NEB Class 11 Work, Energy and Power | Physics Notes PDF

Work:

The product of force and displacement is called Work. It is mathematically given by W=F.d = Fdcos$\theta $.

It is a scalar quantity because it is defined as the dot product of scalar and two vectors force and displacement and the scalar product of two vectors gives the scalar quantity.

Mathematically, W = $\vec F. \vec D$

Some Cases:

(i) When θ = 0⁰,

W = Fdcos$\theta $ = FdCos0 = Fd

(ii) When θ = 90⁰,

W = Fdcos$\theta $ = FdCos90 = Fd.0 = 0

(iii) When θ = 180⁰,

W = Fdcos$\theta $ = FdCos90 = Fd.(-1) = -Fd

Note: Negative sign shows that the work is done in the direction of applied force.

Work Done by Variable Force:

Work is said to be done on the body if the external force applied on that body produces some displacement on the direction of applied force.

Variable Force: The force which vary with position of a body in magnitude, direction or both is called Variable Force.

Let us consider a body is displaced form position A to Position B under the action of varying force as shown in the figure.

Let PQ = dx be the small displacement of the body during the motion. Since the displacement is small and can be considered to be constant.

Small amount of work done (dW) = $\overrightarrow F .\overrightarrow {dx} $  = $PQ . PS $ = area of PQRS

∴ Total work done while moving from Position A to B can be obtained but integrating equation

W =$\mathop {\lim }\limits_{{\rm{dx}} \to 0{\rm{\: }}} \Sigma {\rm{\: Fdx}}$  = $\mathop \smallint \limits_{{{\rm{x}}_{\rm{A}}}}^{{{\rm{X}}_{\rm{B}}}} {\rm{Fdx}}$ =  $\mathop \smallint \limits_{{{\rm{x}}_{\rm{A}}}}^{{{\rm{X}}_{\rm{B}}}} {\rm{area\: of\: PQRS}}$  = Area of ABCDA

Note: Work done by variable force is equal to the area under the force and displacement curve.

Energy:

The capacity of doing work is called energy. The amount of energy that a body has equal to the amount of work that it can do.

Mathematically, Work Done = Energy Stored in it.

Energy can exist in various forms such as chemical energy, nuclear energy, heat energy, light energy etc. This chapter mainly Focuses on Mechanical Energy.

Mechanical Energy: It is the sum of Potential Energy and Kinetic Energy.

Potential Energy: Energy possessed by the body due to virtue of its position is called potential energy. Example: Water stored in the tank.

Mathematically,

Potential Energy (P.E) = mgh

Kinetic Energy: Energy possessed by the body due to virtue of its motion is called kinetic energy.

Example: Bullet is fired from the gun.

Mathematically,

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

Relationship between Kinetic Energy and linear momentum:

The relationship between kinetic energy (K) and momentum (p) can be derived as follows:

Starting with the definition of momentum as the product of mass (m) and velocity (v):

$p=mv$

Or, $v = \frac{p}{m}$ …….(i)

Also, KE = $\frac{1}{2}mv^2$ …… (ii)

From Equation (i) and (ii), We Get

$KE = \frac{1}{2} m \left(\frac{p}{m}\right)^2$

∴$KE = \frac{p^2}{2m}$

Work-Energy Theorem:

Statement: The work done by the net force on a particle is equal to the change in particle’s kinetic energy.

OR

The change in Kinetic Energy of the body is equal to work done.

Let us consider a body of mass m moving with initial speed ‘u' covers a distance ‘s’ under the action of constant force F in the direction of applied force as shown in the figure. After time t it’s velocity becomes ‘v’ at point B.

The work done (W) = Fs [F=ma]

Or, W = mas……..(i)

We Know,

v²=u²+2as

or, $as = \frac{{{v^2} - {u^2}}}{2}$ …..(ii)

From (i) and (ii)

W = m . as = m.$\frac{{{v^2} - {u^2}}}{2}$

Or, W = $\frac{1}{2}m{v^2} - \frac{1}{2}m{u^2}$

Or, W = K.E_final - K.E_initial

Or, W = ΔK.E

Some Special Cases based on study of above equations are:

1. When W is positive, the kinetic energy increases i.e., K.E_final > K.E_initial
2. When W is negative, the kinetic energy decreases i.e., K.E_final < K.E_initial
3. When W is zero, the kinetic energy remains same i.e. K.E_final = K.E_initial

Note: If the body moves in vertical direction under action of gravity, then ΔK.E = ΔP.E.

Thus, W= ΔK.E = ΔP.E

Principle of Conservation of Energy (Mechanical):

Statement: Energy can neither be created nor be destroyed but can be converted from one form to another form.

