(-6C₆) / x⁷ newton; it is an attractive force.
Given:
The potential energy of a pair of hydrogen atoms separated by a large distance x is given by [tex]\boxed{ \ U(x) = - \frac{C_6}{x^6} \ }[/tex], where C₆ is a positive constant.
Question:
The Process:
Let us rewrite the potential energy of the pair of the hydrogen atom:
[tex]\boxed{ \ U(x) = - \frac{C_6}{x^6} \ }[/tex]
The component of a conservative force in a particular direction, equals the negative of the derivative of the corresponding potential energy, properly concerning a displacement in that direction.
For one-dimensional motion, say along the x-axis, the relationship between force and potential energy is as follows:
[tex]\boxed{ \ \overrightarrow{F} = F_x \hat{i} = - \frac{\delta U}{\delta x} \hat{i} \ }[/tex]
Let us determine the force.
[tex]\boxed{ \ F_x \hat{i} = - \frac{\delta}{\delta x} \Big( - \frac{C_6}{x^6} \Big) \ \hat{i} \ }[/tex]
[tex]\boxed{ \ F_x \hat{i} = - (- C_6) \frac{\delta}{\delta x} \Big( \frac{1}{x^6} \Big) \ \hat{i} \ }[/tex]
[tex]\boxed{ \ F_x \hat{i} = C_6 \frac{\delta}{\delta x} (x^{-6}) \ \hat{i}} \ }[/tex]
[tex]\boxed{ \ F_x \hat{i} = (C_6)(-6)x^{-7} \ \hat{i} \ }[/tex]
Thus, we get: [tex]\boxed{\boxed{ \ F_x = - \frac{6C_6}{x^7} \ \hat{i} \ }}[/tex]
From the data problem above, we know that C₆ is a positive constant. Hence, Fₓ is positive when x is negative and vice versa. Thus Fₓ is always directed toward the origin. Besides, a negative sign means that an attractive force occurs.
In essence, the directions are toward the origin, since this is the potential energy for a restoring force.
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Notes:
We know that the difference of potential energy from point 1 to point 2 as the negative of the work done:
[tex]\boxed{ \ \Delta U_{12} = U_2 - U_1 = - W_{12} \ }[/tex]
The work done by the given force as the particle moves from coordinate x to (x + d x) in one dimension is
[tex]\boxed{ \ dW = \overrightarrow{F} \cdot d \vec{r} \ } \rightarrow \boxed{ \ \overrightarrow{F} = \frac{dW}{d \vec{r}} \ }[/tex]
Therefore, for motion along a straight line, a conservative force is the negative derivative of its associated potential energy function, i.e.,
[tex]\boxed{ \ \overrightarrow{F} = F_x \hat{i} = - \frac{\delta U}{\delta x} \hat{i} \ }[/tex]
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Conservative force : force that does work independent of path.
