The potential energy of a body is given by A-Bx^2(where x is the displacement). Where "A
"B" are constants. The magnitude of the force acting on the particle is
(A) Constant
(B) Proportional to x
(C) Proportional to x
(D) Inversely proportional to x
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A conservative force is given as the negative gradient of potential energy, in other words it's the negative of the first order derivative of potential energy.
So, F = -d(U)/dx
In the given question,
U = A - Bx^2
Hence, F = - d(A - Bx^2)/dx
F = - [ dA/dx - d(Bx^2)/dx]
F = - [ 0 - B (2x)]
(since, differentiation of a constant is zero, so dA/dx = 0 and differentiation of x^2 = 2x)
F = 2Bx
Hence, we can see that F is directly proportional to 2X. Since, 2 and B are constants, you can also write it as F is directly proportional to X.
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