Conservation of Mechanical Energy: Definition, Formula and Examples

Conservation of Mechanical Energy: Mechanical energy is known as “movement energy,” since it is found in objects that move or have the ability to move. Except for the gravitational force, it states that the mechanical energy of an object in a closed system remains constant if it is not subjected to any friction, i.e. air resistance. When we use a hammer to, say, strike a nail and drive it into the wall, we are merely exerting force on the nail with the aid of the hammer.

A hammer has no kinetic energy but does have some potential energy when it is at rest. When we swing a hammer up to a certain distance from a nail before hitting it, kinetic energy is released, and the hammer’s combination of kinetic and potential energy, known as mechanical energy, causes the nail to drive into the wall. In this article, we will study and learn about the concept of conservation of mechanical energy in detail.

What is Mechanical Energy?

Conservation of Mechanical Energy will be more clearly understood if we first get acquainted with the concept of mechanical energy. The mechanical energy of an object is the sum of its potential energy and kinetic energy.

In this, the potential energy of an object is the energy possessed by it by virtue of its position (or configuration), whereas kinetic energy is the energy possessed by the object by virtue of its motion.
Thus, the mechanical energy of an object is the energy possessed by it due to the virtue of its position or motion or both.

In the case of a system, mechanical energy is the sum of its potential energy and the kinetic energy of the objects within it.

State the Principle of Conservation of Mechanical Energy

The principle of mechanical energy conservation asserts that if a body or system is only subjected to conservative forces, its mechanical energy remains constant.

What Does Conservation of Mechanical Energy Mean?

Conservation of Mechanical Energy means that the total mechanical energy of the object or the system remains constant at any instant. Here, the kinetic energy or the potential energy at any instant can be changed, but their sum is always constant. But such a situation is possible only if the force causing the transformation of energy is conservative in nature.

A force is considered to be conservative if the work done by it or against it only depends on the initial and the final position of the body and not on the path taken between the initial and the final position. Gravitational force is a conservative force, whereas air resistance is a non-conservative force.

For example, take the case of a body undergoing a fall. During the start of its motion, it will just have potential energy, and just at the end of the fall before it hits the ground, it will just have kinetic energy. Here, the potential energy at the beginning of the fall will be entirely converted into the kinetic energy gained by the object at the end of the fall.

In the intermediate stage, we can find that the decrease in its potential energy will always be equal to the gain in its kinetic energy, and at any instant, their sum will always be the same. Thus, the total mechanical energy of the body always remains constant. Here, we are neglecting the air resistance.

Law of Conservation of Mechanical Energy Class 11

Conservation of Mechanical Energy in Class 11 Physics is well formulated as a law that governs the energy transformations caused by internal conservative forces. The law of conservation of mechanical energy states that “The total mechanical energy of a system remains constant if the internal forces are conservative and the external forces do no work.”

Derivation of Conservation of Mechanical Energy

The conservation of Mechanical Energy can be verified using the case of a simple pendulum. The simple pendulum consists of a string and a bob attached to it. The string of a simple pendulum is fixed at one end about which it is to move. The other end of the string is connected to a bob.

Now, this string with the bob (simple pendulum) is to move about the fixed end in the horizontal direction. Thus, the simple pendulum is to move towards the right and the left side of the mean position.

Let the mass of the bob be \(m\) and the string be massless. So, the mass of the simple pendulum is taken as \(m\). Let the acceleration due to gravity be \(g\) and the maximum height achieved by the simple pendulum while being pulled on either side be \(h\). Thus, at the mean position, the height of the simple pendulum is \(0\).

To verify the law of conservation of mechanical energy, let us now pull the simple pendulum on the right side such that it is at a height \(h\) from the mean position. At this position, the simple pendulum has the maximum potential energy \(\left({ {E_p}} \right)\).

So, at the rightmost side, the potential energy of the simple pendulum is \( {E_p} = mgh\) and its kinetic energy \(\left({ {E_k}} \right)\) is zero as its velocity \((v)\) is zero. So, \({E_k} = \frac{1}{2}\,{\rm{m}}{{\rm{v}}^2} = 0\).

