NEET Important Chapter - Work, Energy and Power

Name of the Concept

Key Points of the Concept

Work and its unit

• Work as a scalar quantity is defined as the product of force and displacement acting on a body.

• Work is classified as positive if both the force and displacement are acting in the same direction or the angle between both the physical quantities is less than $90^{\circ}$ and negative when both the quantities are acting in opposite directions to each other or the angle between both the physical quantities is greater than $90^{\circ}$.

• Similarly, work is considered zero if the force and displacement are perpendicular to each other else the angle between both the physical quantities is equal to $90^{\circ}$.

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•  The S.I. unit of work is Joule with the dimensional formula as $M^{1}L^{1}T^{-2}$

Work done by a Variable force

• The formula of Work done by a variable force is calculated as  $\int_{s_{1}}^{s_{2}}\vec{F}(s)\mathrm{d}s$

• Plotting a graph between force and displacement, we can calculate the work done by variable force by finding the area under the same.

• Or we can say area under force displacement graph =

• $\int_{s_{1}}^{s_{2}}\vec{F}(s)\mathrm{d}s$

Energy and its various types

• Energy is another name of work done. Contrarily, we can say that it refers to the capability of the body to do some kind of work.

• Kinetic Energy - This energy is related to the virtue of motion of a body, having formula as K.E=$\dfrac{1}{2}mv^{2}$

• Potential Energy - This energy is related to the change in position of the body in some field

with the formula =mgh.

Work Energy Theorem

• This is one of the most important theorems stating that the work done by a net force in displacing a body is the same as the change in kinetic energy of the body.

• $W_{}=\dfrac{1}{2}mv^{2}-\dfrac{1}{2}mu^{2}= K_{f}-K_{i}$ This formula depicts the entire statement stated above. 

Conservation of Mechanical Energy

• This theorem states that the sum of kinetic and potential energy of the body is mechanical energy i.e. E = K+V = $\dfrac{1}{2}mv^{2}$ + mgh.

• If the force acting on the body is conservative then we can say that mechanical energy of that body is conserved.

Conservative Forces

• If the work done by a body in displacing it from one position to another is independent of the path followed then that force is called Conservative Force. 

•  Another major characteristic of this force is work done in this case is always zero.

• Examples of conservative forces are gravitational force, spring force and electric force.

• If $\triangle\times\bar{F}$=0 then 

• $\bar{F}=-\dfrac{\partial U}{\partial x}$

• Hence, a conservative force is a negative gradient of potential energy.

Other Forms of Energy:-

The potential energy of spring 

• This energy is related to the application of force done on a spring system. The external force on the spring produces a restoring force inside the spring due to compression or extension. F=-kx

• Hence, the energy stored in spring system is given by relation :

• =$\dfrac{1}{2}kx^{2}$

Where k is the spring constant and x is the compression or extension in the spring.

Gravitational Potential Energy

• This is another example of conservative force stated as whenever a body is raised to some height, it acquires some kind of energy due to a change in its position. This energy is called Gravitational Potential Energy.

• Since the raising of the body is against the direction of earth’s gravity, the force acting here is negative.

• The work done here is written as W = mgh.

• U=-$\dfrac{GM}{m}$ for two bodies having mass M and m separated by distance r

Vertical Circular Motion

• Considering a body of mass ‘m’ is tied with a            string (massless) and imparts a velocity at the lowest  point such that it completes the circular trajectory. 

•  A is the lowest point, VA is the velocity imparted to the particle at the lowest point and the length of the string is ‘R’ 

• Such that tension acting at C(highest point)= $m(v^{2}/r-g)$ whereas tension at lowest point = $(v^{2}/r+g)$.

• Minimum velocity required by a body to complete circular motion is $\geqslant\sqrt{5rg}$

• Minimum velocity required by a body to reach at point B = $\geqslant\sqrt{2rg}$

Power

• Power is defined as a dot product of force and velocity or it is also written as the rate of work done. 

• P= $\dfrac{W}{t}$ or P = $\vec{W\cdot\vec{v}}$

• The dimensional formula of power is $M^{1}L^{2}T^{-3}$ with the measuring unit as Watt.

Collision and its Types

• Collision is defined as the sticking of two bodies after colliding with each other or if the force exerted by one body  affects the path of another body then the two bodies are said to be colliding.

• Collision is classified under two categories 

• Elastic and Inelastic.

 Elastic Collision

• Those collisions in which the laws of total energy and momentum remain conserved along with conservation of kinetic energy are called elastic collisions.

• In elastic collision, the relative velocity of separation is equal to relative velocity of approach.

• Here, the value of coefficient of restitution is 1

i.e. e=1

Inelastic Collision

• Here only the laws of total energy and momentum remain conserved but kinetic energy conservation doesn’t take place.

• In inelastic collision, the value of the coefficient of restitution lies between 0 and 1. 

Sl. No

Name of the Concept

Formula

1. 

Work

W=Fs$\cos\theta$

2.

Work done by a Variable force

$\int_{s_{1}}^{s_{2}}\vec{F}(s)\mathrm{d}s$

3.

Energy and its various types

Kinetic Energy (K.E)=$\dfrac{1}{2}mv^{2}$

Potential Energy =mgh.

4.

Conservative Forces

$\bar{F}=-\dfrac{\partial U}{\partial x}$

5.

Work- Energy Theorem

$W_{}=\dfrac{1}{2}mv^{2}-\dfrac{1}{2}mu^{2}= K_{f}-K_{i}$

6.

Conservation of     Mechanical Energy

E = K+V = $\dfrac{1}{2}mv^{2}$ + mgh.

7. 

Other forms of Energy -

The potential energy of spring 

Work done in spring system

=$\dfrac{1}{2}kx^{2}$

8.

Gravitational Potential Energy

The Work done here is written as W = mgh.

U=-$\dfrac{GM}{m}$ for two bodies having mass M and m separated by distance r

9.

Vertical Circular Motion

• Tension acting at C(highest point)= $m(v^{2}/r-g)$ whereas tension at lowest point = $(v^{2}/r+g)$.

• Minimum velocity required by body to complete circular motion is $\geqslant\sqrt{5rg}$

• Whereas minimum velocity required by body to reach at point B = $\geqslant\sqrt{2rg}$

10.

Power

P=$\dfrac{W}{t}$ or P = $\vec{W\cdot\vec{v}}$

11.

Elastic Collision in one dimension

Here, the formula of final velocity after collision is

$v_{1}$= $(\dfrac{m_{1}-m_{2}}{m_{1}+m_{2}})u_{1}+\dfrac{2m_{2}u_{2}}{m_{1}+m_{2}}$

$v_{2}$= $(\dfrac{m_{2}-m_{1}}{m_{1}+m_{2}})u_{2}+\dfrac{2m_{1}u_{1}}{m_{1}+m_{2}}$