Example To Explain Total Mechanical Energy Of The System

The capacity of doing work by a body is known as energy. In earlier classes, we have learnt that there are different types of energies. For example, heat, chemical, nuclear, mechanical, gravitational, and solar are all types of energy. The SI unit of energy is the joule. Since our topic is mechanical energy, we will learn more about mechanical energy.  

Mechanical energy

The energy of an object due to its motion or position is called the mechanical energy of that body. It is the sum of kinetic and mechanical energy of the body. For e.g., a car moving with any velocity will acquire mechanical energy.

For instance: If an archer stretches a bow, the archer will acquire mechanical energy.

Parts of mechanical energy 

The mechanical energy is divided into two parts:

• Kinetic energy 
• potential energy   

Kinetic energy

The energy in an object due to its motion is called kinetic energy. It is represented simply as, K.E. For example, if a baseball hit by a player moves with the velocity v due to its motion, it will have kinetic energy which can be calculated by the formula given below:

Kinetic energy = ½.mv2      

The SI Unit of kinetic energy is a joule.

Potential energy

The energy by virtue of an object’s position is called potential energy. It is also called energy possessed by the object due to its height. For example, if a person stretches a catapult, the energy possessed in the catapult is called potential energy. It is also called the energy due to the vertical position of an object. It is represented by P.E. The SI unit of potential energy is the joule. 

Ex: Gravitational Potential energy  U= mgh   

Now, the total mechanical energy in a system is defined as the sum of its kinetic energy and mechanical energy and is represented as:

Mechanical energy = kinetic energy + potential energy 

                   M.E. = K.E. + P.E.

Conservation of mechanical energy 

If we calculate the total energy of any system at any point during the motion, we see that the energy of the system always remains constant. This is termed the conservation of energy. It simply states that the energy of interacting bodies in a closed system always remains constant. Whenever the energy transforms, the total energy remains unchanged. Energy can neither be created nor be destroyed; it can be changed from one form to another. It means energy is always conserved. The mechanical energy of the system remains conserved if consvervative force is applied to the system. Under the non-conservative force mechanical energy is not conserved.

Example

Let us understand this with the help of an example. We are assuming a baseball of mass m which is dropped from a building of height H. Also consider the velocity of the baseball at three different points are – v0, v1 and v2. We will calculate the total mechanical energy at three points mentioned in the diagram as per the location of the baseball.                                                                                                                                                                                                   At height H:

Potential energy = m x g x h

Since the velocity of ball at this point is zero, so, kinetic energy = ½.mv02

                    =1/2.m(0)2=0 

and total mechanical energy = mgH+0   = mgH

At height h:

Potential energy =m x g x h

Kinetic energy =1/2(mv2)  

By using the equations of motion:

                              v12= u2+2g(H-h)           

since the initial velocity is zero

                              v12=0+2g(H-h)

                              v1=√2g(H-h)

         Kinetic energy = ½.m(√2gH-h)2

                                              = mgH-mgh

      Total mechanical energy = K.E.+ P.E.

                                               = (mgH-mgh)+mgh

                                                =  mgH

   At height zero:

    Potential energy = 0

    Kinetic energy = ½.mv2

     Using equation of motions

             v22=u2+2gH

                 v22=0+2gH

                v22 =√2gH

   Kinetic energy = ½.m(√2gH)2=mgH

    Total mechanical energy = mgH

    Here we can see that 

      Potential energy + kinetic energy = constant 

                                  mgh + ½.mv2 = constant 

We saw that the total mechanical energy is constant at all three points..

This can also be understood by another example. Suppose a person is travelling on a roller coaster. In the journey, he experiences several highs and lows. The highest point where he starts his journey will have high potential energy and low kinetic energy.

When the coaster reaches the lowest point, it will have maximum kinetic energy and minimum potential energy. After some time, when it attains a low height, it acquires low potential energy and high kinetic energy. This example tells us how one energy is converted into another energy. This is called the conservation of energy.

Conclusion

From the above calculations, we clearly see that the total mechanical energy at every point in the journey of the falling baseball is the same. We added both potential energy and kinetic energy at every point and we got the same total energy i.e., mgH.

The decrease in potential energy, at any point, appears as an equal amount of increase in kinetic energy. This concludes that energy can neither be created nor be destroyed but it changes from one form to another (here the effect of air resistance on the motion of the object is ignored).