The restoring force and P.E. of a particle executing a S.H.M. are F and U respectively when its displacement is x. The relation between F, U ad x is

Options

A. `(F)/(U)+x=0`
B. `(2U)/(F)+x=0`
C. `(F)/(2U)+x=0`
D. `(U)/(F)+x=0`

Explanation

To find the relationship between the restoring force (F), potential energy (U), and displacement (x) of a particle executing Simple Harmonic Motion (SHM), we can follow these steps:

### Step 1: Understanding the Restoring Force
The restoring force (F) in SHM is given by Hooke's Law, which states that the force exerted by a spring is proportional to the displacement from its equilibrium position and acts in the opposite direction. Mathematically, this is expressed as:
\[ F = -k \cdot x \]
where \( k \) is the spring constant and \( x \) is the displacement from the equilibrium position.

### Step 2: Understanding Potential Energy
The potential energy (U) stored in a spring when it is displaced by a distance \( x \) is given by:
\[ U = \frac{1}{2} k x^2 \]

### Step 3: Relating F and U
From the expression for potential energy, we can express the spring constant \( k \) in terms of potential energy and displacement:
\[ k = \frac{2U}{x^2} \]

### Step 4: Substituting k into the Restoring Force Equation
Now, we substitute the expression for \( k \) back into the equation for the restoring force:
\[ F = -k \cdot x = -\left(\frac{2U}{x^2}\right) \cdot x \]
This simplifies to:
\[ F = -\frac{2U}{x} \]

### Step 5: Rearranging the Equation
We can rearrange this equation to express a relationship between U, F, and x:
\[ 2U = -F \cdot x \]
or
\[ \frac{2U}{F} + x = 0 \]

### Conclusion
Thus, the relationship between the restoring force (F), potential energy (U), and displacement (x) is given by:
\[ \frac{2U}{F} + x = 0 \]