Thus, the total mechanical energy \(\left({{E_m}} \right)\)of the pendulum at the right most side is \({E_m} = {E_p} + {E_k} = mgh + 0 = mgh{\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} …\left( 1 \right)\)

Now, when the simple pendulum is released, it starts to move leftwards due to the potential energy possessed by it and reach the mean position.

At the mean position, the potential energy is zero as the height is zero. Thus, \( {E_p} = 0\).
Due to the motion of the simple pendulum, it has gained kinetic energy, which is the maximum at this position. The velocity of the simple pendulum at this position is \(v\). This velocity is gained after travelling a distance \(h\) under the influence of acceleration due to gravity.

Using the third equation of motion \((v^2=2gh-u^2)\), we get, \(v^2=2gh\) by substituting the value of initial velocity \((u)\) as zero as the simple pendulum started from rest.
Putting this value of \(v^2\) in the formula of kinetic energy, we get,
\({E_k} = \frac{1}{2}\,{\rm{m}}{{\rm{v}}^2} = \frac{1}{2}\,{\rm{m}}\left( {2gh} \right) = mgh\)

So, at the mean position, the total mechanical energy of the simple pendulum is
\({E_m}\, = \,{E_p}\, + \,{E_k} = \,0\,\, + \,mgh\, = \,mgh{\mkern 1mu} {\mkern 1mu} {\mkern 1mu} …\left( {\rm{2}} \right)\)
Now, this simple pendulum has velocity, due to which it will move leftwards and eventually reach the leftmost side and will stop for a moment. At this location, its kinetic energy will be zero as its velocity will be zero. Thus, \({E_k} = \frac{1}{2}\,{\rm{m}}{{\rm{v}}^2} = 0\).

But as it reaches at the height \(h\), its potential energy will be \(E_p = mgh\) which will be the maximum like that at the rightmost side.
Thus, the total mechanical energy of the simple pendulum at the leftmost side will be
\({E_m} = \,{E_p} + {E_k} = \,mgh\, + \,0\, = \,mgh{\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} …\left( {\rm{3}} \right)\)

Hence, from the equations, \((1),\,(2)\), and \((3)\) it is clear that, the total mechanical energy of the simple pendulum remains constant throughout its swing from the rightmost side to its leftmost side. From this, we can conclude that the total mechanical energy of the simple pendulum remains conserved throughout its motion. Here, we are ignoring the loss of energy due to friction at the fixed end as well as the air resistance while swinging.

Conservation of Mechanical Energy Formula

Conservation of Mechanical Energy \((E_m)\) can be utilised in the numerical problems using the formula,
\({E_m} = {E_p} + {E_k} = mgh + \frac{1}{2}\,{\rm{m}}{{\rm{v}}^2} = {\rm{Constant}}\)
Where,
\(E_p\) is the potential energy
\(E_k\) is the kinetic energy
\(m\) is the mass
\(g\) is the acceleration due to gravity
\(h\) is the height and
\(v\) is the velocity.

Solved Conservation of Mechanical Energy Examples

Example 1: An object of mass \(200{\mkern 1mu} \,{\rm{g}}\) is raised to a height \(5\,{\mkern 1mu} {\rm{m}}\) above the ground. Calculate it potential energy at this height. If the object is made to fall, what will be its kinetic energy halfway down? Take \(g = 10\,{\mkern 1mu} {\rm{m}}\,{{\rm{s}}^{\rm{2}}}\).
Solution: Given that,
The mass of the object is \(m = 200\,{\mkern 1mu} {\rm{g}} = 0.2{\mkern 1mu} \,{\rm{kg}}\)
The height of the object is \(h = 5{\mkern 1mu} \,{\rm{m}}\)
The acceleration due to gravity is \(g = 10{\mkern 1mu} \,{\rm{m}}\,{{\rm{s}}^2}\)
The potential energy of the object at the height \(\left( {h = 5{\mkern 1mu} \,{\rm{m}}} \right)\) is
\({E_p} = mgh = 0.2 \times 10 \times 5 = 10{\mkern 1mu} \,{\rm{J}}\)