For a body falling freely, the total mechanical energy remains constant throughout the motion.

Let us consider a body with mass ‘m’ is initially at rest at height ‘h’ is dropped from point A. Also let B and C be the position of the body during the fall as shown in figure below.

At position A

Height = h and velocity = V_A = 0 (Body at rest)

K.E_A=0 [Body at rest]

P.E_A = mgh

Total Energy = K.E_A + P.E_A = 0 + mgh = mgh

At position B

Height = h-x and velocity = V_B

We have V_B² = V_A² + 2aS =0+2gx=2gx

K.E_B= $\frac{1}{2}m{V_B}^2\$=$\frac{1}{2}m.(2gx) = mgx$

P.E_B=mg(h-x)=mgh-mgx

Total Energy = K.E_A + P.E_A = mgx + mgh-mgx = mgh

At position C

Height = 0 and velocity = V_C 

We have, V_C² = V_A² + 2gS

or, V_C² = 0 + 2gh 

K.E_C = $\frac{1}{2}m{V_C}^2 = \frac{1}{2}m \times 2gh = mgh$

P.E_B = mgh=mg.0 = 0

Total Energy = K.E_A + P.E_A = mgh + 0 = mgh

Total Energy _{At A} = Total Energy _{At B} = Total Energy _{At C}

Finally, this shows that the total energy remains constant throughout the motion.  

Conservative and Non-Conservative Force:

Property	Conservative Forces	Non-conservative Forces
Definition	A force is said to be conservative if the work done by or against the force in moving body depends on the initial and final positions of the body and independent on the nature of the path followed.	A force is said to be non-conservative if the work done by or against the force in moving body depends on nature of the path followed between these points.
Effect on Mechanical Energy	Do not change the total mechanical energy of a system.	Change the total mechanical energy of a system.
Work Done	Path-independent.	Path-dependent.
Potential Energy	Exists and is associated with conservative forces.	Does not exist for non-conservative forces.
Energy Conservation	Total mechanical energy is conserved.	Total mechanical energy is not conserved.
Examples	Gravitational force, electrostatic force, spring force.	Frictional force, air resistance, tension in a rope.
Force Field	Has a scalar potential function.	Does not have a scalar potential function.

Power:

The rate at which the work is done by a force is said to be the power due to that force.

Mathematically,

Power =$\frac{Work}{Time Taken}\$

$P = \frac{{\overrightarrow F .\overrightarrow S }}{t} = \overrightarrow F .\frac{{\overrightarrow S }}{t} = \overrightarrow F .\overrightarrow V  = FV{\mathop{\rm Cos}\nolimits} \theta $

Where, V is called instantaneous velocity and P is called instantaneous Power. The Practical unit of power is horse power.

1H.P = 746W

Collision:

Collision is defined as the mutual interaction of the particle for relatively short interval of a time as a result of which the energy and momentum of interacting particles changes.

In physics, two bodies do not necessarily need physical touch for collision. Example: Whenever two marbles collide with each other there is a physical contact but when two electrons collide there may not be physical contact, they actually repel each other.

On the basis of conservation of kinetic energy collision or classified under two categories. They are

1. Elastic collision: A collision is said to be elastic whenever the linear momentum and kinetic energy remains conserved which results in no loss of energy.
2. Inelastic collision: A collision is said to be inelastic whenever the linear momentum is conserved but not the kinetic energy.

Elastic Collision in One Dimension

Let us consider two particles, with mass m_1­ and m₂ and initial velocities u₁ and u₂ before collision, and v₁ and v₂ velocities after collision. Since the elastic collision has conserved linear momentum and kinetic energy.

From the conservation of linear momentum:

m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂

Or, m₁(u₁−v₁)=m₂(v₂−u₂) ……..(i)

Also, Kinetic energy remains conserved. So,

Kinetic Energy Before Collison = Kinetic energy after collision

$\frac{1}{2}{m_1}{u_1}^2 + \frac{1}{2}{m_2}{u_2}^2\; = \frac{1}{2}{m_1}{v_1}^2 + \frac{1}{2}{m_2}{v_2}^2\;$

${m_1}({u_1}^2 - {v_1}^2) = {m_2}({u_2}^2 - {v_2}^2)\;$ ……..(ii)

Solving Above equations (i) and (ii)

u₁ + v₁ = u₂+v₂

u₁-u₂=v₁-v₂

Thus we see that in one dimensional elasticity elastic collision the relative velocity of approach before collision is equal to relative velocity of recession(separation) after collision.

