Law of Conservation of Energy - A Complete Guide

The Law of Conservation of Energy is a fundamental principle in physics that states energy cannot be created or destroyed, only transformed from one form to another. This law governs everything around us, from the way our bodies use food to produce energy, to how machines operate and even how stars shine in the universe. Understanding this law helps us appreciate the seamless flow of energy through various systems, showing us the interconnectedness and balance of the natural world. By exploring the Law of Conservation of Energy, we can see how energy transfers and transformations impact our everyday lives, ensuring that nothing is ever truly lost, just changed.

In this article, we are going to cover the Law of Conservation of Energy and Chain, which is placed on a table. which belongs to the chapter work, energy, and power, which is an important chapter in Class 11 physics. It is not only essential for board exams but also for competitive exams like the Joint Entrance Examination (JEE Main), National Eligibility Entrance Test (NEET), and other entrance exams such as SRMJEE, BITSAT, WBJEE, BCECE and more. Over the last ten years of the JEE Main exam (from 2013 to 2023), more than ten questions have been asked on this concept. And for NEET seven questions were asked from this concept.

Let's read this entire article to gain an in-depth understanding of law of conservation of energy

Law of Conservation of Energy

Conservation of Mechanical Energy

Mechanical energy is the sum of potential energy and kinetic energy.

According to Conservation of Mechanical Energy, If only conservative forces act on a system, the total mechanical energy remains constant.

By work-energy theorem, we have
W=kf−ki or △K=∫rirff→⋅ds→

The change in potential energy in a conservative field is
Ui−Uf=∫rirff→⋅ds→

Or,
−△U=∫rirff→⋅ds→

From equation (1) and (2)
We get, ΔK=−ΔU
ΔK+ΔU=0

Means, K+U=E (constant)
Or,E is constant in a conservative field

i.e.; if the kinetic energy of the body increases its potential energy will decrease by an equal amount and vice versa.

Proof of Law of Conservation of Energy

Let us consider the motion of a bob in a simple pendulum and also suppose that it is in simple harmonic motion.

In this case, we need to prove that the motion of the pendulum is conserved.

Let us consider the following motion of the bob of a simple pendulum.

Learn Law Of Conservation Of Energy Better From the Video given Below

Law of Conservation of Energy Derivation

Now, let us find the total energy conservation at point A

If we consider point A, it is the extreme position and is at rest for a while. At this point the velocity of the bob will be zero. Since, at point ‘A’ the bob is at rest, so there won’t be any motion in it. Ultimately, the Kinetic energy will be zero. But, the bob is at a height ‘h’, so, the Potential energy at this point will be maximum.

So, we have K.E = 0

And P.E =mgh

No, the total energy = K.E + P.E

= 0 + mgh

Therefore, total energy at point A = mgh

We can conclude that the total energy at point A is potential energy.

Similarly let us find the total energy conservation at point M.

Now, when the bob of the pendulum is released from the point ‘A’, then the velocity of the bob is gradually increased and the height of the bob is continuously decreased. It means that at point ‘M’ the velocity will be maximum and the height will be minimum or we can say it will be zero.

So, we have, K.E. = $\frac{1}{2} m v^2$

And P.E. = 0

Total Energy at point ‘M’ = $\frac{1}{2} m v^2+0=\frac{1}{2} m v^2$

It means that the potential energy of the bob is converted into kinetic energy at point ‘M’.

Also, let us compare the energy conservation at point ‘B’.

As the bob of the pendulum is in motion, it will not stop at point ‘M’ due to the inertia in it and the bob will move toward the point ‘B’. Again when the bob moves from point ‘M’ to ‘B’, its velocity will gradually decrease but the height will increase. So, at point ‘B’ again the velocity becomes zero.

So, we again have K.E = 0

And P.E =mgh

No, the total energy = K.E + P.E

= 0 + mgh

Therefore, total energy at point B = mgh

Now, from this we can conclude that the total energy at point ‘B’ is again potential energy.

So, we can see here in the case of a simple pendulum that total energy is constant.

Law Of Conservation Of Total Energy

If some non-conservative force like friction is also acting on the particle, the mechanical energy is no longer constant.

It changes by the amount of work done by non-conservative forces.

\text { i.e; } \Delta K+\Delta U=\Delta E=W_{f n c}

The lost energy is transformed into heat or in other forms of energy. But the total energy remains constant.

So, according to the Law of conservation of total energy “Energy may be transformed from one kind to another but it cannot be created or destroyed. The total energy in an isolated system is constant.”

To further comprehend the above concept, consider the solved example provided below.

Example: Two hydrogen atoms are in an excited state with electrons residing in n = 2. The first one is moving towards the left and emits a photon of energy towards the right. The second one is moving towards the right with the same speed and emits a photon of energy towards the right. Taking the recoil of the nucleus into account during the emission process

1)
E1>E2
2)
E1<E2
3)
E1=E2
4) none

Solution:

If only conservative forces act on a system, total mechanical energy remains constant -
K+U=E (constant )ΔK+ΔU=0ΔK=−ΔU

In the first case, K.E. of H-atom increases due to recoil whereas in the second case, K.E. decreases due to recoil but
E1+KE1=E2+KE2∴E2>E1

Chain Placed On A Table

Work Done in Pulling the Chain Against Gravity

A chain of mass M and length L is held on a frictionless table with (1n)th of its length hanging over the edge.

Suppose we take a dy length of chain at a distance y from the table

Let μ=ML mass per unit length of the chain and the mass of that dy length of chain be dm.