Now, halfway down, the kinetic energy of the body can be calculated using the law of conservation of mechanical energy. At this position, the kinetic energy gained will be equal to the potential energy lost which will be half of \(10\,{\mkern 1mu} {\rm{J}}.\)

Thus, the kinetic energy of the object will be \({E_k} = \frac{{{E_p}}}{2} = \frac{{10}}{2} = 5\,{\mkern 1mu} {\rm{J}}.\) Hence, the potential energy of the object at the highest point will be \(10\, {\text{J}}\) and its kinetic energy halfway down will be \(5{\mkern 1mu} \,{\rm{J}}.\)

Example 2: A body of mass \(5{\mkern 1mu} \,{\rm{kg}}\) is projected vertically upwards with a speed of \(2\,{\mkern 1mu} {\rm{m}}\,{{\rm{s}}^{ – 1}}\). What will be its maximum gravitational potential energy?
Solution: Given that,
The mass of the body is \(m = 5{\mkern 1mu} \,{\rm{kg}}\)
The velocity of the body is \(v = 2\,{\rm{m}}\,{{\rm{s}}^{ – 1}}\)
The kinetic energy of the body while throwing is \({E_k} = \frac{1}{2}\,{\rm{m}}{{\rm{v}}^2} = \frac{1}{2} \times 5 \times {2^2} = 10{\mkern 1mu} \,{\rm{J}}\)

Now, according to the law of conservation of mechanical energy, the maximum gravitational potential energy gained by the body will be equal to its kinetic energy while throwing. So, its maximum gravitational potential energy will be \( {E_p} = {E_k} = 10\,{\mkern 1mu} {\rm{J}}\)

Hence, the maximum gravitational potential energy of the body will be \(10\, {\text{J}}\).

Conservation of Mechanical Energy Applications

Conservation of Mechanical Energy can be applied where there are transformations of energy by the action of internal conservative forces. Some of the examples are as given below:

1. In a roller coaster ride, we see the interconversions of potential energy into kinetic energy and vice versa. Throughout the ride, the total mechanical energy of the system remains constant, and we can enjoy the ride for a pretty long duration along a thrilling path giving an exhilarating experience.
2. The water falling from a dam can rotate the blades of the turbine. Due to the conservation of mechanical energy, the potential energy of the water stored in the dam gets converted into kinetic energy while the water falls from the height, which is finally converted to electrical energy through the turbine which runs the generator.
3. Swing in the playground is yet another example. Once the swing is taken to the maximum position on one side and then released, it is capable of swinging on both sides one after the another continuously if the friction at its fixed end is taken care of by greasing or oiling.

FAQs on Conservation of Mechanical Energy

The most commonly asked queries on Mechanical Energy are answered here:

Q.1: Give one example where mechanical energy is conserved.
A: When a body undergoes fall, the total mechanical energy of the body remains constant, and thus, the mechanical energy remains conserved in this situation.

Q.2: What is a non-conservative force?
A: If the work done by a force depends on the path taken between the initial and the final position, it is called a non-conservative fore. The frictional force is a non-conservative force.

Q.3: What is Mechanical Energy?
A: Mechanical energy is the sum of kinetic energy and potential energy.

Q.4: What is the statement of conservation of mechanical energy?
A: The total mechanical energy of a system remains constant if the internal forces are conservative and the external forces do no work.

Q.5: What is a conservative force?
A: If the work done by a force depends only on the initial and the final position and not on the path taken, it is called a conservative force. Magnetic force is a conservative force.

Q.6: What is the formula for the conservation of mechanical energy?
A: The formula representing the conservation of mechanical energy is ({E_m} = {E_p} + {E_k} = mgh + \frac{1}{2}\,{\rm{m}}{{\rm{v}}^2} = {\rm{Constant}}).

Q.7: When is conservation of mechanical energy not applicable?
A: Conservation of mechanical energy is not applicable when the forces acting on the system are non-conservative in nature, like frictional force.