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Question: Potential energy function U (r) corresponding to the central force \[F\;{\text{ }} = {\text{ }}K/{\t...

Potential energy function U (r) corresponding to the central force F   = K/ r2F\;{\text{ }} = {\text{ }}K/{\text{ }}{r^2} would be:
A. −K/r - K/r
B. −2K/r - 2K/r
C. −r/K - r/K
D. None of these

Explanation

Solution

First of all, we will find the expression for the amount of work done. We will replace the value of the force by the central force. After that we will directly apply integration, to find the total force over the complete region.

Formula used:
We know that as per definition of potential energy (Only defined for conservative forces) is integration of central force (F) with respect to dr  dr\; .
U= ∫−Fdr  U = {\text{ }}\smallint - Fdr\;

Complete step by step answer:
Given: Central Force (F) = K/r2 = {\text{ }}K/{r^2}
Central force may be a force (possibly negative) that points from the particle directly towards a hard and fast point in space, the middle, and whose magnitude only depends on the space of the object to the center.A central force may be a force (possibly negative) that points from the particle directly towards a hard and fast point in space, the middle, and whose magnitude only depends on the space of the object to the center.

If the central force is a conservative force, then the magnitude F(r) of a central force can always be expressed as the derivative of a time-independent potential energy function U(r)
The expression for the force is given below:
U= ∫−Fdr  U = {\text{ }}\smallint - Fdr\; …… (i)
Where F   = K/ r2F\;{\text{ }} = {\text{ }}K/{\text{ }}{r^2} is given and KK= constant
U    =∫ −K/r2drU{\text{ }}\;{\text{ }} = \smallint {\text{ }} - K/{r^2}dr
⇒U   = −K ∫ 1/r2dr\Rightarrow U\;{\text{ }} = {\text{ }} - K{\text{ }}\smallint {\text{ }}1/{r^2}dr
∴U   = −K. r\therefore U\;{\text{ }} = {\text{ }} - K.{\text{ }}r
Hence potential energy corresponding to central force K/r2 = −K. rK/{r^2}{\text{ }} = {\text{ }} - K.{\text{ }}r

Hence, option A is correct.

Note: A central force may be a force (possibly negative) that points from the particle directly towards a hard and fast point in space, the middle, and whose magnitude only depends on the space of the object to the center. It is important to note that the negative sign indicates the work done by the system.