Conservation of Mechanical Energy

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 interrelated branches of physics. In fact, every work done or force applied is in some or other form 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 virtue of its motion. Potential energy 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 energies in a system. Mechanical energy is basically a combination of potential and kinetic energy.

Principle of Conservation of Mechanical Energy
According to this law, in an isolated system, that is, in the absence of non-conservative forces like friction, the initial total energy of the system equals to the total energy of the system. 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 remain constant. Conservative forces are those that don't depend on the path taken by an object. For example, gravity, spring and electrical forces are examples of mechanical energy.

Conservation of Mechanical Energy 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_i = Initial kinetic energy
K_f = Final kinetic energy
P_i = Initial potential energy
P_f = Final potential energy
W_ext = External work done

This is a general equation for mechanical energy conservation. In case, there are some external or internal forces acting on the object, that is the forces are non-conservative like friction, air resistance, etc, then only W_ext is considered. In absence of such forces, W_ext = 0 and so the mechanical energy conservation equation takes the form:

U_T = K_i + P_i = K_f + P_f

Conservation of Mechanical Energy: Mathematical Problem
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 of stone when 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 external work done zero, that means we can easily apply the law of conservation of mechanical energy formula.

Total energy in the beginning of the event E_i = K_i + Gravitational potential energy (mgh) + spring force (½ kx²). Here,

K_i = (0.5 mv²) = (0.5)m (0)² = 0 (Since v = 0 initially)
Gravitational potential energy = mg(0) = 0 (since h = 0 initially)
Spring force = ½ kx² = (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 mv² + mgh + ½ kx² = (0.5)(0.02)(v)² + mg(0) + (0.5)k(0)² = 0.001v²

Since E_i = E_f
Therefore, 24.5 J = 0.001v² = 24,500 = v². 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 mv² + mgh + ½ kx²
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 universal law of energy conservation according to which, "energy can neither be created nor be destroyed but can be transferred from one body 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.