Two isolated point masses and are located on the -axis at and respectively

PHYSICS · Multi-part · A Level · Physics · Fields and their consequences · Fields and their consequences

Problem

Two isolated point masses M1M_1 and M2M_2 are located on the xx-axis at x=0x = 0 and x=dx = d respectively. The gravitational potential at a general point P(x)P(x) on the axis (taking zero at infinity) is V(x)=−G(M1∣x∣+M2∣x−d∣).V(x) = -G\Bigg(\frac{M_1}{|x|} + \frac{M_2}{|x-d|}\Bigg). Using this information, analyse the following: (i) Show that dVdx=0\dfrac{dV}{dx}=0 at any point where the net gravitational field on a test mass is zero. (ii) Prove that, along the line joining the two masses, there can be at most one point where dVdx=0\dfrac{dV}{dx}=0. (iii) By evaluating d2Vdx2\dfrac{d^2V}{dx^2} at that point, demonstrate that the point corresponds to a maximum (least negative) value of potential on the line. (iv) Comment briefly on the physical significance of this result for a spacecraft attempting to remain at that position without active control systems.

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