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Question: An astronaut is standing at the north pole of a newly discovered planet of radius R. He holds a cont...

An astronaut is standing at the north pole of a newly discovered planet of radius R. He holds a container full of liquid having mass m and volume V. At the surface of the liquid pressure is p0 and at a depth d below the surface, pressure is p. From this information determine the mass of the planet.

A

(a) rac(p+p0)VR2Gdm rac{\left( p + p_{0} \right)VR^{2}}{Gdm}

A

(b) rac(p+p0)R2V2Gdm rac{\left( p + p_{0} \right)R^{2}V}{2Gdm}

A

(c) rac(p−p0)R2V2Gdm rac{\left( p - p_{0} \right)R^{2}V}{2Gdm}

A

(d) rac(p−p0)R2VGdm rac{\left( p - p_{0} \right)R^{2}V}{Gdm}

Explanation

Solution

(d)

ρgd = (p – p0)

or g = (p−p0)d(m/V)\frac{\left( p - p_{0} \right)}{d(m/V)}

or GMR2=(p−p0)Vdm\frac{GM}{R^{2}} = \frac{\left( p - p_{0} \right)V}{dm}

or M = (p−p0)VR2Gdm\frac{\left( p - p_{0} \right)VR^{2}}{Gdm}