So the mass of dy length of chain:-

dm=μdy⇒dm=(ML)dy

Here we are taking the top of the table as the reference point and below the reference point, the potential energy will be negative.

We know that potential energy against gravity is given by mgh

and here, mass = dm, h=y.

So for small mass, the potential energy will be small as dU:

dU=−(dm)g(y)⇒dU=−(ML)gy(dy)⇒dU=−dWc⇒∫0Ln(ML)gydy And also, ⇒Wc=(ML)g∫0Lnydy=(ML)g[y22]0Ln⇒Wc=MgL2n2

Law of Conservation of Energy Examples

The Law of Conservation of Energy is followed everywhere. Let us take a few examples from our daily life.

1. During winters we often rub our hands. In doing so, we feel warmth. We know this fact that by rubbing the hands friction is created and friction is mechanical energy. While rubbing, the mechanical energy is converted into heat energy making us feel warm.

2. The cells or the battery that we use in our remote controls, torches, toys, etc use the chemical energy stored in them and convert them into electrical energy. This electrical energy can further be converted into other forms of energy such as light, motion etc. Here, also we can see that the total energy is conserved.

3. When we burn the fuels, then the chemical energy stored in them is converted into heat and light energy.

4. The food that we eat has chemical energy stored in it. It is further converted into thermal energy when it is broken down inside the body and later converted into other forms due to which we are able to do other mechanical works such as walking, running, pulling, pushing, etc.

5. In motors the electrical energy is converted into mechanical energy.

Also read -

• NCERT Solutions for Class 11 Physics
• NCERT Solutions for Class 12 Physics
• NCERT Solutions for All Subjects

NCERT Physics Notes:

• NCERT Notes Class 11th Physics
• NCERT Notes Class 12th Physics
• NCERT Notes For All Subjects

Let's see some solved examples to understand this concept in a better way.

Solved Example Based on Law of Conservation of Energy

Example 1: A uniform chain of length L and mass M is lying on a smooth table and one-third of its length is hanging vertically down over the edge of the table. If g is the acceleration due to gravity, the work required to pull the hanging part onto the table is

1) MgL
2) MgL3
3) MgL9
4) MgL18

Solution:

Work done to raise the centre of mass of the chain on the table is given by

W=\frac{M g L}{2 n^2}

As 1/3 part of the chain is hanging from the edge of the table. So by substituting n = 3 in standard expression

We get : W=MgL2n2=MgL2(3)2=MgL18

Hence, the answer is the option 4.

The velocity of the Chain While Leaving the Table

Taking the surface of the table as a reference level (zero potential energy)

The potential energy of the chain when (1n)th length hanging from the edge =−MgL2n2
The potential energy of the chain when it leaves the table =−MgL2 (here n=1) By the law of conservation of mechanical energy,
K⋅E⋅+Ui=K⋅E⋅f+Uf⇒0−MgL2n2=12Mv2−MgL2⇒12Mv2=MgL2−MgL2n2⇒v=gL[1−1n2]

Example 2: A man places a chain (of mass ‘m’ and length ‘’) on a table slowly. Initially, the lower end of the chain just touches the table. The man drops the chain when half of the chain is in a vertical position. Then work done by the man in this process is :

1) −mgl2
2) −mgl4
3) −3mgl8
4) −mgl8

Solution:

Potential Energy stored when a particle is displaced against gravity -

U=−∫fdx=−∫(mg)dxcos⁡180∘ - wherein m= mass of body g= acceleration due to gravity dx= small displacement

The work done by man is negative of the magnitude of the decrease in potential energy of the chain

ΔU=mgL2−m2gL4=3mgL8∴W=−3mgl8

Example 3: A ball of mass 4kg moving with a velocity of 10 ms^-1, collides with a spring of length 8m and force constant 100 Nm^-1. The length of the compressed spring is x m. The value of x, to the nearest integer is _______.

1) 6

2) 2

3) 8

4) 10

Solution:

Let the spring be compressed by y.

Applying the energy conservation principle,

⇒12mv2=12ky2⇒y=mk⋅v⇒y=4100×10⇒y=2m

Therefore, the final length of the spring =8−2 = 6m

Hence, the answer is (6).

Example: Two identical blocks A and B each of mass m resting on the smooth horizontal floor are connected by a light spring of natural length L and spring constant K. A third block C of mass m moves with a speed v along the line joining A and B collides with A. The maximum compression in the spring is

1) m2K
2) vm2K
3) mvK
4) mv2K

Solution:

If only conservative forces act on a system, total mechanical energy remains constant -

K+U=E( constant )V=x44−x22.

First, we need to calculate the minimum potential energy.
When potential energy is minimum
dVdx=0⇒x3−x=0∴x=0 or x=1 V(x=0)=0V(x=1)=14−12=−14J So minimum potential energy =−14J

So minimum potential energy
K.E. + P.E. = Total mechanical energy.
K.E.|max+ P.E. |min= Total mechanical energy
K⋅E⋅max−14=2JK⋅E⋅max=94J

⇒12mVmax2=94 or Vmax2=92Vmax=32

Hence, the answer is option (2).

Conclusion

Every process is nothing but energy transformation, be it in natural phenomena or man-made machines. It is defined as a law that requires us to understand the relationship between various energy forms, energies and the efficient use of energy. Recognising that energy is only transformed, we can start to create more green technologies that focus on energy conservation and sustainability. The law, therefore, ultimately affirms the assumption that energy is unequivocal and integral imbued in every physical interaction, providing safety, and continuity in the natural world.