From the above results,

v₂=u₁-u₂+v₁

Using this value in equation (i), we get:

m₁(u₁−v₁)=m₂(u₁-u₂+v₁−u₂)

or, m₁u₁- m₁v₁₌ m₂u₁-m₂u₂ + m₂v₁- m₂u₂

or, (m₁-m₂)u₁ +2m₂u₂ = (m₁+m₂)v₁

$[\therefore {v_1} = \frac{{{m_1} - {m_2}}}{{{m_1} + {m_2}}}{u_1} + \frac{{2{m_2}}}{{{m_1} + {m_2}}}{u_2}$

Similarly, Final Velocity (v₂) of mass m₂ can be represented as

$\therefore {v_2} = \frac{{{m_2} - {m_1}}}{{{m_1} + {m_2}}}{u_2} + \frac{{2{m_1}}}{{{m_1} + {m_2}}}{u_1}$

Some Special Cases:

(i) When two bodies are if same masses i.e., m₁=m₂=m (say)

$\therefore {v_1} = 0 + \frac{{2m}}{{2m}}{u_2} = {u_2}$

Note: When two bodies of equal masses collide elastically in one dimension, they simply exchange their velocities after collision.

(ii) When one body of mass (say m₂) is at rest i.e. (u₂=0)

$\therefore {v_1} = \frac{{{m_1} - {m_2}}}{{{m_1} + {m_2}}}{u_1} + 0 = \frac{{{m_1} - {m_2}}}{{{m_1} + {m_2}}}{u_1}$

$\therefore {v_2} = 0 + \frac{{2{m_1}}}{{{m_1} + {m_2}}}{u_1} = \frac{{2{m_1}}}{{{m_1} + {m_2}}}{u_1}$

Inelastic Collision in one dimension:

Let us consider a body of mass m₁ moving with initial velocity u₁ collides with another body of mass m₂ initially at rest u₂=0. Let the two bodies permanently stick together after collision and move with common velocity v as shown in the figure. Since inelastic collision have only conserved linear momentum not kinetic energy.

From Conservation of Linear Momentum,

m₁u₁ + m₂u₂ = m₁v + m₂v

Or, m₁u₁ + 0 = (m₁ + m₂)v [∵ m₂ is initially at rest]

$\therefore \;v = \frac{{{m_1}{u_1}}}{{{m_1} + {m_2}}}$......(i)

Kinetic energy before collision = K.E.₁ = $\frac{1}{2}{m_1}{u_1}^2 + \frac{1}{2}{m_2}{u_2}^2\;$

$\therefore \;K.E{._1}\; = \frac{1}{2}{m_1}{u_1}^2$ …..(ii)

And Kinetic Energy after Collision = K.E.₂= $\frac{1}{2}\left( {{m_1}{v^2} + {m_2}{v^2}} \right)$

$\therefore \;K.E{._2} = \frac{1}{2}\left( {{m_1} + {m_2}} \right){v^2}$……(iii)

Dividing Equation (ii) by (iii)

$\frac{{K.E{._2}\;}}{{K.E{._1}}} = \frac{{\left( {{m_1} + {m_2}} \right){v^2}}}{{{m_1}{u_1}^2}}$

Using equation (i)

$\begin{array}{l}\frac{{K.E{._2}\;}}{{K.E{._1}}} = \frac{{\left( {{m_1} + {m_2}} \right)}}{{{m_1}{u_1}^2}}{(\frac{{{m_1}{u_1}}}{{{m_1} + {m_2}}})^2}\\or,\frac{{K.E{._2}\;}}{{K.E{._1}}} = \frac{{{m_1}}}{{{m_1} + {m_2}}}\\or,\frac{{K.E{._2}\;}}{{K.E{._1}}} < 1\\\therefore K.E{._2}\; < K.E{._1}\;\end{array}$

i.e., Kinetic Energy Before Collision> Kinetic Energy after Collision, which means there is loss in kinetic energy in inelastic collision.

Coefficient of Restitution (e):

The ability of a body to temporarily deform rapidly is called resilience.

Coefficient of Restitution is defined as the number expression a ration of the relative velocity of separation to the relative velocity of approach.

If two bodies move with velocities u₁ and u₂ before collision and v₁ and v₂ after collision, then

Coefficient of Restitution (e) = $\frac{{relative{\rm{ }}\;velocity{\rm{ }}\;of\;{\rm{ }}separation}}{{relative\;{\rm{ }}\;velocity{\rm{ }}\;of{\rm{ \; approach}}}}$

$\therefore e = \frac{{{v_2} - {v_1}}}{{{u_1} - {u_2}}}$

Some Cases for Coefficient of Restitution:

1. For elastic collision, e=1
2. For perfectly inelastic collision e=0
3. Practice in collision 0<e<1