1
If we assume that the density of the Earth stays the same, then doubling the radius increases the planet's mass eight-fold. Surface gravity is now doubled, so most plants and trees promptly fall over. Any animal dog-sized or bigger can't run without breaking a leg, so large predators can't move fast enough to catch prey.
2
Angular momentum is proportional to mass x radius squared. As angular momentum is conserved, the planet now rotates 32 times slower. A day lasts over a month! This creates huge temperature imbalances between the light and dark side of the planet, powering wind storms that flatten all buildings.
3
If the atmosphere doubles in size too, then air pressure is doubled. This is still breathable and the extra oxygen helps to offset the greater effort needed to pump blood up to your brain. The thicker air actually makes flying easier for birds, although most need to land on water to cushion the impact caused by the stronger gravity.
4
The greater volume of radioactive elements in the mantle means hotter magma. Even though the crust is thicker, hundreds of new volcanoes erupt, pumping more CO2 into the atmosphere and creating a runaway greenhouse effect. This leads to the biggest extinction event Earth has ever seen.
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10 Questions 10 Marks 10 Mins
Concept:
Newton's law of gravitation:
- "Any particle of matter in the universe attracts any other with a force varying directly as the product of the masses and inversely as the square of the distance between them".
- It is directed along the line joining the point masses.
- It is a conservative force field, i.e mechanical energy remains conserved.
- It is a central force i.e angular momentum remains conserved.
For two mass m1 and m2, it is given by
\(F = \frac{Gm_1 \times m_2}{r^2}\)
- Gravity Equation or Newton’s universal law of gravitation:
- where G is the gravitational constant, F is the force due to gravity between two masses (m1 and m2), which are a distance r apart.
- The expression for acceleration due to gravity (g) on the surface of a planet is given by:
F = mg
\(mg= {GMm \over r^2}\)
\(g= {GM \over r^2}\)
where r is the radius of the planet; G is the gravitational constant, M is the mass of the planet.
Calculation:
Given that,
Final mass of earth, M’ = 2 × initial mass of earth, 2M
Final radius of earth, R’ = 2× initial radius of earth, 2R
Thus, the ratio of final acceleration and initial acceleration can be given as
\(\frac{{g'}}{g} = \frac{{\frac{{GM'}}{{{R^{'2}}}}}}{{\frac{{GM}}{{{R^2}}}}} = \frac{{\frac{{2GM}}{{ {{\left( {2R} \right)}^2}}}}}{{\frac{{GM}}{{{R^2}}}}} = \frac{1}{2}\)
\(\therefore g' = \frac{g}{2}\)
Hence the weight becomes half of the original weight.
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Text Solution
Solution : Earth's gravitational acceleration , ` g = (GM)/R^(2)` <br> ` g_(1) = (GM_(1))/(R_(1)^(2)) and g_(2) = (GM_(2))/(R_(3)^(2))` <br> ` g_(2)/g_(1) = ((M_(2))/M_(1)) (R_(1)/R_(2))^(2) = 2(2)^(2) = 8` <br> `g_(2) = 8 g_(1)` Thus , the value of g on the surface of the earth would be eight times the present value.