Potential Energy - Practice Questions & MCQ

• Definition-

           Potential energy is defined only for conservative forces.

           In the space occupied by conservative forces, every point is associated with certain energy which is called the energy of

           position or potential energy.

• Change in potential energy -

          Change in potential energy between any two points is defined as the work done by the associated conservative force

               in  displacing the particle between these two points without any change in kinetic energy. 

$
\begin{aligned}
U_i-U_f & =\int_{r_i}^{r_f} \vec{f} \cdot \overrightarrow{d s} \\
\text { Where, } U_f & - \text { final potential energy } \\
U_i & - \text { initial potential energy } \\
f & - \text { force } \\
d s & - \text { small displacement } \\
r_i & - \text { initial position } \\
r_f & - \text { final position }
\end{aligned}
$

We can define a unique value of potential energy only by assigning some arbitrary value to a fixed point called the reference point.

Whenever and wherever possible, we take the reference point at infinite and assume potential energy to be zero there. i.e; if take $r_i=\infty$ and $r_f=r_{\text {then from equation (1) }}$

$
U_r=-\int_{\infty}^r \vec{f} \cdot \overrightarrow{d r}=-W
$
 

         In the case of conservative force (field) potential energy is equal to negative of work done in shifting the body from

               reference position to given position. 

• Types of potential energy-

 Potential energy generally is of three types:

            Elastic potential energy, Electric potential energy, and Gravitational potential energy, etc.

1. Potential Energy stored when particle displaced against gravity

$
\begin{aligned}
& U=-\int f d x=-\int(m g) d x \cos 180^{\circ} \\
& \text { Where } m=\text { mass of body } \\
& g=\text { acceleration due to gravity } \\
& d x=\text { small displacement }
\end{aligned}
$

2. Potential Energy stored in the spring-
- Restoring force $=f=-k x$ (or spring force)

Where $\mathbf{k}$ is called spring constant.
- Work done by restoring the force

$
W=-\frac{1}{2} k x^2
$

- Potential Energy

$
\begin{aligned}
& \qquad \begin{array}{l}
U \\
\\
\qquad
\end{array} \\
& \text { Where } K=\text { spring constant } \\
& \qquad \\
& x \\
& x
\end{aligned}
$

3. The relation between Conservative Force and Change in potential energy -

For only conservative fields $F$ equals the Negative of the rate of change of potential energy with respect to position.

$
F=\frac{-d U}{d r}
$

The three-dimensional formula for potential energy-
For only conservative fields $F$ equals the negative gradient $(-\vec{\nabla})$ of the potential energy.

$
F=-\vec{\nabla} U
$

Where $\vec{\nabla}$ is del operator

And,

$
\vec{\nabla}=\frac{d}{d x} \vec{i}+\frac{d}{d y} \vec{j}+\frac{d}{d z} \vec{k}
$

So,

$
F=-\left[\frac{d U}{d x} \vec{i}+\frac{d U}{d y} \vec{j}+\frac{d U}{d z} \vec{k}\right]
$

Where $\frac{d U}{d x}=$ Partial derivative of $U$ w.r.t. $\times$ (keeping y and z constant)
$\frac{d U}{d y}=$ Partial derivative of U w.r.t. y (keeping x and z constant)
$\frac{d U}{d z}=$ Partial derivative of U w.r.t. Z (keeping x and y constant)

• Potential energy curve 

  A graph plotted between the potential energy of a particle and its displacement from the center of force is called potential energy curve. 

 The figure shows a graph of the potential energy function U(x) for one-dimensional motion. As we know that negative gradient of the potential energy gives force.

                    $-\frac{d U}{d x}=F$

•   Nature of force-

1.  Attractive force -

• If  $\frac{d U}{d x}$  is positive (means on increasing x, U is increasing)

           Then F is negative in direction i.e. force is attractive in nature. 

•  In the graph, this is represented in region BC.

2. Repulsive force-

• If  $\frac{d U}{d x}$  is negative (means on increasing x, U is decreasing)

           Then F is positive in direction i.e. force is repulsive in nature.

• In the graph, this is represented in the region AB.

3. Zero force 

• If  $\frac{d U}{d x}$  is zero  (means on increasing x, U is not changing )  then F is zero

• Points B, C, and D represent the point of zero force.

•  These points can be termed as a position of equilibrium.

• Types of equilibrium 

           If the net force acting on a particle is zero, it is said to be in equilibrium.

           Means For equilibrium $\frac{d U}{d x}=0$

           Equilibrium of particle can be of three types-

1. Stable equilibrium

• When a particle is displaced slightly from a position, then a force acting on it brings it back to the initial position, it is said to be in the stable equilibrium position.

• $\frac{d^2 U}{d x^2}>0$ is positive.

           i.e; the rate of change of $\frac{d U}{d x}$  is positive

• Potential energy is minimum.

• A marble is placed at the bottom of a hemispherical bowl.

2. Unstable equilibrium

• When a particle is displaced slightly from a position, then a force acting on it tries to displace the particle further away from the equilibrium position, it is said to be in unstable equilibrium.

• $\frac{d^2 U}{d x^2}$ is negative

           i.e; rate of change of $\frac{d U}{d x}$  is negative

• Potential energy is maximum.

• A marble balanced on top of a hemispherical bowl.

3. Neutral equilibrium 

• When a particle is slightly displaced from a position then it does not experience any force acting on it and continues to be in equilibrium in the displaced position, it is said to be in neutral equilibrium.

• $\frac{d^2 U}{d x^2}>0$

           i.e; the rate of change of $\frac{d U}{d x}$  is zero.

• Potential energy is constant.

• A marble is placed on a horizontal table.