Which of the following most closely depicts the correct variation of the gravitation potential \({V(r)}\) due to a large planet of radius \({R}\) and uniform mass density? (figures are not drawn to scale)
1. 3.
2. 4.

 
Subtopic:  Gravitational Potential |
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On the \(x\text-\)axis at a distance \(x\) from the origin, the gravitational field due to a mass distribution is given by \(\frac{A x}{\left(x^2+a^2\right)^{3 / 2}}\) in the \(x\)-direction. The magnitude of gravitational potential on the \(x\)-axis at a distance \(x\), taking its value to be zero at infinity, is:
1. \( \frac{A}{\left(x^2+a^2\right)^{3 / 2}} \)
2. \( A\left(x^2+a^2\right)^{1 / 2} \)
3. \( A\left(x^2+a^2\right)^{3 / 2} \)
4. \(\frac{A}{\left(x^2+a^2\right)^{1 / 2}}\)

Subtopic:  Gravitational Potential |
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An asteroid is moving directly toward the center of the Earth. When at a distance of \(10 R\) (\(R\) is the radius of the earth) from the earth's center, it has a speed of \(12 ~\text{km/s}.\) Neglecting the effect of earth's atmosphere, what will be the speed of the asteroid when it hits the surface of the earth (escape velocity from the earth is \(11.2 ~\text{km/s})?\) 
1. \(12~\text{km/s}\)
2. \(14~\text{km/s}\)
3. \(16~\text{km/s}\)
4. \(18~\text{km/s}\)
 
Subtopic:  Gravitational Potential |
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A mass of \(50~\text{kg}\) is placed at the center of a uniform spherical shell of mass \(100~\text{kg}\) and radius \(50~\text{m}.\) If the gravitational potential at a point, \(25~\text{m}\) from the center is \(V~\text{kg/m}.\) The value of \({V}\) is:
1. \({-60~G}\)
2. \({+2~G}\)
3. \({-20~G}\)
4. \({-4~G}\)
Subtopic:  Gravitational Potential |
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