**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*, where,

_{T}= K_{i}+ P_{i}+ W_{ext}= K_{f}+ P_{f}*U*= Total mechanical energy

_{T}*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.