gravitational Energy. Show all posts

The period of the moon revolving under the gravitational force of the earth is $27.3$ days. Find the distance of the moon from the center of the earth if the mass of the earth is $5.97 \;*\;10^{24}kg$.


Given,
Time $(T) = 27.3$ days = $27.3 \; * \; 24 \; * \; 60 \; * \; 60  = 2358720 \; Sec$
Mass of earth $(M) = 5.97 \; *\; 10^{24}\; kg$
Height of the Moon from the center of the earth $(h) = \;?$

We know that,
$h = $ $\left ( \frac{T^2 \; * \; R^2\; * \;g}{4\; \pi^2} \right )^{1/3}$ $ - \;R$
$h =$ $ \left ( \frac{(2358720)^2 \; * \; {6400000}^2\; * \; 9.8}{4\; \pi^2} \right )^{1/3}$ $ - \; 6400000 = 3.77 * 10^8 \; m$

∴ Height of the Moon from the center of the earth $(h) = \;3.77 * 10^8 \; m$

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An earth satellite moves in a circular orbit with a speed of $6.2\;km/s$. Find the time of one revolution and its centripetal acceleration.

Given,

Orbital velocity $v_o = 6.2\;km/s = 6200\;m/s$
Radius of earth $(R) = 6400000\;m$
Time Period $(T) = \; ?$
Centripetal Acceleration $(a) = \;?$

We have,
$ v_{0} = R \sqrt { \frac{g}{R+h}} $
or, $6200 = 6400000\;*\; \sqrt{\frac{10}{6400000 \; + \; h}}$
or, $9.38 \; *\; 10^{-7} = \sqrt{\frac{10}{6400000 + h}}$
or, $h = 10655301 - 6400000$
h = $4255301\;m$

Now, 
$T = 2 \pi \; \sqrt{\frac{(R + h)^3}{g * R^2}}$
or, $2\pi \; \sqrt{\frac{(4255301 + 6400000)^3}{10 * 6400000}}$
or, $2 \; \pi \;*\;1718.57 = 10792.65\;Sec = 2.99\;hrs$
∴ Time of revolution = $2.99\;hrs$

Again,
Centripetal Acceleration $a = \frac{v_0^2}{h + R} = \frac{(6200)^2}{4255301 + 6400000} = 3.6\;m/s^2$
∴ Centripetal Acceleration = $3.6\;m/s^2$

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Taking the earth to be the uniform sphere of radius $6400 \; km$, and the value of $g$ at the surface of earth $10\;m/s^2$, calculate the total energy needed to raise a satellite of mass $2000 \; kg$ to a height of $800 \; km$ above the ground and to set it into a circular orbit at that altitude.


Given;
Radius of earth (R) = 6400 km = 6400000 m
Mass of satellite (m) = 2000 kg
Height of satellite (h) = 800 km = 800000 m
Total Energy needed (E) = ?

We have,
Energy needed = Increase in Potential Energy + Kinetic Energy at Orbit

$\left [ -\;\frac{G\;M\;m}{r}\;-\;\left (\frac{-\;G\;M\;m}{R}\right )\; \right ]$

= $-$$\frac{G\;M\;m}{r}$$\;-\;$$\frac{-\;G\;M\;m}{R}$$\;+\;$$\frac{1}{2}$$mv^2$
= $-$$\frac{G\;M\;m}{r}$$\;-\;$$\frac{-\;G\;M\;m}{R}$$\;+\;$$\frac{1}{2}$$m$$ \frac{G\;M}{r}$                               [∵ r = R+h]

= $g\;R^2\;m[$$\frac{1}{R}$$\;-\;$$ \frac{1}{2(R+h)}$$]$     =     $g\;m[R\;-\;$$\frac{R^2}{2(R+h)}$$]$
= $2000\;*\;10\;[6400000 \;-\; $$\frac{(6400000)^2}{2(6400000 \;+\; 800000)}]$
= $7.12\;*\;10^{10}\;J$

∴ The total Energy needed (E) = $7.12\;*\;10^{10}\;J$

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What Came Before the Big Bang?


It is cosmology’s most fundamental question: How did the universe begin?
“We have very good evidence that there was a Big Bang, so the universe as we know it almost certainly started some 14 billion years ago. But was that the absolute beginning, or was there something before it?” asks Alexander Vilenkin, a cosmologist at Tufts University near Boston.

It seems like the kind of question that can never be truly answered because every time someone proposes a solution, someone else can keep asking the annoying question: What happened before that?


A universe with a beginning begs the vexing question: Just how did it begin? Vilenkin’s answer is by no means confirmed, and perhaps never can be, but it’s still the best solution he’s heard so far: Maybe our fantastic, glorious universe spontaneously arose from nothing at all.

This heretical statement clashes with common sense, which admittedly fails us when talking about the birth of the universe, an event thought to occur at unfathomably high energies. It also flies in the face of the Roman philosopher Lucretius, who argued more than 2,000 years ago that “nothing can be created from nothing.”


“The way the universe gets around that problem is that gravitational energy is negative,” Vilenkin says. That’s a consequence of the fact, mathematically proven, that the energy of a closed universe is zero: The energy of matter is positive, the energy of gravitation is negative, and they always add up to zero. “Therefore, creating a closed universe out of nothing does not violate any conservation laws.”

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