Question 27: Gravitation

Question

Energy required for moving a body of mass $$\mathrm{m}$$ from a circular orbit of radius 3R to a higher orbit of radius 4R around the earth is.

Accepted Answer

To determine the energy required to move a body of mass $$m$$ from a circular orbit of radius $$3R$$ to a higher circular orbit of radius $$4R$$ around the Earth, we need to calculate the difference in total mechanical energy between the two orbits.
HereÃ¢ÂÂs how to do it step-by-step:
$$	extbf{Energy in a circular orbit:}$$
For an object in a circular orbit, the total mechanical energy (kinetic plus potential) is given by:
$$E = -\frac{1}{2}\frac{GMm}{r},$$
where
$$G$$ is the gravitational constant,
$$M$$ is the mass of the Earth, and
$$r$$ is the radius of the orbit.
$$	extbf{Energy at radius } 3R:$$
Substitute $$r = 3R$$ into the equation:
$$E_1 = -\frac{1}{2}\frac{GMm}{3R} = -\frac{GMm}{6R}.$$
$$	extbf{Energy at radius } 4R:$$
Substitute $$r = 4R$$:
$$E_2 = -\frac{1}{2}\frac{GMm}{4R} = -\frac{GMm}{8R}.$$
$$	extbf{Energy difference (required energy):}$$
The energy required to move the body is the difference between the energies of the two orbits:
$$\Delta E = E_2 - E_1.$$
Substitute our values:
$    \begin{aligned}    \Delta E &= \left(-\frac{GMm}{8R}ight) - \left(-\frac{GMm}{6R}ight) \    &= \frac{GMm}{6R} - \frac{GMm}{8R}.    \end{aligned}    $
To combine the fractions, find a common denominator:
$$\frac{GMm}{6R} = \frac{4GMm}{24R},\quad \frac{GMm}{8R} = \frac{3GMm}{24R}.$$
Therefore,
$$\Delta E = \frac{4GMm}{24R} - \frac{3GMm}{24R} = \frac{GMm}{24R}.$$
Thus, the energy required is
$$\frac{GMm}{24R}.$$
Comparing with the given options, the correct answer is:
Option B
$$\frac{GMm}{24R}.$$

Answer

$$
\frac{\mathrm{GMm}}{\mathrm{R}}
$$

Answer

$$
\frac{\mathrm{GMm}}{24 \mathrm{R}}
$$

Answer

$$
\frac{\mathrm{GMm}}{4 \mathrm{R}}
$$

Answer

$$
\frac{\mathrm{GMm}}{12 \mathrm{R}}
$$

Gravitation

Energy required for moving a body of mass $\mathrm{m}$ from a circular orbit of radius 3R to a higher orbit of radius 4R around the earth is.

Choose the correct answer: