Q.

A particle which is constrained to move along the x-axis, is subjected to a force in the same direction which varies with the distance x of the particle from the origin as F(x)=−kx+ax3. Here k and a are positive constants. For  x≥0, the functional from of the potential energy  U(x) of the particle is

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a

b

c

d

answer is D.

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Detailed Solution

F=−dUdx⇒dU=−F  dx ⇒U=−∫0x(−Kx + ax3)dx=kx22−ax44∴ We get U = 0 at x = 0 and x = 2k​/​aand also U = negative for  x>2k​/​a.So F = 0 at x = 0 i.e. slope of U – x graph is zero at x = 0.
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A particle which is constrained to move along the x-axis, is subjected to a force in the same direction which varies with the distance x of the particle from the origin as F(x)=−kx+ax3. Here k and a are positive constants. For  x≥0, the functional from of the potential energy  U(x) of the particle is