The principle of conservation of mechanical energy is a fundamental law in physics and forms the foundation of various other laws and interrelated concepts. Let us know more about this concept.

The principle of conservation of mechanical energy forms an integral part of physics and a proper understanding of the basics of this concept can help students gain mastery in mechanics and other related branches of physics. In fact, every work done or force applied is related to energy. It won’t be an exaggeration if it’s stated that energy is the essence of this Universe. Mechanical energy is stored in an object due to the virtue of its motion.

Potential and kinetic energy are the two types of mechanical energy, and the law we are going to discuss is associated with the conservation of these two in a system. Mechanical energy is basically a combination of potential and kinetic energy.

### Principle

According to this law, *in an isolated system, i.e., in the absence of non-conservative forces like friction, the initial total energy of the system remains constant.* Simply stated, the total mechanical energy of a system is always constant in case of absence of non-conservative forces. For instance, if a ball is rolled down a frictionless roller coaster, the initial and final energies will remain constant. Conservative forces are those that don’t depend on the path taken by an object.

### Equation

The quantitative relationship between work and energy is stated by the mechanical energy equation.

*U _{T} = K_{i} + P_{i} + W_{ext} = K_{f} + P_{f}*, where,

*U _{T}* = Total mechanical energy

*K*= Initial kinetic energy

_{i}*K*= Final kinetic energy

_{f}*P*= Initial potential energy

_{i}*P*= Final potential energy

_{f}*W*= External work done

_{ext}This is the general equation for conservation of mechanical energy. In case there are some external or internal forces acting on the object, i.e., if the forces are non-conservative like friction, air resistance, etc., then only W_{ext}is considered. In the absence of such forces, W_{ext} = 0, so the mechanical energy conservation equation takes the form:

*U _{T} = K_{i} + P_{i} = K_{f} + P_{f}*

### Mathematical Example

Let us consider a mathematical problem that involves the use of this law in finding the values of unknown quantities.

*Question:* A 20 g stone is put in a sling shot with a spring constant of 100 N/m and it is stretched back to 0.7 m. Determine the maximum velocity that the stone will acquire and the speed at which it is shot straight up.

*Solution:* In this problem, we ignore the air resistance and heat effects that are present while operating the sling shot. This makes the external work done zero, which means we can easily apply the law of conservation of mechanical energy.

Total energy in the beginning of the event is,

E_{i} = K_{i} + Gravitational potential energy (mgh) + spring force (½ kx^{2}).

Here,

K_{i} = (0.5 mv^{2}) = (0.5)m (0)^{2} = 0 (Since v = 0 initially)

Gravitational potential energy = mg(0) = 0 (since h = 0 initially)

Spring force = ½ kx^{2} = (0.5)(100)(0.7)^{2} = 24.5 J = E_{i}

Once out of the sling shot, the stone gains some maximum velocity before it reaches some altitude.

E_{f} = 0.5 mv^{2} + mgh + ½ kx^{2} = (0.5)(0.02)(v)^{2} + mg(0) + (0.5)k(0)^{2} = 0.001v^{2}

Since E_{i} = E_{f},

24.5 J = 0.001v^{2} = 24,500 = v^{2}. Therefore, v = 156.1 m/s (approximate value)

At the highest point, the velocity of stone is zero.

Therefore, E_{f} = 24.5 J = 0.5 mv^{2} + mgh + ½ kx^{2}

24.5 J = 0.5mv(0)^{2} + mgh + 1/2k(0)^{2} = 24.5 J = (0.02)(9.8 N/Kg)h

= 125 m.

*Answer:* Velocity attained = 156.1 m/s and height attained = 125 m

Almost every phenomena of the universe is governed by the universal law of energy conservation according to which, “energy can neither be created nor be destroyed, but can be transferred from one form to the other”. Gravitational energy, nuclear energy, electrical energy, and mechanical energy are various types of energy and they can all be transformed from one state to other.