Mechanical energy

An object that possesses mechanical energy is able to do work. In fact, mechanical energy is often defined as the ability to do work. Any object that possesses mechanical energy - whether it is in the form of potential energy or kinetic energy is able to do work.

Example :

A hammer is a tool that utilizes mechanical energy to do work. The mechanical energy of a hammer gives the hammer its ability to apply a force to a nail in order to cause it to be displaced. Because the hammer has mechanical energy (in the form of kinetic energy) it is able to do work on the nail. Mechanical energy is the ability to do work.

The mechanical energy of a bowling ball gives the ball the ability to apply a force to a bowling pin in order to cause it to be displaced. Because the massive ball has mechanical energy (in the form of kinetic energy), it is able to do work on the pin. Mechanical energy is the ability to do work.

Kinetic energy

kinetic energy is a form of energy that an object or a particle has by reason of its motion. If work, which transfers energy, is done on an object by applying a net force, the object speeds up and thereby gains kinetic energy.

Kinetic energy is a property of a moving object or particle and depends not only on its motion but also on its mass.

Kinetic energy  K.E= 1/2 mv^2

Potential Energy

The energy possessed by the object due to its position and configuration is called Potential Energy.

Potential energy is defined with the conservative force. For example, for the gravitational force, we have gravitational potential energy and for the electric force, we have electrostatic potential energy.

  • When a man picks an object of mass ‘m’ from the ground and puts that at height ‘h’. As gravity is pulling the object downward so the man has to do some work to raise the object to a height ‘h’. The work done by the man gets converted into the Potential energy of the object.

gravitational potential energy = mgh

  • To compress a spring an external force needs to do work. The work done by external force gets converted into the potential energy of the spring.

Elastic  potential energy=1/2 kx^2, where K is the spring constant of the spring and x= compression/ elongation

Conservation of total mechanical energy.

For motion under conservative force, we have a conservation law of mechanical energy. When the motion is subjected to only conservative force, the total mechanical energy ( kinetic + potential) of the system remains conserved.

Potential energy and Kinetic energy gets converted into each other such that total mechanical energy remains conserved.

In the first figure given above, the Kinetic energy of the bicycle at the ground gets converted into potential energy at the top and then again gets converted into Kinetic energy at the ground.

In the second figure, a girl at some height holds a ball and then drops it. The potential energy of the ball at height gets converted into kinetic energy as it falls under the effect of gravity.

In the above figure, the potential stored in the string of the bow due to stretching gets converted into the Kinetic energy of the arrow

In the case of a pendulum, the motion of the bob is under the force of gravity. At the extreme position on either side Potential energy of the bob is maximum and at the mean position, Kinetic energy is maximum. Total mechanical energy at any instant is constant in the case of Simple harmonic motion.

Work-Energy Theorem

Statement: The work-energy theorem states that the net work done by the forces on an object equals the change in its kinetic energy.

Net work done= change in Kinetic energy= K.E_f-K.E_i

Sometimes people forget that the work-energy theorem only applies to the network, not the work done by a single force.

Derivation of the work-energy theorem for the constant force

we have Work done = Fd= Fd cosθ, where F is force, d is displacement and θ is the angle between force and displacement.

When F || d , θ=0  then   W= Fd= (ma)d  …(1)

From the third equation of motion we have

v^2= u^2  + 2 a d  ;     2ad= v^2-u^2   ;  d=(v^2-u^2)/ 2a

So now we put the value of d in equation 1

W= ma ( (v^2-u^2)/2a) = m/2(v^2-u^2)= K.E_f-K.E_i  

Derivation of the work-energy theorem for variable force

Work done by small displacement  dW= Fds= ma ds=m(dv/dt)ds=m(ds/dt)dv=m v dv

Total work done will be the integration of the dW

Work done in changing the velocity from u to v

W= dW=_u^v mv dv= m_u^v v dv= m/2  [v^2-u^2 ]= Δ K.E

Conservative  forces

 According to the law of conservation of energy in a closed system, i.e., a system that is isolated from its surroundings, the total energy of the system is conserved”. Conservative force abides by the law of conservation of energy. A conservative force is a force that does zero work done in a closed path. If only these forces act then the mechanical energy of the system remains conserved.

Work done by conservative force= -(change of potential Energy)

Wcons. = -(P.Ef- P.Ei)

Non-conservative forces

Non-conservative forces are dissipative forces such as friction or air resistance. These forces take energy away from the system as the system progresses, the energy that you can’t get back. These forces are path-dependent; therefore it matters where the object starts and stops.

Mechanical energy is not conserved in non-conservative forces.

Work done by non conservative force = change in total Energy

W_(non cons. )= T.E_f- T.E_i=(K.E+P.E)_f-(K.E+P.E)_i

Work done by all forces ( conservative+ non conservative)

W_net= K.E_f- K.E_i

Various form of Energy

In addition to mechanical energy, there are various other forms of energy.

Chemical Energy: Energy stored in the bonds of chemical compounds. Chemical energy may be released during a chemical reaction, often in the form of heat; such reactions are called exothermic. Reactions that require an input of heat to proceed may store some of that energy as chemical energy in newly formed bonds.

 Heat: Heat is the form of energy that is transferred between systems or objects with different temperatures (flowing from the high-temperature system to the low-temperature system). Also referred to as heat energy or thermal energy. Heat is typically measured in Btu, calories, or joules.

Nuclear Energy: Nuclear energy comes from splitting atoms in a reactor to heat water into steam, turn a turbine and generate electricity. Ninety-three nuclear reactors in 28 states generate nearly 20 percent of the nation’s electricity, all without carbon emissions because reactors use uranium, not fossil fuels. These plants are always on: well-operated to avoid interruptions and built to withstand extreme weather, supporting the grid 24/7.

Electrical Energy:  Energy is the ability to do work, where work is done when a force moves an object. We need and we use energy every day, and energy is available in all different forms. Electrical energy is energy that's stored in charged particles within an electric field. Electric fields are simply areas surrounding a charged particle. In other words, charged particles create electric fields that exert force on other charged particles within the field. The electric field applies the force to the charged particle, causing it to move - in other words, to do work.

Principle of conservation of Energy

We have discussed that the total mechanical energy of the system is conserved if the force doing work on it is conservative.

If some of the forces are non-conservative then mechanical energy conservation does not hold. Some energy of the system gets converted into some other form like heat, light and sound.

But the total energy of an isolated system does not change, as long as we can account for all forms of energy.

Energy may be transformed from one form to another but the energy of an isolated system remains conserved.

There is no violation of this principle. Since the universe as a whole may be viewed as an isolated system, the total energy of the universe remains the same. If one part of the universe loses energy, another part must gain an equal amount of energy.

Power

We can define power as the rate of doing work, it is the work done in unit time. The SI unit of power is Watt (W) which is joules per second (J/s). Power is a time-based quantity. Which is related to how fast a job is done.

Power= Energy/ time

The SI unit of power is Watt (W) which is joules per second (J/s).

  • The power of motor vehicles and other machines is given in terms of Horsepower (hp), which is approximately equal to 745.7 watts.
  • The commercial unit of energy is 1 kWh. One kilowatt-hour is defined as the amount of energy consumed by a device in one working hour at a constant rate of one kilowatt.

The SI unit of energy is Joule.

  • Therefore, the relationship between commercial and SI unit of energy is: 1 kWh = 1kW x 1h = 1000W x 1h = 1000(J/s) x 3600 s = 3.6 x10^6 J

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