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# Potential Energy - Practice Questions & MCQ

Edited By admin | Updated on Sep 18, 2023 18:34 AM | #JEE Main

## Quick Facts

• Potential energy is considered one the most difficult concept.

• Potential energy curve is considered one of the most asked concept.

• 31 Questions around this concept.

## Solve by difficulty

A spring of force constant 800 N/m has an extension of 5 cm. The work done in extending it from 5 cm to 15 cm is

A block of mass ‘ m ‘ is attached to a spring in natural length of spring constant ‘ k ‘ . The other end A of the spring is moved with a constant velocity v away from the block . Find the maximum extension in the spring.

A spring of spring constant  $5\times 10^{3} N/m$ is stretched initially by  5 cm from the unstretched position.Then the work (in Nm) required to stretch it further by another  5 cm is :

This question has statement 1 and statement 2. Of the four choices given after the statements, choose the one that best describes the two statements.

If two springs S1 and S2 of force constants k1 and k2, respectively, are stretched by the same force, it is found that more work is done on spring S1 than on spring S2.

Statement 1: If stretched by the same amount, work done on S1, will be more than that on S2.

Statement 2: k1 < k2

A spring has a spring constant $k$. If the spring is compressed by a distance $x$, the potential energy stored in the spring is :

What is the energy stored in an object called due to its position or height from the ground?

A porter lifts a 10 kg suitcase from the platform and places it on his head 3.0 m above the platform. Calculate the work done by the porter on the suitcase.

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A spring with spring constant $k$ when stretched through 1cm, the potential energy is $U$. If it is stretched by 4cm. The potential energy will be :

## Concepts Covered - 2

Potential energy
• 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.

$\small U_{i}-U_{f}= \int_{r_{i}}^{r_{f}}\vec{f}\cdot \vec{ds}$  ......(1)

Where,  $U_{f}-final\: potential\: energy$

$U_{i}-initial \: potential\: energy$

$f-force$

$ds-small \: displacement$

$r_{i}-initial \: position$

$r_{f}-final\: position$

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  $r_{f} = r$ then from equation (1)

$\small U_{r}= -\int_{\infty}^{r}\vec{f}\cdot \vec{dr} = -W$

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

$\small U= -\int fdx= -\int \left ( mg \right )dx\: cos180^{0}$

Where $\small m=mass \: of \: body$

$\small g= acceleration \: due \: to\: gravity$

$\small dx= small\: displacement$

1. Potential Energy stored in the spring-

• Restoring force =  $\small f=-kx$ (or spring force)

Where k is called spring constant.

• Work done by restoring the force

$\small W= -\frac{1}{2}\: kx^{2}$

• Potential Energy

$\small U= \frac{1}{2}\: kx^{2}$

Where $\small K=spring \: constant$

$\small x= elongation\ or\ compression\ of\ spring\ from\ natural\ position.$

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.

$\small F=\frac{-dU}{dr}$

The three-dimensional formula for potential energy-

For only conservative fields F  equals the negative gradient $\small (-\vec{\bigtriangledown })$ of the potential energy.

$\small F = -\vec{\bigtriangledown }U$

Where $\small \vec{\bigtriangledown}$ is del operator

And,  $\small \vec{\bigtriangledown} = \frac{d}{dx}\vec{i}+\frac{d}{dy}\vec{j}+\frac{d}{dz}\vec{k}$

So, $\small F = -[\frac{dU}{dx}\vec{i}+\frac{dU}{dy}\vec{j}+\frac{dU}{dz}\vec{k}]$

Where   $\small \frac{dU}{dx}$ =Partial derivative of U w.r.t. x (keeping y and z constant)

$\small \frac{dU}{dy}$ =Partial derivative of U w.r.t. y (keeping x and z constant)

$\small \frac{dU}{dz}$ =Partial derivative of U w.r.t. Z (keeping x and y constant)

Potential energy curve
• 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  $\frac{dU}{dx}$  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.

1. Repulsive force-

• If  $\frac{dU}{dx}$  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.

1. Zero force

• If  $\frac{dU}{dx}$  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 $\frac{dU}{dx} = 0$

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.

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

i.e; the rate of change of $\frac{dU}{dx}$  is positive

• Potential energy is minimum.

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

1. 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.

• $\frac{d^{2}U}{dx^{2}}$ is negative

i.e; rate of change of $\frac{dU}{dx}$  is negative

• Potential energy is maximum.

• A marble balanced on top of a hemispherical bowl.

1. 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.

• $\frac{d^{2}U}{dx^{2}} = 0$

i.e; the rate of change of $\frac{dU}{dx}$  is zero.

• Potential energy is constant.

• A marble is placed on a horizontal table.

## Study it with Videos

Potential energy
Potential energy curve

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