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1. The problem statement, all variables and given/known data
Think that we have only conservative force in this question and no nonconservative force like friction exists. Also## U ##means potential energy (eg Gravitational Potential Energy),##K## means Kinetic Energy (##1/2 mv^2##) and ##E = K + U = constant##
We have a arbitrary Potential energy (U) in terms of position (x) diagram like the picture. The object is at first, in a position like ##x_0## and wants to get to point ##x_1##. At ##x_1## we have ##dU / dx = 0## and Also the initial Mechanical Energy (E = K + U) of the object at first is equal to the U at point ##x_1##. We want to prove that it takes infinite time theoretically for this object to get to the point ##x_1##. (The movement is 1 Dimensional)
2. Relevant Equations
##E = K + U = Constant##
##K = 1/2 m v^2##
##F = m a##
##F = dU/dx##
Taylor Series at ##x = x_1## for function ##f(x)##: ##f(x) = f(x_1) + f'(x) (x  x_1) + f''(x) (x  x_1)^2 / 2! + ...##
All forces are conservative in this question.
3. The attempt at a solution
At first we know that the K at ##x_1## equals to zero. Intuitively, I can understand the at the end of the path, the acceleration goes to zero and also the velocity goes to zero (because ## K > 0 ==> V > 0##). But I don't know how to prove this. I know that F = dU/dx.
For the start, I get this from taylor series but I don't know what to do with it:
##U(x) = U(x1) + dU/dx (x  x1) + O(x^2)## > ##U(x) = U(x1) + (m a) (x  x1)
Think that we have only conservative force in this question and no nonconservative force like friction exists. Also## U ##means potential energy (eg Gravitational Potential Energy),##K## means Kinetic Energy (##1/2 mv^2##) and ##E = K + U = constant##
We have a arbitrary Potential energy (U) in terms of position (x) diagram like the picture. The object is at first, in a position like ##x_0## and wants to get to point ##x_1##. At ##x_1## we have ##dU / dx = 0## and Also the initial Mechanical Energy (E = K + U) of the object at first is equal to the U at point ##x_1##. We want to prove that it takes infinite time theoretically for this object to get to the point ##x_1##. (The movement is 1 Dimensional)
2. Relevant Equations
##E = K + U = Constant##
##K = 1/2 m v^2##
##F = m a##
##F = dU/dx##
Taylor Series at ##x = x_1## for function ##f(x)##: ##f(x) = f(x_1) + f'(x) (x  x_1) + f''(x) (x  x_1)^2 / 2! + ...##
All forces are conservative in this question.
3. The attempt at a solution
At first we know that the K at ##x_1## equals to zero. Intuitively, I can understand the at the end of the path, the acceleration goes to zero and also the velocity goes to zero (because ## K > 0 ==> V > 0##). But I don't know how to prove this. I know that F = dU/dx.
For the start, I get this from taylor series but I don't know what to do with it:
##U(x) = U(x1) + dU/dx (x  x1) + O(x^2)## > ##U(x) = U(x1) + (m a) (x  x1)
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